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Term-StructureModels

DamirFilipovic

Outlines

Part 1: InterestRates andRelatedContracts

Part 2:Estimating theTerm-Structure

Part 3:Arbitrage Theory

Part 4: ShortRate Models

Part 5: Heath–Jarrow–Morton(HJM)Methodology

Part 6: ForwardMeasures

Part 7: Forwardsand Futures

Part 8:ConsistentTerm-StructureParametrizations

Part 9: AffineProcesses

Part 10: MarketModels

Term-Structure ModelsA Graduate Course

Damir Filipovic

Version 5 November 2009


DamirFilipovic

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Course Book


DamirFilipovic

Outlines











Outline

Part 1: Interest Rates and Related ContractsPart 2: Estimating the Term-StructurePart 3: Arbitrage TheoryPart 4: Short Rate ModelsPart 5: Heath–Jarrow–Morton (HJM) MethodologyPart 6: Forward MeasuresPart 7: Forwards and FuturesPart 8: Consistent Term-Structure ParametrizationsPart 9: Affine ProcessesPart 10: Market Models


DamirFilipovic

Outlines











Outline of Part 1

1 Zero-Coupon Bonds

2 Interest Rates

3 Money-Market Account and Short Rates

4 Coupon Bonds, Swaps and YieldsFixed Coupon BondsFloating Rate NotesInterest Rate SwapsYield and Duration

5 Market Conventions

6 Caps and FloorsBlack’s Formula

7 SwaptionsBlack’s Formula


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 2

8 A Bootstrapping Example

9 Non-parametric Estimation MethodsBond MarketsMoney MarketsProblems

10 Parametric Estimation MethodsEstimating the Discount Function with Cubic B-splinesSmoothing SplinesExponential–Polynomial Families

11 Principal Component AnalysisPrincipal Components of a Random VectorSample Principle ComponentsPCA of the Forward CurveCorrelation


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 3

12 Stochastic CalculusStochastic Differential Equations

13 Financial MarketSelf-Financing PortfoliosNumeraires

14 Arbitrage and Martingale MeasuresMartingale MeasuresMarket Price of RiskAdmissible StrategiesThe First Fundamental Theorem of Asset Pricing

15 Hedging and PricingComplete MarketsArbitrage Pricing


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 4

16 Generalities

17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve

18 Affine Term-Structures

19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 5

20 Forward Curve Movements

21 Absence of Arbitrage

22 Implied Short-Rate Dynamics

23 HJM ModelsProportional Volatility

24 Fubini’s Theorem


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 6

25 T -Bond as Numeraire

26 Bond Option PricingExample: Vasicek Short-Rate Model

27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 7

28 Forward Contracts

29 Futures ContractsInterest Rate Futures

30 Forward vs. Futures in a Gaussian Setup


DamirFilipovic

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DamirFilipovic

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Outline of Part 8

31 Multi-factor Models

32 Consistency Condition


34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1

35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family


DamirFilipovic

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Outline



DamirFilipovic

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Outline of Part 9

36 Definition and Characterization of Affine Processes

37 Canonical State Space

38 Discounting and Pricing in Affine ModelsExamples of Fourier DecompositionsBond Option Pricing in Affine ModelsHeston Stochastic Volatility Model

39 Affine Transformations and Canonical Representation

40 Existence and Uniqueness of Affine Processes

41 On the Regularity of Characteristic Functions

42 Auxiliary Results for Differential Equations


DamirFilipovic

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DamirFilipovic

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Outline of Part 10

43 Heuristic Derivation From HJM

44 LIBOR Market ModelLIBOR Dynamics Under Different Measures

45 Implied Bond Market

46 Implied Money-Market Account

47 Swaption PricingForward Swap MeasureAnalytic Approximations

48 Monte Carlo Simulation of the LIBOR Market Model

49 Volatility Structure and CalibrationPrincipal Component AnalysisCalibration to Market Quotes

50 Continuous-Tenor Case


DamirFilipovic

Zero-CouponBonds

Interest Rates

Money-MarketAccount andShort Rates

CouponBonds, Swapsand Yields

Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Part I

Interest Rates and Related Contracts


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Overview

• Bond = securitized form of a loan

• Bonds: primary financial instruments in the market wherethe time value of money is traded

• This chapter: basis concepts of interest rates and bondmarkets:

• zero-coupon bonds• related interest rates• market conventions• market practice for pricing caps, floors and swaptions


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Zero-Coupon Bonds

• 1 euro today is worth more than 1 euro tomorrow

• zero-coupon bond pays 1 euro at maturity T

• time t value denoted by P(t,T )

Figure: Cash flow of a T -bond.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Standing Assumptions

In theory we will assume that:

• there exists a frictionless market for all T -bonds

• P(T ,T ) = 1 for all T

• P(t,T ) is differentiable in T

In reality these assumptions are not always satisfied!Q: why not assuming P(t,T ) ≤ 1?


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Term-Structure

The term-structure of zero-coupon bond prices (or discountcurve) T 7→ P(t,T ) is smooth:

Figure: Term-structure T 7→ P(t,T ).


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Trajectories

Note: t 7→ P(t,T ) is stochastic process:

Figure: T -bond price process t 7→ P(t,T ).


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Forward Rate Agreement (FRA)FRA: current date t, expiry date T > t, maturity S > T :

• At t: sell one T -bond and buy P(t,T )P(t,S) S-bonds: zero net

investment.

• At T : pay one euro.

• At S : receive P(t,T )P(t,S) euros.

Figure: Net cash flow

Net effect: forward investment of one euro at time T yieldingP(t,T )P(t,S) euros at S with certainty.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Simply Compounded Interest Rates

• simple forward rate for [T , S ] prevailing at t:

F (t; T ,S) =1

S − T

(P(t,T )

P(t, S)− 1

),

which is equivalent to

1 + (S − T )F (t; T , S) =P(t,T )

P(t, S).

• simple spot rate for [t,T ]:

F (t,T ) = F (t; t,T ) =1

T − t

(1

P(t,T )− 1

).


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Continuously CompoundedInterest Rates

• Continuously compounded forward rate for [T ,S ]prevailing at t:

R(t; T ,S) = − log P(t,S)− log P(t,T )

S − T,


eR(t;T ,S)(S−T ) =P(t,T )

P(t,S).

• continuously compounded spot rate for [t,T ]:

R(t,T ) = R(t; t,T ) = − log P(t,T )

T − t.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Instantaneous Interest Rates

• let S ↓ T :

• forward rate with maturity T prevailing at time t:

f (t,T ) = limS↓T

R(t; T , S) = −∂ log P(t,T )

∂T


P(t,T ) = e−∫ Tt f (t,u) du.

T 7→ f (t,T ) is called forward curve at time t.

• short rate at time t:

r(t) = f (t, t) = limT↓t

R(t,T ).


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Market Example: LIBOR

• LIBOR (London Interbank Offered Rate): rate at whichhigh-credit financial institutions can borrow in interbankmarket.

• maturities: from overnight to 12 months

• quoted on a simple compounding basis. E.g.:three-months forward LIBOR for period [T ,T + 1/4] attime t is

L(t,T ) = F (t; T ,T + 1/4).

• under normal conditions considered as risk-free, but . . .

• . . . LIBOR may reflect liquidity and credit risk (August2007!)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Simple vs. ContinuousCompounding

• annual rate R

• m-times compounded per year:(1 + R

m

)m• limit as m→∞: (

1 +R

m

)m

→ eR

continuous compounding

• Taylor: eR = 1 + R + o(R)

• Caution: e0.04 − 1.04 = 8.1× 10−4 = 8.1 bp. Basis points(bp) matter!


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Forward vs. Future Rates

• Can forward rates predict future spot rates?

• Thought experiment: deterministic world: all future ratesare known today (t)

• Consequence: P(t, S) = P(t,T )P(T , S) for all t ≤ T ≤ S

• This is equivalent to shifting forward curve:

f (t, S) = f (T ,S) = r(S), t ≤ T ≤ S

• In reality (non-deterministic): forecast of future short rateby forward rate have little predictive power


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Money-Market Account

• money-market account: instantaneous return r(t):

dB(t) = r(t)B(t)dt

• with B(0) = 1 this is equivalent to: B(t) = e∫ t

0 r(s) ds

• B is risk-free asset, r is risk-free rate of return

• B as numeraire: relate amounts of euro at different times


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Proxies for the Short Rate

• r(t) cannot be directly observed

• overnight interest rate not considered a good proxy(liquidity and microstructure effects)

• practiced solution: use longer rates as proxies, e.g. one- orthree months LIBOR (liquid)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Fixed Coupon BondsA (fixed) coupon bond is specified by

• coupon dates T1 < · · · < Tn (Tn=maturity)

• fixed coupons c1, . . . , cn

• a nominal value N

Figure: cash flow

The price at t ≤ T1 is

p(t) =n∑

i=1

P(t,Ti )ci + P(t,Tn)N.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Floating Rate NotesA floating rate note is specified by

• reset/settlement dates T0 < · · · < Tn (T0=first resetdate, Tn=maturity)

• a nominal value N• floating coupon payments at T1, . . . ,Tn

ci = (Ti − Ti−1)F (Ti−1,Ti )N

Figure: cash flow

Price at t ≤ T0 (replicate cash flow by buying N T0-bonds):

p(t) = NP(t,T0)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Interest Rate SwapsExchange of fixed and floating coupon paymentsA payer interest rate swap settled in arrears is specified by:

• reset/settlement dates T0 < T1 < · · · < Tn (T0=firstreset date, Tn=maturity)

• a fixed rate K

• a nominal value N

• for notational simplicity assume: Ti − Ti−1 ≡ δAt Ti , i ≥ 1, the holder of contract

• pays fixed KδN,

• receives floating F (Ti−1,Ti )δN.

Figure: net cash flow


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swap Value

• value of payer interest rate swap at t ≤ T0:

Πp(t) = N

(P(t,T0)− P(t,Tn)− Kδ

n∑i=1

P(t,Ti )

)

= Nδn∑

i=1

P(t,Ti ) (F (t; Ti−1,Ti )− K )

• value of receiver interest rate swap at t ≤ T0:Πr (t) = −Πp(t)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swap Rate

• forward (or par) swap rate makes Πr (t) = −Πp(t) = 0:

Rswap(t) =P(t,T0)− P(t,Tn)

δ∑n

i=1 P(t,Ti )

=n∑

i=1

wi (t)F (t; Ti−1,Ti )

with weights wi (t) = P(t,Ti )∑nj=1 P(t,Tj )


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Market Quotes for Par Swap Rates

source: WestLBURL: www.westlbmarkets.de/cms/sitecontent/ib/investmentbankinginternet/de/services/new swapindikationen.standard.gid

Figure: forward swap rates from 25 Sep 09 (for illustration)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swap Example

Swaps were developed because different companies couldborrow at fixed or at floating rates in different markets.Example:

• company A is borrowing fixed at 5 12 %, but could borrow

floating at LIBOR plus 12 %;

• company B is borrowing floating at LIBOR plus 1%, butcould borrow fixed at 6 1

2 %.

By agreeing to swap streams of cash flows both companiescould be better off, and a mediating institution would alsomake money:

• company A pays LIBOR to the intermediary in exchangefor fixed at 5 3

16 % (receiver swap);

• company B pays the intermediary fixed at 5 516 % in

exchange for LIBOR (payer swap).


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swap Example cont’d

The net payments are as follows:

• company A is now paying LIBOR plus 516 % instead of

LIBOR plus 12 %;

• company B is paying fixed at 6 516 % instead of 6 1

2 %;

• the intermediary receives fixed at 18 %.

Figure: A swap with mediating institution.

Everyone seems to be better off (why?)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Interest Rate Swap Markets

• interest rate swap markets are over the counter

• but swap contracts exist in standardized form, e.g. by theISDA (International Swaps and Derivatives Association,Inc.).

• swap markets are extremely liquid

• maturities from 1 to 30 years are standard, swap ratequotes available up to 60 years

• gives market participants, such as life insurers, opportunityto create synthetically long-dated investments


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Zero-Coupon Yield

• zero-coupon yield is the continuously compounded spotrate R(t,T ):

P(t,T ) = e−R(t,T )(T−t).

• T 7→ R(t,T ) is called (zero-coupon) yield curve

Figure: Yield curve T 7→ R(t,T ).

• note: term “yield curve” is ambiguous


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Yield-to-Maturity

• consider fixed coupon bond (short hand: cn contains N)

p =n∑

i=1

P(0,Ti )ci

• bond’s “internal rate of interest”: (continuouslycompounded) yield-to-maturity y : unique solution to

p =n∑

i=1

cie−yTi .

• Schaefer [47]: yield-to-maturity is inadequate statistic forbond market:

• coupon payments occurring at the same point in time arediscounted by different discount factors, but

• coupon payments at different points in time from the samebond are discounted by the same rate.

In reality, one would wish to do exactly the opposite !


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Macaulay duration

• bond price change as function of y : Macaulay duration:

DMac =

∑ni=1 Ticie−yTi

p

= weighted average of the coupon dates T1, . . . ,Tn (“meantime to coupon payment”)

= first-order sensitivity of bond price w.r.t. changes in theyield-to-maturity:

dp

dy=

d

dy

(n∑

i=1

cie−yTi

)= −DMacp

(interest rate risk management!)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Duration

• write yi = R(0,Ti )

• duration of the bond

D =

∑ni=1 Ticie−yiTi

p=

n∑i=1

ciP(0,Ti )

pTi

= first-order sensitivity of bond price w.r.t. parallel shifts ofyield curve:

d

ds

(n∑

i=1

cie−(yi +s)Ti

)|s=0 = −Dp.

→ duration is essentially for bonds (w.r.t. parallel shift of theyield curve) what delta is for stock options


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Convexity

• bond equivalent of gamma is convexity:

C =d2

ds2

(n∑

i=1

cie−(yi +s)Ti

)|s=0 =

n∑i=1

cie−yiTi (Ti )2

→ second-order approximation for bond price change ∆pw.r.t. parallel shift ∆y of yield curve:

∆p ≈ −Dp∆y +1

2C (∆y)2


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Day-Count Conventions

• convention: measure time in units of years

• market evaluates year fraction between t < T in differentways

• examples of day-count conventions δ(t,T ):• actual/365: year has 365 days

δ(t,T ) =actual number of days between t and T

365.

• actual/360: as above but year counts 360 days• 30/360: months count 30 and years 360 days. Let

t = d1/m1/y1 and T = d2/m2/y2

δ(t,T ) =min(d2, 30) + (30− d1)+

360+

(m2 −m1 − 1)+

12+y2−y1.

Example: t = 4 January 2000 and T = 4 July 2002:

δ(t,T ) =4 + (30− 4)

360+

7− 1− 1

12+ 2002− 2000 = 2.5.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Coupon Bonds

Coupon bonds issued in the American (European) marketstypically have semiannual (annual) coupon payments.Debt securities issued by the US Treasury are divided into threeclasses:

• Bills: zero-coupon bonds with time to maturity less thanone year.

• Notes: coupon bonds (semiannual) with time to maturitybetween 2 and 10 years.

• Bonds: coupon bonds (semiannual) with time to maturitybetween 10 and 30 years.1

STRIPS (separate trading of registered interest and principal ofsecurities): synthetically created zero-coupon bonds, tradedsince August 1985

130-year Treasury bonds were not offered from 2002 to 2005.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Accrued Interest, Clean Price andDirty Price

• recall coupon bond price formula

p(t) =∑Ti≥t

ciP(t,Ti )

→ systematic discontinuities of price trajectory at t = Ti

• accrued interest at t ∈ (Ti−1,Ti ] is defined by

AI (i ; t) = cit − Ti−1

Ti − Ti−1

• clean price (quoted) of coupon bond at t ∈ (Ti−1,Ti ] is

pclean(t) = p(t)− AI (i ; t)

→ dirty price (to pay) is

p(t) = pclean(t) + AI (i ; t)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Yield-to-Maturity

see course book Section 2.5.4


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Caplets

• caplet with reset date T and settlement date T + δ: paysthe holder difference between simple market rateF (T ,T + δ) (e.g. LIBOR) and strike rate κ

• cash flow at time T + δ:

δ(F (T ,T + δ)− κ)+


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Caps

• cap: strip of caplets, specified by• reset/settlement dates T0 < T1 < · · · < Tn (T0=first reset

date, Tn=maturity)• a cap rate κ• for notational simplicity assume: Ti − Ti−1 ≡ δ

• cap price at t ≤ T0 is

Cp(t) =n∑

i=1

Cpl(t; Ti−1,Ti )

where Cpl(t; Ti−1,Ti ) is price of ith caplet


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

CapsAt Ti , the holder of the cap receives

δ(F (Ti−1,Ti )− κ)+,

which is equivalent (. . . ) to cash flow

(1 + δκ)

(1

1 + δκ− P(Ti−1,Ti )

)+

at Ti−1 (= (1 + δκ) times put option on Ti -bond with strikeprice 1/(1 + δκ) and maturity Ti−1)

→ protects against rising interest rates

Figure: cash flow of cap


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Floors

• floor: converse to a cap, protects against low rates

• strip of floorlets with cash flow at time Ti :

δ(κ− F (Ti−1,Ti ))+

• ith floorlet price: Fll(t; Ti−1,Ti )

• floor price at t ≤ T0 is

Fl(t) =n∑

i=1

Fll(t; Ti−1,Ti )


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Caps, Floors and Swaps

• parity relation:

Cp(t)− Fl(t) = Πp(t)

value of a payer swap with rate κ, nominal one and sametenor structure as cap and floor

• cap/floor is . . . at-the-money (ATM) if

κ = Rswap =P(0,T0)− P(0,Tn)

δ∑n

i=1 P(0,Ti )

• . . . in-the-money (ITM) if κ < Rswap

• . . . out-of-the-money (OTM) if κ > Rswap


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Black’s Formula

Black’s formula for ith caplet value is

Cpl(t; Ti−1,Ti ) = δP(t,Ti )

× (F (t; Ti−1,Ti )Φ(d1)− κΦ(d2))

where

d1,2 =log(

F (t;Ti−1,Ti )κ

)± 1

2σ(t)2(Ti−1 − t)

σ(t)√

Ti−1 − t,

• Φ: standard Gaussian cumulative distribution function

• σ(t): cap (implied) volatility (same for all capletsbelonging to a cap)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Black’s Formula cont’d

Black’s formula assumes F (Ti−1,Ti ) = X (Ti−1) where

dX = σX dW , X (t) = F (t; Ti−1,Ti )

and

Cpl(t; Ti−1,Ti ) = δP(t,Ti )E[(X (Ti−1)− κ)+ | Ft

](to be justified later: in Market Models)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Black’s Formula cont’d

Black’s formula for ith floorlet is

Fll(t; Ti−1,Ti ) = δP(t,Ti ) (κΦ(−d2)− F (t; Ti−1,Ti )Φ(−d1))

• cap/floor prices are quoted in the market in terms of theirimplied volatilities

• typically: t = 0, T0 = δ = Ti − Ti−1 = three months (USmarket) or half a year (euro market)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Example of Cap Quotes

Table: US dollar ATM cap volatilities, 23 July 1999

Maturity (in years) ATM vols (in %)

1 14.12 17.43 18.54 18.85 18.96 18.77 18.48 18.2

10 17.712 17.015 16.520 14.730 12.4


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Example of Cap Quotes

Figure: US dollar ATM cap volatilities, 23 July 1999.

It is a challenge for any market realistic interest rate model tomatch the given volatility curve.


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swaptions

• payer (receiver) swaption with strike rate K : right to entera payer (receiver) swap with fixed rate K at swaptionmaturity

• usually, swaption maturity = first reset date T0 ofunderlying swap

• tenor of the swaption: underlying swap length Tn − T0


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swaption Payoff

• swaption payoff at maturity

N

(n∑

i=1

P(T0,Ti )δ(F (T0; Ti−1,Ti )− K )

)+

= Nδ(Rswap(T0)− K )+n∑

i=1

P(T0,Ti )

cannot be decomposed into more elementary payoffs!

→ dependence between different forward rates will entervaluation procedure

• payer (receiver) swaption is ATM, ITM, OTM if

K = Rswap(t), K < (>)Rswap(t), K > (<)Rswap(t)

• x × y -swaption: maturity in x years, underlying swap yyears long


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Application: Callable Bond

Swaptions can be used to synthetically create callable bonds:

• company has issued 10-year bond with 4% coupon

• wants to add right to call bond (i.e. prepay bond) at parafter 5 years

• cannot change original bond

Solution: buy a 5× 5 receiver swaption with strike rate 4%:

• swaption cancels fixed coupon payments

• exchange of notionals between t = 5 and T = 10 isequivalent to paying floating


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Outline

1 Zero-Coupon Bonds

2 Interest Rates







DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Black’s Formula

Black’s price formula for payer and receiver swaption is

Swptp(t) = Nδ (Rswap(t)Φ(d1)− K Φ(d2))n∑

i=1

P(t,Ti ),

Swptr (t) = Nδ (K Φ(−d2)− Rswap(t)Φ(−d1))n∑

i=1

P(t,Ti ),

with

d1,2 =log(

Rswap(t)K

)± 1

2σ(t)2(T0 − t)

σ(t)√

T0 − t,

• Φ: standard Gaussian cumulative distribution function

• σ(t): swaption implied volatility


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Swaption Quotes

• swaption prices are quoted in terms of implied volatilitiesin matrix form

• note: accrual period δ = Ti − Ti−1 for underlying swapcan differ from prevailing δ for caps within the samemarket region!

• e.g. euro zone: caps are written on semiannual LIBOR(δ = 1/2), while swaps pay annual coupons (δ = 1)


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Example of Swaption Quotes

Table: Black’s implied volatilities (in %) of ATM swaptions on May16, 2000. Maturities are 1,2,3,4,5,7,10 years, swaps lengths from 1 to10 years

1y 2y 3y 4y 5y 6y 7y 8y 9y 10y1y 16.4 15.8 14.6 13.8 13.3 12.9 12.6 12.3 12.0 11.72y 17.7 15.6 14.1 13.1 12.7 12.4 12.2 11.9 11.7 11.43y 17.6 15.5 13.9 12.7 12.3 12.1 11.9 11.7 11.5 11.34y 16.9 14.6 12.9 11.9 11.6 11.4 11.3 11.1 11.0 10.85y 15.8 13.9 12.4 11.5 11.1 10.9 10.8 10.7 10.5 10.47y 14.5 12.9 11.6 10.8 10.4 10.3 10.1 9.9 9.8 9.6

10y 13.5 11.5 10.4 9.8 9.4 9.3 9.1 8.8 8.6 8.4


DamirFilipovic

Zero-CouponBonds

Interest Rates



Fixed CouponBonds

Floating RateNotes

Interest RateSwaps

Yield andDuration

MarketConventions

Caps andFloors

Black’s Formula

Swaptions

Black’s Formula

Example of Swaption Quotes

Figure: Black’s implied volatilities (in %) of ATM swaptions on May16, 2000.

An interest rate model for swaptions valuation must fit today’svolatility surface.


DamirFilipovic

ABootstrappingExample

Non-parametricEstimationMethods

Bond Markets

Money Markets

Problems

ParametricEstimationMethods

Estimating theDiscountFunction withCubic B-splines

SmoothingSplines

Exponential–PolynomialFamilies

PrincipalComponentAnalysis

PrincipalComponents of aRandom Vector

Sample PrincipleComponents

PCA of theForward Curve

Correlation

Part II

Estimating the Term-Structure


DamirFilipovic



Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

Overview

• in theory: assume given initial term-structure for all T

• reality: finitely many (possibly noisy) market quoteobservations

• pricing exotic derivatives: cash flow dates possibly do notmatch the predetermined finite time grid

→ interpolate the term-structure

• simples method: build up term structure from shortermaturities to longer maturities (“bootstrapping”)


DamirFilipovic



Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

Outline






DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Bootstrapping Example

Table: Yen data, 9 January 1996

LIBOR (%) Futures Swaps (%)

o/n 0.49 20 Mar 96 99.34 2y 1.141w 0.50 19 Jun 96 99.25 3y 1.601m 0.53 18 Sep 96 99.10 4y 2.042m 0.55 18 Dec 96 98.90 5y 2.433m 0.56 7y 3.01

10y 3.36

• spot date t0: 11 January, 1996

• day-count convention: actual/360 (note: 1996 was leapyear):

δ(T ,S) =actual number of days between T and S

360


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Maturity Overlap

0 2 4 6 8 10 12

Time to maturity

Figure: Overlapping maturity segments (from bottom up) of LIBOR,futures and swap markets.


DamirFilipovic



Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

First Column: LIBOR

• maturities S1, . . . ,S5 = 12/1/96, 18/1/96, 13/2/96,11/3/96, 11/4/96

• zero-coupon bonds are

P(t0, Si ) =1

1 + δ(t0,Si )F (t0,Si ).


DamirFilipovic



Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

Second Column: Futures

• quoted as: futures price for settlement day Ti =100(1− FF (t0; Ti ,Ti+1))

• FF (t0; Ti ,Ti+1): futures rate for period [Ti ,Ti+1]prevailing at t0

• reset/settlement dates

T1, . . . ,T5 = 20/3/96, . . . 19/3/97

hence δ(Ti ,Ti+1) ≡ 91/360

• proxy: F (t0; Ti ,Ti+1) = FF (t0; Ti ,Ti+1) (see later)


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Second Column: Futures

• for P(t0,T1): use geometric interpolation (S4 < T1 < S5)

P(t0,T1) = P(t0, S4)q P(t0, S5)1−q

which is equivalent to linear interpolation of yields

R(t0,T1) = q R(t0,S4) + (1− q) R(t0, S5)

where

q =δ(T1, S5)

δ(S4,S5)=

22

31= 0.709677

• to derive P(t0,T2), . . . ,P(t0,T5) use:

P(t0,Ti+1) =P(t0,Ti )

1 + δ(Ti ,Ti+1) F (t0; Ti ,Ti+1)


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Third Column: Swaps

• semiannual cash flows at dates

U1, . . . ,U20 =

11/7/96, 13/1/97,11/7/97, 12/1/98,13/7/98, 11/1/99,12/7/99, 11/1/00,11/7/00, 11/1/01,11/7/01, 11/1/02,11/7/02, 13/1/03,11/7/03, 12/1/04,12/7/04, 11/1, 05,11/7/05, 11/1/06

• from data: Rswap(t0,Ui ) for i = 4, 6, 8, 10, 14, 20


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Third Column: Swaps

• recall

Rswap(t0,Un) =1− P(t0,Un)∑n

i=1 δ(Ui−1,Ui ) P(t0,Ui )(set U0 = t0)

• overlap: T2 < U1 < T3 and T4 < U2 < T5

• linear interpolation of yields → R(t0,U1), R(t0,U2)

→ P(t0,U1), P(t0,U2) and hence Rswap(t0,U1), Rswap(t0,U2)

• remaining swap rates by linear interpolation, e.g.

Rswap(t0,U3) =1

2(Rswap(t0,U2) + Rswap(t0,U4))

• inversion of above formula:

P(t0,Un) =1− Rswap(t0,Un)

∑n−1i=1 δ(Ui−1,Ui ) P(t0,Ui )

1 + Rswap(t0,Un)δ(Un−1,Un)

gives P(t0,Un) for n = 3, . . . , 20


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Bond Markets

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Obtain Term Structure

• set P(t0, t0) = 1

→ have constructed term structure P(t0, ti ) for 30 points:

ti = t0, S1, . . . ,S4,T1, S5,T2,U1,T3,T4,U2,T5,U3, . . . ,U20

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10 12

Time to maturity

Figure: Zero-coupon bond curve.


DamirFilipovic



Bond Markets

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Correlation

Yield and Forward Rate Curves

• continuously compounded yield and forward rates:R(t0, ti ) and R(t0, ti , ti+1)

• “sawtooth”: linear interpolation of swap ratesinappropriate for implied forward rates

0

0,01

0,02

0,03

0,04

0,05

0,06

0 2 4 6 8 10 12

Time to maturity

Figure: yields (lower curve), forward rates (upper curve)


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Larger Time Scale

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

0 0,1 0,2 0,3 0,4 0,5 0,6

Time to maturity

Figure: yields (lower curve), forward rates (upper curve)

• “sawtooth”: systematic inconsistency of our use of LIBORand futures rates data (we have treated futures rates asforward rates)


DamirFilipovic



Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

Futures vs. Forward Rates

• “sawtooth”: systematic inconsistency of our use of LIBORand futures rates data (we have treated futures rates asforward rates)

• in reality: futures rates often greater than forward rates

• difference is called convexity adjustment (modeldependent)

• example: forward rate = futures rate −12σ

2τ2 where• τ = time to maturity of futures contract• σ = volatility parameter

(later: we will derive a more general formula)


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Bond Markets

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Correlation

Summary

• we constructed entire term-structure from relatively fewinstruments

• method exactly reconstructs market prices (desirable forinterest rate option traders: marking to market), but . . .

• . . . forward curve (derivative −∂T log P(t0,T )!) irregular,sensitive to bond price variations/errors

• three curves resulting from LIBOR, futures and swaps: notcoincident to common underlying curve

• bootstrapping: example of non-parametric estimationmethod


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Outline






DamirFilipovic



Bond Markets

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Correlation

Non-parametric EstimationMethods: General Problem

• finding today’s (t0) discount curve (term-structure)x 7→ D(x) = P(t0, t0 + x)

• can be formulated as p = C d + ε where• p = column vector of n market prices• C = related cash flow matrix• d = (D(x1), . . . ,D(xN))>

• cash flow dates t0 < T1 < · · · < TN , Ti − t0 = xi

• ε = vector of pricing errors, subject to being minimized

• including errors reasonable: prices never exactsimultaneously quoted, bid ask spreads, allows forsmoothing

• next: bring data from bond and money markets into aboveformat


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Bond Markets

Money Markets

Problems



SmoothingSplines






Correlation

Outline






DamirFilipovic



Bond Markets

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Correlation

Data

Table: Market prices for UK gilts, 4/9/96

Coupon Next Maturity Dirty price(%) coupon date (pi )

Bond 1 10 15/11/96 15/11/96 103.82Bond 2 9.75 19/01/97 19/01/98 106.04Bond 3 12.25 26/09/96 26/03/99 118.44Bond 4 9 03/03/97 03/03/00 106.28Bond 5 7 06/11/96 06/11/01 101.15Bond 6 9.75 27/02/97 27/08/02 111.06Bond 7 8.5 07/12/96 07/12/05 106.24Bond 8 7.75 08/03/97 08/09/06 98.49Bond 9 9 13/10/96 13/10/08 110.87


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Bond Markets

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Correlation

Formalization

• basic instruments: coupon bonds

• vector of quoted market bond prices p = (p1, . . . , pn)>,

• dates of all cash flows t0 < T1 < · · · < TN ,

• bond i = 1, . . . , n with cash flows ci ,j at Tj (may be zero),form n × N cash flow matrix

C = (ci ,j) 1≤i≤n1≤j≤N


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Bond Markets

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Correlation

Example

• UK government bond (gilt) market on 4 September 1996,semiannual coupons, spot date 4/9/96, day-countconvention actual/365

→ n = 9, N = 1 + 3 + 6 + 7 + 11 + 12 + 19 + 20 + 25 = 104,T1 = 26/09/96, T2 = 13/10/96, T3 = 06/11/97, . . .

• note: no bonds have cash flows at same date ⇒ N large

C =

0 0 0 105 0 0 0 0 0 0 . . .0 0 0 0 0 4.875 0 0 0 0 . . .

6.125 0 0 0 0 0 0 0 0 6.125 . . .0 0 0 0 0 0 0 4.5 0 0 . . .0 0 3.5 0 0 0 0 0 0 0 . . .0 0 0 0 0 0 4.875 0 0 0 . . .0 0 0 0 4.25 0 0 0 0 0 . . .0 0 0 0 0 0 0 0 3.875 0 . . .0 4.5 0 0 0 0 0 0 0 0 . . .


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Bond Markets

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Outline






DamirFilipovic



Bond Markets

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Correlation

Data

Table: US money market, 6 October 1997

Period Rate Maturity date

LIBOR o/n 5.59375 9/10/971m 5.625 10/11/973m 5.71875 8/1/98

Futures Oct 97 94.27 15/10/97Nov 97 94.26 19/11/97Dec 97 94.24 17/12/97Mar 98 94.23 18/3/98Jun 98 94.18 17/6/98Sep 98 94.12 16/9/98Dec 98 94 16/12/98

Swaps 2 6.012533 6.108234 6.165 6.227 6.32

10 6.4215 6.5620 6.5630 6.56


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Bond Markets

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Formalization

• LIBOR L, maturity T : p = 1 and c = 1 + (T − t0)L at T• forward rate F for [T ,S ]: p = 0, c1 = −1 at T1 = T ,

c2 = 1 + (S − T )F at T2 = S• swap (receiver, swap rate K , tenor t0 ≤ T0 < · · · < Tn,

Ti − Ti−1 ≡ δ): since

0 = −D(T0−t0)+δKn−1∑j=1

D(Tj−t0)+(1+δK )D(Tn−t0),

we can choose• if T0 = t0: p = 1, c1 = · · · = cn−1 = δK , cn = 1 + δK ,• if T0 > t0: p = 0, c0 = −1, c1 = · · · = cn−1 = δK ,

cn = 1 + δK .

→ at t0: LIBOR and swaps have notional price 1, FRAs andforward swaps have notional price 0.


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Bond Markets

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Example

• US money market on 6 October 1997, spot date 8/10/97,day-count convention actual/360

• LIBOR is for o/n (1/360), 1m (33/360), and 3m (92/360)

• futures are three-month rates (δ = 91/360) taken asforward rates: quote 100(1− F (t0; T ,T + δ))

• swaps are annual (δ = 1) with first payment date 8/10/98

→ n = 3 + 7 + 9 = 19, N = 3 + (14− 4) + 30 = 43,T1 = 9/10/97, T2 = 15/10/97 (first future),T3 = 10/11/97, . . .


DamirFilipovic



Bond Markets

Money Markets

Problems



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Correlation

Cash Flow Matrixfirst 14 columns of the 19× 43 cash flow matrix C :

c11 0 0 0 0 0 0 0 0 0 0 0 0 00 0 c23 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 c36 0 0 0 0 0 0 0 0

0 −1 0 0 0 0 c47 0 0 0 0 0 0 00 0 0 −1 0 0 0 c58 0 0 0 0 0 00 0 0 0 −1 0 0 0 c69 0 0 0 0 00 0 0 0 0 0 0 0 −1 c7,10 0 0 0 00 0 0 0 0 0 0 0 0 −1 c8,11 0 0 00 0 0 0 0 0 0 0 0 0 −1 0 c9,13 00 0 0 0 0 0 0 0 0 0 0 0 −1 c10,14

0 0 0 0 0 0 0 0 0 0 0 c11,12 0 00 0 0 0 0 0 0 0 0 0 0 c12,12 0 00 0 0 0 0 0 0 0 0 0 0 c13,12 0 00 0 0 0 0 0 0 0 0 0 0 c14,12 0 00 0 0 0 0 0 0 0 0 0 0 c15,12 0 00 0 0 0 0 0 0 0 0 0 0 c16,12 0 00 0 0 0 0 0 0 0 0 0 0 c17,12 0 00 0 0 0 0 0 0 0 0 0 0 c18,12 0 00 0 0 0 0 0 0 0 0 0 0 c19,12 0 0

with

c11 = 1.00016, c23 = 1.00516, c36 = 1.01461,

c47 = 1.01448, c58 = 1.01451, c69 = 1.01456, c7,10 = 1.01459,

c8,11 = 1.01471, c9,13 = 1.01486, c10,14 = 1.01517

c11,12 = 0.060125, c12,12 = 0.061082, c13,12 = 0.0616,

c14,12 = 0.0622, c15,12 = 0.0632, c16,12 = 0.0642,

c17,12 = c18,12 = c19,12 = 0.0656.


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Problems

• typically n N and many entries of C are zero

⇒ quadratic optimization problem

mind∈RN

‖p − C d‖2

ill-posed: first-order condition C>C d = C>p, anddimension of solution space =dim ker(C>C ) = dim ker(C ) ≥ N − n

• moreover: as many parameters as there are cash flowdates, and there is nothing to regularize the discount curvefound from the regression ⇒ discount factors of similarmaturity can be very different ⇒ ragged yield and forwardcurves

• alternative and better method: estimate a smooth yieldcurve parametrically from market rates . . .


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Parametric Estimation Methods

• reduction of parameters and smooth term-structure ofinterest rates by using parameterized families of smoothcurves

• class of linear families: fix a set of basis functions,preferably with compact support (why?)

• examples: B-splines (next), smoothing splines (later)

• non-linear families: Nelson–Siegel, Svensson (later)


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Cubic Splines

• piecewise cubic polynomial, everywhere twice differentiable

• interpolates values at q + 1 knot points ξ0 < · · · < ξq

• general form is

σ(x) =3∑

i=0

aixi +

q−1∑j=1

bj(x − ξj)3+

→ q + 3 parameters a0, . . . , a3, b1, . . . , bq−1 (akth-degree has q + k parameters)

• uniquely characterized by σ′ or σ′′ at ξ0 and ξq


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Cubic B-splines• introduce six extra knot points

ξ−3 < ξ−2 < ξ−1 < ξ0 < · · · < ξq < ξq+1 < ξq+2 < ξq+3

to obtain a basis for the cubic splines on [ξ0, ξq]: q + 3B-splines

ψk(x) =k+4∑j=k

k+4∏i=k,i 6=j

1

ξi − ξj

(x−ξj)3+, k = −3, . . . , q−1

• B-spline ψk is zero outside [ξk , ξk+4]

Figure: B-spline with knot points 0, 1, 6, 8, 11


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Estimating the Discount Functionwith Cubic B-splines

• Steeley [50]: use B-splines to estimate the discount curve:

D(x ; z) = z1ψ1(x) + · · ·+ zmψm(x)

• quadratic optimization problem: minz∈Rm ‖p − C Ψz‖2

with

d(z) =

D(x1; z)...

D(xN ; z)

=

ψ1(x1) · · · ψm(x1)...

...ψ1(xN) · · · ψm(xN)

z1

...zm

=: Ψ z

• if the n ×m matrix A = C Ψ has full rank m, the uniqueunconstrained solution is z∗ = (A>A)−1A>p

• a reasonable constraint:D(0; z) = ψ1(0)z1 + · · ·+ ψm(0)zm = 1


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Example

• UK government bond market data from last section

• maximum time to maturity x104 = 12.11 [years]

• first bond is a zero-coupon bond: exact yield

y = −365

72log

103.822

105= − 1

0.197log 0.989 = 0.0572

• three different estimates with B-splines: 8, 7, 5 . . .


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Example

Figure: 8 (resp. first 7) B-splines with knots−20,−5,−2, 0, 1, 6, 8, 11, 15, 20, 25, 30.

Figure: 5 B-splines with knot points−10,−5,−2, 0, 4, 15, 20, 25, 30.


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Example• 8 B-splines: minz∈R8 ‖p − C Ψz‖ = ‖p − C Ψz∗‖ = 0.23

with

z∗ =

13.864111.46658.496297.697416.980666.23383−4.9717855.074

• first 7 B-splines:

minz∈R7 ‖p − C Ψz‖ = ‖p − C Ψz∗‖ = 0.32 with

z∗ =

17.801911.36038.579927.565627.288535.387664.9919

• 5 B-splines: minz∈R5 ‖p − C Ψz‖ = ‖p − C Ψz∗‖ = 0.39

with

z∗ =

15.652

19.438512.98867.402966.23152


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Example: 8 B-splines

Figure: Discount curve, yield and forward curves for estimation with8 B-splines. The dot is the exact yield of the first bond.


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Summary: B-splines

• trade-off between quality (regularity) and the correctnessof fit

• unstable and irregular yield and forward curves: short- andlong-term maturities

• B-spline fits extremely sensitive to number and location ofknot points

→ need criteria asserting smooth yield and forward curvesthat flatten towards long end

→ directly estimate the yield or forward curve

→ number and location of the knot points for splinesadjusted to the data

→ smoothing splines . . .


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Problem

• data: N observed yields Yi at Ti

• fitting requirement for forward curve f (u):∫ Ti

0f (u) du + εi/

√α = TiYi ,

• smoothness criterion: ∫ T

0(f ′(u))2 du

→ optimization problem: minf ∈H F (f ) where

F (f ) =

∫ T

0(f ′(u))2 du + α

N∑i=1

(YiTi −

∫ Ti

0f (u) du

)2

• α: tunes trade-off smoothness–correctness of fit

• H = Hilbert space of absolutely continuous functions withscalar product 〈g , h〉H = g(0)h(0) +

∫ T0 g ′(u)h′(u) du


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Lorimier’s Theorem

Theorem (Lorimier [38])

The unique solution f is a second-order spline characterized by

f (u) = f (0) +N∑

k=1

akhk(u)

where hk ∈ C 1[0,T ] is a second-order polynomial on [0,Tk ]with

h′k(u) = (Tk − u)+, hk(0) = Tk , k = 1, . . . ,N,

and f (0) and ak solve the linear system of equations

N∑k=1

akTk = 0,

α

(YkTk − f (0)Tk −

N∑l=1

al〈hl , hk〉H

)= ak , k = 1, . . . ,N.


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Lorimier’s Theorem cont’d

Figure: The function hk and its derivative h′k for Tk = 1.


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Exponential–Polynomial Families

• parametric curve families used by most central banks forterm-structure estimation

• estimate forward curve x 7→ f (t0, t0 + x) = φ(x) = φ(x ; z)

• calibration to bond prices: nonlinear optimization problem

minz∈Z‖p − C d(z)‖ ,

withdi (z) = e−

∫ xi0 φ(u;z) du

for payment tenor 0 < x1 < · · · < xN

• obvious modification for yield fitting

• examples: Nelson–Siegel and Svensson


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Nelson–Siegel and SvenssonFamilies

• Nelson–Siegel (NS): φNS(x ; z) = z1 + (z2 + z3x)e−z4x

• Svensson (S): φS(x ; z) = z1 + (z2 + z3x)e−z5x + z4xe−z6x

• both belong to general exponential–polynomial functions

p1(x)e−α1x + · · ·+ pn(x)e−αnx

where pi denote polynomials of degree ni

Figure: NS curves for z1 = 7.69, z2 = −4.13, z4 = 0.5 and 7 differentvalues for z3 = 1.76, 0.77, −0.22, −1.21, −2.2, −3.19, −4.18.


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Nelson–Siegel and SvenssonFamilies

Table: Overview of estimation procedures by several central banks.Bank for International Settlements (BIS) 1999 [5]. NS is forNelson–Siegel, S for Svensson, wp for weighted prices

Central bank Method Minimized error

Belgium S or NS wp

Canada S wp

Finland NS wp

France S or NS wp

Germany S yields

Italy NS wp

Japan smoothing pricessplines

Norway S yields

Spain S wp

Sweden S yields

UK S yields

USA smoothing bills: wpsplines bonds: prices


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Desirable Features for CurveFamilies

• Flexible: curves shall fit wide range of term structures

• Parsimonious: number of factors not too large (curse ofdimensionality).

• Regular: prefer smooth yield/forward curves that flattenout towards the long end

• Consistent: curve families compatible with dynamicinterest rate models (explained and exploited in moredetail below)


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Principal Component Analysis

• major problem in term-structure estimation: highdimensionality

• aim: find basis shapes of the yield curve (increments)

• principal component analysis (PCA): dimension reductiontechnique in multivariate analysis

• key mathematical principle: spectral decomposition forsymmetric n × n matrix Q: Q = ALA> where

• L = diag(λ1, . . . , λn) is the diagonal matrix of eigenvaluesof Q with λ1 ≥ λ2 ≥ · · · ≥ λn;

• A is an orthogonal matrix (that is, A−1 = A>) whosecolumns a1, . . . , an are the normalized eigenvectors of Q(Qai = λiai ), which form orthonormal basis of Rn.

A> denotes the transpose of A


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PCA of Random Vector (1)

• X : n-dimensional random vector with mean µ = E[X ],covariance matrix Q = cov[X ]

• Q symmetric and positive semi-definite: Q = ALA> withλ1 ≥ · · · ≥ λn≥ 0 and eigenvectors a1, . . . , an

• principal components transform of X : Y = A>(X − µ)(recentering and rotation of X )

• Yi = a>i (X − µ) = ith principal component of X

• ai = ith vector of loadings of X

• note: Yi = projection of X − µ onto ai

→ obtain decomposition

X = µ+ AY = µ+n∑

i=1

Yiai


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• can show: E[Y ] = 0 andCov[Y ] = A>QA = A>ALA>A = L

⇒ principal components of X are uncorrelated, havevariances Var[Yi ] = λi

• can show: Var[a>1 X ] = maxVar[b>X ] | b>b = 1

: Y1

has maximal variance among all standardized linearcombinations of X . For i = 2, . . . , n: Yi has maximalvariance among all such linear combinations orthogonal tofirst i − 1 linear combinations


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• observe that

n∑i=1

Var[Xi ] = trace(Q) =n∑

i=1

λi =n∑

i=1

Var[Yi ]

• hence∑k

i=1 λi∑ni=1 λi

= amount of variability in X explained by

the first k principal components Y1, . . . ,Yk

• application: X = high-dimensional stationary model for(daily changes of) the forward curve. If first k principalcomponents Y1, . . . ,Yk explain significant amount (e.g.99%) of variability in X then approximate

X ≈ µ+k∑

i=1

Yiai .

⇒ loadings a1, . . . , ak are main components of stochasticforward curve movements


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Sample PCs

• multivariate data observations x = [x(1), . . . , x(N)]

• each column x(t) = (x1(t), . . . , xn(t))> is samplerealization of random vector X (t)

• X (t) identically distributed as X : mean µ = E[X ],covariance matrix Q = cov[X ]

• empirical mean µ = 1N

∑Nt=1 x(t)

• empirical n × n covariance matrix (positive semi-definite)

Qij = Cov[xi , xj ] =1

N

N∑t=1

(xi (t)− µi )(xj(t)− µj)

• PCA decomposition: x = µ+∑n

i=1 yi ai with

• Q = ALA>, loadings A = (a1 | · · · | an)• empirical principal components y = A>(x − µ)

• Cov[yi , yj ] = 1N

∑Nt=1 yi (t)yj(t) =

λi , if i = j ,

0, else


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Sample PCs cont’d

• empirical mean µ and covariance matrix Q standardestimators for true parameters µ and Q, if observationsX (t) are either independent or at least seriallyuncorrelated (i.e. Cov[X (t),X (t + h)] = 0 for all h 6= 0)

• if this kind of stationarity of time series X (t) is in doubt,the standard practice is to differentiate and to consider theincrements ∆X (t) = X (t)− X (t − 1)

• as illustrated in the following example . . .


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PCA of Forward Curve

• increments of forward curve:

xi (t) = R(t+∆t; t+∆t+τi−1, t+∆t+τi )−R(t; t+τi−1, t+τi )

for times to maturity 0 = τ0 < · · · < τn

• example (Rebonato [44, Section 3.1]): UK market datafrom 1989–1992, original maturity spectrum divided intoeight distinct buckets, i.e. n = 8


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PCA of Forward Curve (2)

Figure: First three forward curve loadings

Stylized facts:

• first loading roughly flat: parallel shifts of the forwardcurve (affects the average rate);

• second loading upward sloping: tilting of the forward curve(affects the slope)

• third loading hump-shaped: flex (affecting the curvature)


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PCA of Forward Curve (3)

Table: Explained variance of the principal components

PC Explainedvariance (%)

1 92.172 6.933 0.614 0.245 0.03

6–8 0.01

• First 3 PCs explain more than 99% of variance of x :forward curves can be approximated by linear combinationof first three loadings, with small relative error

• these features are very typical (stylized facts), to beexpected in most PCA of forward curve (increments). Seealso Carmona and Tehranchi [14, Section 1.7]. PCA offorward curve goes back to Litterman andScheinkman [37].


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Correlation

• typical example (Brown and Schaefer [13]): US Treasuryterm structure 1987–1994

• correlation for changes of forward rates of maturities 0,0.5, 1, 1.5, 2, 3 years:

1 0.87 0.74 0.69 0.64 0.61 0.96 0.93 0.9 0.85

1 0.99 0.95 0.921 0.97 0.93

1 0.951


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Correlation (2)

• first row of this correlation matrix:

Figure: Correlation between the short rate and instantaneousforward rates for the US Treasury curve 1987–1994.

• in stylized fashion: de-correlation occurs quickly ⇒exponentially decaying correlation structure is plausible


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The FirstFundamentalTheorem ofAsset Pricing

Hedging andPricing

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Part III

Arbitrage Theory


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Overview

• Recall fundamental arbitrage principles in aBrownian-motion-driven financial market

• Basics of stochastic calculus (without proofs)

• Standard terminology: used without further explanation.Consult text books on stochastic calculus.

• Main pillars for financial applications:• Ito’s formula• Girsanov’s change of measure theorem• Martingale representation theorem


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Outline






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Usual Stochastic Setup

• Filtered probability space (Ω,F , (Ft)t≥0,P)

• Usual conditions:• completeness: F0 contains all of the null sets• right-continuity: Ft = ∩s>tFs for t ≥ 0

• d-dimensional (Ft)-Brownian motionW = (W1, . . . ,Wd)>

• Infinite time horizon (w.l.o.g.): F = F∞ = ∨t≥0Ft

• Sometimes also : Ft = FWt (for hedging/completeness)


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Stochastic Processes

• Convention: “X = Y ” means “X = Y a.s.”(P[X = Y ] = 1)

• Borel σ-algebras: B[0, t], B(R+), or simply B, etc.

• Stochastic process X = X (ω, t) is called:• adapted if Ω 3 ω 7→ X (ω, t) is Ft-measurable ∀t ≥ 0,• progressively measurable if Ω× [0, t] 3 (ω, s) 7→ X (ω, s) isFt ⊗ B[0, t]-measurable ∀t ≥ 0.

• Progressive implies adapted

• Progressive ⇒∫ t

0 X (s) ds, X (t ∧ τ) (stopping time τ),etc. are adapted

• Notation: Prog = progressive σ-algebra, generated by allprogressive processes, on Ω× R+. Fact: progressive ⇔Prog-measurable (Proposition 1.41 in [42])


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Stochastic Integrands

• L2 := set of Rd -valued progressive processesh = (h1, . . . , hd) with

E[∫ ∞

0‖h(s)‖2 ds

]<∞

• L := set of Rd -valued progressive processesh = (h1, . . . , hd) with∫ t

0‖h(s)‖2 ds <∞ for all t > 0

• Obvious: L2 ⊂ L


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Stochastic Integral

Theorem (Stochastic Integral)

For every h ∈ L one can define the stochastic integral

(h •W )t =

∫ t

0h(s) dW (s) =

d∑j=1

∫ t

0hj(s) dWj(s).

with the following properties:

1 The process h •W is a continuous local martingale.

2 Linearity: (λg + h) •W = λ(g •W ) + h •W , for g , h ∈ Land λ ∈ R.

3 For any stopping time τ , the stopped integral equals∫ t∧τ

0h(s) dW (s) =

∫ t

01s≤τh(s) dW (s) for all t > 0.


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Stochastic Integral cont’d

Theorem (Stochastic Integral cont’d)

4 If h ∈ L2 then h •W is a martingale and the Ito isometryholds:

E

[(∫ ∞0

h(s) dW (s)

)2]

= E[∫ ∞

0‖h(s)‖2 ds

].

5 Dominated convergence: if (hn) ⊂ L is a sequence withlimn hn = 0 pointwise and such that |hn| ≤ k for somefinite constant k then limn sups≤t |(hn •W )s | = 0 inprobability for all t > 0.

Proof.See [45, Section 2, Chapter IV].


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Ito processes• Convention: stochastic integrands = row vectors,

Brownian motion = column vector• Ito process := drift + continuous local martingale

X (t) = X (0) +

∫ t

0a(s) ds +

∫ t

0ρ(s) dW (s)

where ρ ∈ L and a is progressive process with∫ t0 |a(s)| ds <∞ ∀t > 0

LemmaThe above decomposition of X is unique in the sense that

X (t) = X (0) +

∫ t

0a′(s) ds +

∫ t

0ρ′(s) ds

implies a′ = a and ρ′ = ρ dP⊗ dt-a.s.

Proof.This follows from Proposition (1.2) in [45, Chapter IV].


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Ito processes cont’d

• Notation:dX (t) = a(t) dt + ρ(t) dW (t)

or, shorter,dX = a dt + ρ dW

• L2(X ) := set of progressive h = (h1, . . . , hd) withE[∫∞

0 |h(s)a(s)|2 ds]<∞ and hρ ∈ L2

• L(X ) := set of progressive h = (h1, . . . , hd) with∫ t0 |h(s)a(s)| ds <∞ for all t > 0 and hρ ∈ L

• For h ∈ L(X ): define stochastic integral w.r.t. X as∫ t

0h(s) dX (s) =

∫ t

0h(s)a(s) ds +

∫ t

0h(s)ρ(s) dW (s)


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Quadratic Variation andCovariation

• let Y (t) = Y (0) +∫ t

0 b(s) ds +∫ t

0 σ(s) dW (s) be anotherIto process

• Define covariation process of X and Y as

〈X ,Y 〉t =

∫ t

0ρ(s)σ(s)>ds

• 〈X ,X 〉 called quadratic variation process of X

• Fact (Theorem (1.8) and Definition (1.20) in [45, ChapterIV]):

〈X ,Y 〉t = limm∑

i=0

(Xti+1 − Xti )(Yti+1 − Yti ) in probability,

for any sequence of partitions 0 = t0 < t1 < · · · < tm = twith maxi |ti+1 − ti | → 0


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Levy’s Characterization Theorem

• Fact ([49, Section 6.4] or Exercise (1.27) in [45, ChapterIV]): 〈Wi ,Wj〉t = δij t

Theorem (Levy’s Characterization Theorem)

An Rd -valued continuous local martingale X vanishing at t = 0is a Brownian motion if and only if 〈Xi ,Xj〉t = δij t for every1 ≤ i , j ≤ d.

Proof.See Theorem (3.6) in [45, Chapter IV].


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Multivariate Ito Processes

• Definition: X = (X1, . . . ,Xn)> n-dimensional Ito process ifevery Xi is an Ito process

• Definition: L2(X ) (L(X )) := set of progressive processesh = (h1, . . . , hn) such that hi is in L2(Xi ) (L(Xi )) for all i

• In this sense: L2 = L2(W ) and L = L(W )

• Stochastic integral of h ∈ L(X ) w.r.t. to X is definedcoordinate-wise as

(h • X )t =

∫ t

0h(s) dX (s) =

n∑i=1

∫ t

0hi (s) dXi (s)


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Ito’s Formula

Core formula of stochastic calculus:

Theorem (Ito’s Formula)

Let f ∈ C 2(Rn). Then f (X ) is an Ito process and

f (X (t)) = f (X (0)) +n∑

i=1

∫ t

0

∂f (X (s))

∂xidXi (s)

+1

2

n∑i ,j=1

∫ t

0

∂2f (X (s))

∂xi∂xjd〈Xi ,Xj〉s .

Proof.See Theorem (3.3) in [45, Chapter IV].


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Integration by Parts Formula

Corollary (for f (x , y) = xy): integration by parts formula

X (t)Y (t) = X (0)Y (0) +

∫ t

0X (s) dY (s) +

∫ t

0Y (s) dX (s)

+ 〈X ,Y 〉t .


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Definitions

Ingredients:

• b : Ω× R+ × Rn → Rn Prog ⊗ B(Rn)-measurable

• σ : Ω× R+ × Rn → Rn×d Prog ⊗ B(Rn)-measurable

• ξ: some F0-measurable initial value


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Definitions

DefinitionA process X is solution of the stochastic differential equation

dX (t) = b(t,X (t)) dt + σ(t,X (t)) dW (t)

X (0) = ξ(1)

if X is an Ito process satisfying

X (t) = ξ +

∫ t

0b(s,X (s)) ds +

∫ t

0σ(s,X (s)) dW (s).

We say that X is unique if any other solution X ′ of (1) isindistinguishable from X , that is, X (t) = X ′(t) for all t ≥ 0 a.s.If b(ω, t, x) = b(t, x) and σ(ω, t, x) = σ(t, x): solution X of(1) is also called a (time-inhomogeneous) diffusion withdiffusion matrix a(t, x) = σ(t, x)σ(t, x)> and drift b(t, x).


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Existence and Uniqueness

TheoremSuppose b(t, x) and σ(t, x) satisfy the Lipschitz and lineargrowth conditions

‖b(t, x)− b(t, y)‖+ ‖σ(t, x)− σ(t, y)‖ ≤ K‖x − y‖,‖b(t, x)‖2 + ‖σ(t, x)‖2 ≤ K 2(1 + ‖x‖2),

for all t ≥ 0 and x , y ∈ Rn, where K is some finite constant.Then, for every time–space initial point (t0, x0) ∈ R+ × Rn,there exists a unique solution X = X (t0,x0) of the stochasticdifferential equation

dX (t) = b(t0 + t,X (t)) dt + σ(t0 + t,X (t)) dW (t)

X (0) = x0.(2)

Proof.See [35, Theorem 5.2.9].


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Existence and UniquenessNote: existence and uniqueness hold sometimes withoutLipschitz condition on σ(t, x), see “Affine Processes” below


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Markov Property

TheoremSuppose b(t, x) and a(t, x) = σ(t, x)σ(t, x)> are continuous in(t, x), and assume that for every time–space initial point(t0, x0) ∈ R+ × Rn, there exists a unique solution X = X (t0,x0)

of the stochastic differential equation (2). Then X has theMarkov property. That is, for every bounded measurablefunction f on Rn, there exists a measurable function F onR+ × R+ × Rn such that

E[f (X (T )) | Ft ] = F (t,T ,X (t)), t ≤ T .

In words, the Ft-conditional distribution of X (T ) is a functionof t, T and X (t) only.

Proof.Follows from [35, Theorem 4.20].


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Remarks

• Continuity assumption on the diffusion matrix a(t, x)rather than on σ(t, x), since a(t, x) determines law of X .Ambiguity with σ(t, x): σDD>σ> = σσ> for anyorthogonal d × d-matrix D.

• For most practical purposes: assume σ(t, x) itself iscontinuous in (t, x)

• Time-inhomogeneous diffusion X in (2) can be regardedas R+ × Rn-valued homogeneous diffusion(X ′0, . . . ,X

′n)(t) = (t0 + t,X (t)). Calendar time at

inception (t = 0) is then X ′0(0) = t0. Accordingly, tmeasures relative time with respect to t0.


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Stochastic ExponentialDefine: stochastic exponential of an Ito process X by

Et(X ) = eX (t)− 12〈X ,X 〉t

LemmaLet X and Y be Ito processes.

1 U = E(X ) is a positive Ito process and the unique solutionof the stochastic differential equation

dU = U dX , U(0) = eX (0).

2 E(X ) is continuous local martingale if X is localmartingale.

3 E(0) = 1.4 E(X )E(Y ) = E(X + Y ) e 〈X ,Y 〉.5 E(X )−1 = E(−X ) e 〈X ,X 〉.

Proof.Stochastic calculus (exercise).


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Financial Market

Financial market S = (S0, . . . ,Sn)> with

• a risk-free asset (money-market account)

dS0 = S0 r dt, S0(0) = 1,

• n risky assets (i = 1, . . . , n)

dSi = Si (µi dt + σi dW ) , Si (0) > 0


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Financial Market

Standing assumption: short rates r , appreciation rates µi ,volatility σi = (σi1, . . . , σid) progressive and s.t.

X0(t) =

∫ t

0r(s) ds

Xi (t) =

∫ t

0µi (s) ds +

∫ t

0σi (s) dW (s)

are well-defined Ito processes (i = 1, . . . , n)

Corollary

For all i : Si (t) = Si (0)Et(Xi ) are positive Ito processes.


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Self-Financing Portfolios

• Definition: a portfolio/trading strategy is an Rn+1-valuedprogressive process φ = (φ0, . . . , φn)

• Corresponding value process: V = φ S =∑n

i=0 φi Si

• Portfolio φ is self-financing for S if φ ∈ L(S) and

dV = φ dS =n∑

i=0

φi dSi

In words: there is no in- or outflow of capital


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• All prices are in terms of a numeraire (e.g. euros)

• Any traded instrument qualifies as a numeraire: S0

(often), or Sp for some p ≤ n

• Notation: calligraphic letters for discounted values

S =S

Sp

V =V

Sp=

n∑i=0

φi Si


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Invariance of Self-FinancingProperty

LemmaLet φ ∈ L(S) ∩ L(S). Then φ is self-financing for S if and onlyif it is self-financing for S, in particular

dV = φ dS =n∑

i=0i 6=p

φi dSi . (3)

Proof.Exercise.

Note: Since dSp ≡ 0, the number of summands in (3) reducesto n.


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Construct Self-Financing PortfolioGiven

• initial wealth V (0)

• any (φ0, . . . , φp−1, φp+1, . . . , φn) ∈L(S0, . . . ,Sp−1,Sp+1, . . . ,Sn)

Aim: construct φp s.t. φ = (φ0, . . . , φn) is self-financing

• From above lemma: discounted value process is

V(t) = V (0) +n∑

i=0i 6=p

∫ t

0φi (s) dSi (s)

• Hence define φp(t) = V(t)−∑n

i=0i 6=p

φi (t)Si (t)

• Since Sp ≡ 1 ⇒ φ = (φ0, . . . , φn) ∈ L(S): φ isself-financing for S

• To be checked from case to case: φ ∈ L(S)


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Arbitrage Portfolios

• Definition: arbitrage portfolio := self-financing portfolio φwith value process satisfying

V (0) = 0 and V (T ) ≥ 0 and P[V (T ) > 0] > 0

for some T > 0.

• If no arbitrage portfolios exist for any T > 0 we say themodel is arbitrage-free.


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Example of Arbitrage

LemmaSuppose there exists a self-financing portfolio with valueprocess

dU = U k dt,

for some progressive process k. If the market is arbitrage-freethen necessarily

r = k, dP⊗ dt-a.s.


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Proof I

After discounting with S0 we obtain

U(t) =U(t)

S0(t)= U(0)e

∫ t0 (k(s)−r(s)) ds .

Then

ψ(t) = 1k(t)>r(t)

yields a self-financing strategy with discounted value process

V(t) =

∫ t

0ψ(s) dU(s)

=

∫ t

0

(1k(s)>r(s)(k(s)− r(s))U(s)

)ds ≥ 0.


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Proof II

Hence absence of arbitrage requires

0 = E[V(T )]

=

∫N

(1k(ω,t)>r(ω,t)(k(ω, t)− r(ω, t))U(ω, t)

)︸︷︷︸>0 on N

dP⊗ dt

where

N = (ω, t) | k(ω, t) > r(ω, t)

is a measurable subset of Ω× [0,T ]. But this can only hold ifN is a dP⊗ dt-nullset. Using the same arguments withchanged signs proves the lemma (→ exercise).


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Equivalent Martingale Measures

• When is a model arbitrage-free?

• For simplicity of notation: fix S0 as a numeraire

DefinitionAn equivalent (local) martingale measure (E(L)MM) Q ∼ Phas the property that the discounted price processes Si areQ-(local) martingales for all i .

• Need to understand how W transforms under equivalentchange of measure . . .


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Girsanov’s Theorem

Theorem (Girsanov’s Change of Measure Theorem)

Let γ ∈ L be such that the stochastic exponential E(γ •W ) isa uniformly integrable martingale with E∞(γ •W ) > 0. Then

dQdP

= E∞ (γ •W )

defines an equivalent probability measure Q ∼ P, and theprocess

W ∗(t) = W (t)−∫ t

0γ(s)>ds

is a Q-Brownian motion.

Proof.See Theorem (1.12) in [45, Chapter VIII].

Note: dQdP |Ft = Et (γ •W )


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Novikov’s Condition

Theorem (Novikov’s Condition)

IfE[e

12

∫∞0 ‖γ(s)‖2 ds

]<∞

then E(γ •W ) is a uniformly integrable martingale withE∞(γ •W ) > 0.

Proof.See Proposition (1.15) in [45, Chapter VIII] for uniformintegrability of E(γ •W ), and Proposition (1.26) in [45,Chapter IV] for finiteness of (γ •W )∞ which is equivalent toE∞(γ •W ) > 0.

Note: Novikov’s condition is only sufficient but not necessary,see exercise in “Affine Processes”.


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Market Price of RiskAssume:

• Q ELMM of the form dQdP = E∞ (γ •W )

• Girsanov transformed Brownian motionW ∗(t) = W (t)−

∫ t0 γ(s)>ds

Integration by parts ⇒ S-dynamics (i = 1, . . . , n):

dSi = Si (µi − r) dt + Si σi dW

= Si

(µi − r + σiγ

>)

dt + Siσi dW ∗

S is a Q-local martingale ⇒ dQ⊗ dt-a.s.

− σi γ> = µi − r for all i = 1, . . . , n.

Economic interpretation:

• right: excess of return over the risk free rate r for asset i

• left: linear combination of volatilities σij (risk factor Wj)with factor loadings −γj


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Market Price of Risk cont’d

⇒ −γ := the market price of risk vector

• Main point: −γ the same for all risky assets i = 1, . . . , n

• Conversely: if γ ∈ L satisfies −σi γ> = µi − r and

Novikov’s condition, then dQdP = E∞ (γ •W ) defines an

ELMM Q• Final note: Si = Si (0)E(σi •W ∗) ⇒ if σi satisfies Novikov

condition ∀i = 1, . . . , n then Q an EMM


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An Arbitrage Strategy

• Local martingales ⇒ be alert to pitfalls!

• Example: ∃φ ∈ L s.t. local martingaleM(t) =

∫ t0 φ(s) dW (s) satisfies M(1) = 1 (Dudley’s

Representation Theorem 12.1 in [49])

• Looks like discounted value process of a self-financingstrategy (in the Bachelier model [3] S = W )

⇒ arbitrage!

• Solution: M is unbounded from below (a true localmartingale). In reality, no lender would provide us with aninfinite credit line.


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Admissible Strategies

DefinitionA self-financing strategy φ is admissible if its discounted valueprocess V is a Q-martingale for some ELMM Q.

• Caution: admissibility is sensitive with respect to thechoice of numeraire (see Delbaen and Schachermayer [22])

Useful local martingale property result (generalizing“Stochastic Integral” Theorem):

LemmaThe discounted value process V of an admissible strategy is aQ-local martingale under every ELMM Q.


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Proof

Proof.By assumption, dV = φ dS is the stochastic integral withrespect to the continuous Q-local martingale S. The statementnow follows from Proposition (2.7) in [45, Chapter IV] andProposition (1.5) in [45, Chapter VIII].


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First Fundamental Theorem ofAsset Pricing

LemmaSuppose there exists an ELMM Q. Then the model isarbitrage-free, in the sense that there exists no admissiblearbitrage strategy.

Proof.Indeed, let V be the discounted value process of an admissiblestrategy, with V(0) = 0 and V(T ) ≥ 0. Since V is aQ-martingale for some ELMM Q, we have

0 ≤ EQ[V(T )] = V(0) = 0,

whence V(T ) = 0.


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First Fundamental Theorem ofAsset Pricing

As for converse statement:

• Absence of arbitrage among admissible strategies notsufficient for the existence of an ELMM

• Delbaen and Schachermayer [21]: “no free lunch withvanishing risk” (some form of asymptotic arbitrage) isequivalent to the existence of an ELMM

• Technical details far from trivial, beyond this course

• Custom in financial engineering: consider existence of anELMM as “essentially equivalent” to the absence ofarbitrage


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Contingent Claims

DefinitionA contingent claim due at T (or T -claim) is an FT -measurablerandom variable X .

• Examples: cap, floor, swaption

Main problems:

• How can one hedge against the financial risk involved intrading contingent claims?

• What is a fair price for a contingent claim X ?


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Attainable Claims

DefinitionA contingent claim X due at T is attainable if ∃ admissiblestrategy φ which replicates, or hedges, X . That is, its valueprocess V satisfies V (T ) = X

• Simple example: S1 = price process of T -bond.Contingent claim X ≡ 1 due at T : attainable bybuy-and-hold strategy φ0 ≡ 0, φ1 = 1[0,T ], with valueprocess V = S1.


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Representation TheoremWe know: stochastic integral w.r.t. W is a local martingale.The converse holds true if the filtration is not too large:

Theorem (Representation Theorem)

Assume that the filtration

(Ft) is generated by the Brownian motion W . (4)

Then every P-local martingale M has a continuous modificationand there exists ψ ∈ L such that

M(t) = M(0) +

∫ t

0ψ(s) dW (s).

Consequently, every equivalent probability measure Q ∼ P canbe represented in the form (15.3) for some γ ∈ L.

Proof.See Theorem (3.5) in [45, Chapter V]. The last statement

follows since M(t) = E[

dQdP | Ft

]is a positive martingale.


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Complete Markets

DefinitionThe market model is complete if, on any finite time horizonT > 0, every T -claim X with bounded discounted payoffX/S0(T ) is attainable.

• Note: completeness 6⇔ absence of arbitrage

• However . . .


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Second Fundamental Theorem ofAsset Pricing

Theorem (Second Fundamental Theorem of Asset Pricing)

Assume Ft = FWt and there exists an ELMM Q. Then the

following are equivalent:

1 The model is complete.

2 The ELMM Q is unique.

3 The n × d-volatility matrix σ = (σij) is dP⊗ dt-a.s.injective.

4 The market price of risk −γ is dP⊗ dt-a.s. unique.

Under any of these conditions, every T -claim X with

EQ

[|X |

S0(T )

]<∞ is attainable.


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Second Fundamental Theorem ofAsset Pricing

Note: Property 3 ⇒ number of risk factors d ≤ n number ofrisky assets: randomness generated by the d noise factors dWcan be fully absorbed by the n discounted price incrementsdS1, . . . , dSn


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Proof I

1⇒ 2: Let A ∈ FT . By definition there exists an admissiblestrategy φ with discounted value process V satisfyingV(t) = EQ[1A | Ft ] for some ELMM Q. This implies that|V| ≤ 1. In view of Lemma 15.6, V is thus a martingale underany ELMM. Now let Q′ be any ELMM. ThenQ′[A] = V(0) = Q[A], and hence Q = Q′.2⇒ 3: See Proposition 8.2.1 in [41].3⇒ 4⇒ 2: This follows from the linear market price of riskequation (2) and the last statement of the representationtheorem.3⇒ 1: Let X be a claim due at some T > 0 satisfying

EQ

[|X |

S0(T )

]<∞ for some ELMM Q (this holds in particular if

X/S0(T ) is bounded). We define the Q-martingale

Y (t) = EQ

[X

S0(T )| Ft

], t ≤ T .


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StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Proof II

By Bayes’ rule we obtain

Y (t)D(t) = D(t)EQ[Y (T ) | Ft ] = E[Y (T )D(T ) | Ft ],

with the density process D(t) = dQ/dP|Ft = Et(γ •W ).Hence YD is a P-martingale and by the representation theoremthere exists some ψ ∈ L such that

Y (t)D(t) = Y (0) +

∫ t

0ψ(s) dW (s).

Applying Ito’s formula yields

d

(1

D

)= − 1

Dγ dW +

1

D‖γ‖2 dt,


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Proof IIIand

dY = d

((YD)

1

D

)= YD d

(1

D

)+

1

Dd(YD) + d

⟨YD,

1

D

⟩=

(1

Dψ − Y γ

)dW −

(1

Dψ − Y γ

)γ>dt

=

(1

Dψ − Y γ

)dW ∗

where dW ∗ = dW − γ dt denotes the Girsanov transformedQ-Brownian motion. Note that we just have shown that themartingale representation property also holds for W ∗ under Q.Since σ is injective, there exists some d × n-matrix-valuedprogressive process σ−1 such that σ−1σ equals thed × d-identity matrix. If we define φ = (φ1, . . . , φn) via

φi =

((1Dψ − Y γ

)σ−1

)i

Si,


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Proof IV

it follows that

dY =

(1

Dψ − Y γ

)σ−1σ dW ∗ =

n∑i=1

φiSiσi dW ∗ =n∑

i=1

φi dSi .

Hence φ yields an admissible strategy with discounted valueprocess satisfying

V(t) = Y (t) = EQ

[X

S0(T )

]+

n∑i=1

∫ t

0φi (s) dSi (s),

and in particular V(T ) = Y (T ) = X/S0(T ). Notice that φ isadmissible since V is by construction a true Q-martingale. Thisalso proves the last statement of the theorem.


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Outline






DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Arbitrage Price

Complete model: unique arbitrage price

Π(t) = S0(t)V(t) = S0(t)EQ

[X

S0(T )| Ft

].

Any other price would yield arbitrage!Illustration for t = 0 (see Proposition 2.6.1 in [41]): supposeprice p > Π(0). Then

• at t = 0: sell short the claim and receive p. Investp − Π(0) in the money-market account, replicate claimwith remaining initial capital Π(0)

• at T : clear short position in the claim, we are left withp − Π(0) > 0 units of S0(T ) ⇒ arbitrage (similar forp < Π(0))


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

Incomplete Markets

• Our bond markets are often complete: infinitely manytraded assets

• But: real markets are generically incomplete (e.g. pricejumps)

• Vast literature on pricing and hedging in incompletemarkets

Our approach (custom):

• exogenously specify a particular ELMM Q (the marketprice of risk −γ)

• price a T -claim X by Q-expectation:

Π(t) = S0(t)EQ

[X

S0(T ) | Ft

]⇒ Consistent arbitrage price: S0, . . . ,Sn,Π is arbitrage-free


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

State-Price Density

Define

π(t) =1

S0(t)

dQdP|Ft .

Bayes’ rule implies

Π(t) = S0(t)EQ

[X

S0(T )| Ft

]= S0(t)

E[

XS0(T )

dQdP |FT

| Ft

]dQdP |Ft

=E [Xπ(T ) | Ft ]

π(t)

In particular Π(0) = E[Xπ(T )]

Definitionπ is called the state-price density process


DamirFilipovic

StochasticCalculus


FinancialMarket


Numeraires


MartingaleMeasures

Market Price ofRisk



Hedging andPricing

CompleteMarkets

Arbitrage Pricing

State-Price Density

• Example: T -bond price

P(t,T ) = E[π(T )

π(t)| Ft

]= EQ

[S0(t)

S0(T )| Ft

]• Show: If Q is an EMM then Siπ are P-martingales.


DamirFilipovic

Generalities

DiffusionShort-RateModels

Examples

Inverting theForward Curve

Affine Term-Structures

SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Part IV

Short Rate Models


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Overview

• Earliest stochastic interest rate models: models of theshort rates

• Introduction to diffusion short-rate models in general

• Survey of some standard models

• Particular focus is on affine term-structures


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

General Assumptions

• Usual stochastic setup (Ω,F , (Ft)t≥0,P)

• P objective probability measure

• short rates follow Ito process

dr(t) = b(t) dt + σ(t) dW (t)

• money-market account B(t) = e∫ t

0 r(s) ds

• no arbitrage: ∃ EMM Q of the form dQ/dP = E∞(γ •W ),s.t. discounted bond price process P(t,T )/B(t) is aQ-martingale and P(T ,T ) = 1 for all T > 0

• Girsanov: W ∗(t) = W (t)−∫ t

0 γ(s)>ds is Q-Brownianmotion

• for strategies involving a continuum of bonds see Bjork etal. [6] and Carmona and Tehranchi [14]


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Consequences

• Q is true EMM: P(t,T ) = EQ

[e−

∫ Tt r(s) ds | Ft

]• under Q: dr(t) =

(b(t) + σ(t) γ(t)>

)dt + σ(t) dW ∗(t)

• If Ft = FWt then for any T > 0 ∃ progressive Rd -valued

process v(·,T ) s.t. (. . . )

dP(t,T )

P(t,T )= r(t) dt + v(t,T ) dW ∗(t).

Hence under objective probability P:

dP(t,T )

P(t,T )=(

r(t)− v(t,T ) γ(t)>)

dt + v(t,T ) dW (t)

→ market price of risk: −γ = excess of instantaneousreturn over r(t) in units of volatility v(t,T )


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Specification of Market Price ofRisk

• How to specify market price of risk (mpr)?

• general equilibrium: market price of risk endogenous, seeCox, Ingersoll and Ross [19]

• arbitrage theory: unable to identify market price of risk:money-market account B alone cannot be used toreplicate bond payoffs: model is incomplete

• in line with non-uniqueness of EMM: can be anyequivalent probability measure Q ∼ P


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Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Summary

• A short-rate model is not fully determined without theexogenous specification of the market price of risk.

• custom: postulate Q-dynamics of r

→ implies Q-dynamics of all bonds

→ contingent claims priced by Q-expectation of discountedpayoffs

• market price of risk (and hence P): inferred by statisticalmethods from historical observations of price movements


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Diffusion Short-Rate Models

• stochastic basis (Ω,F , (Ft)t≥0,Q) with martingalemeasure Q

• W ∗ one-dimensional Q-Brownian motion (d = 1)

• Z ⊂ R closed interval with non-empty interior

• b and σ continuous functions on R+ ×Z• Assumption: for all (t0, r0) ∈ Z there exists a uniqueZ-valued solution r = r (t0,r0) of SDE

dr(t) = b(t0+t, r(t)) dt+σ(t0+t, r(t)) dW ∗(t), r(0) = r0

• sufficient conditions for existence and uniqueness available

• recall: Markov property of r


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Term-Structure Equation

LemmaLet T > 0, and Φ continuous function on Z. AssumeF = F (t, r) in C 1,2([0,T ]×Z) is a solution to boundary valueproblem on [0,T ]×Z (term-structure equation for Φ):

∂tF (t, r) + b(t, r)∂r F (t, r) +1

2σ2(t, r)∂2

r F (t, r)− rF (t, r) = 0

F (T , r) = Φ(r).

Then M(t) = F (t, r(t))e−∫ t

0 r(u) du, t ≤ T , is a localmartingale.If in addition either:

1 EQ

[∫ T0

∣∣∣∂r F (t, r(t))e−∫ t

0 r(u) duσ(t, r(t))∣∣∣2 dt

]<∞, or

2 M is uniformly bounded,

then M is a true martingale, and

F (t, r(t)) = EQ

[e−

∫ Tt r(u) duΦ(r(T )) | Ft

], t ≤ T .


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Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Term-Structure Equation: Proof

Proof.We can apply Ito’s formula to M and obtain

dM(t) =(∂tF (t, r(t)) + b(t, r(t))∂r F (t, r(t))

+1

2σ2(t, r)∂2

r F (t, r(t))− r(t)F (t, r(t)))e−

∫ t0 r(u) du dt

+ ∂r F (t, r(t))e−∫ t

0 r(u) duσ(t, r(t)) dW ∗(t)

= ∂r F (t, r(t))e−∫ t

0 r(u) duσ(t, r(t)) dW ∗(t).

Hence M is a local martingale.Either Condition 1 or 2 implies M is true martingale. Since

M(T ) = Φ(r(T ))e−∫ T

0 r(u) du

we obtain

F (t, r(t))e−∫ t

0 r(u) du = M(t) = EQ

[e−

∫ T0 r(u) duΦ(r(T )) | Ft

].

Multiplying both sides by e∫ t

0 r(u) du yields the claim.


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Term-Structure Equation: PricingEfficiency

• term-structure equation for Φ: solution F (t, r(t) = priceof T -claim Φ(r(T ))

• for Φ ≡ 1: P(t,T ) = F (t, r(t); T )

• pricing algorithm computationally efficient?

• ok: solving PDEs in less than three space dimensionsnumerically feasible, but . . .

• . . . nuisance: have to solve a PDE for every T > 0, just toget bond prices

• problem: term-structure calibration

⇒ short-rate models admitting closed-form solutions for bondprices favorable


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Examples

1 Vasicek [52]: Z = R,

dr(t) = (b + βr(t)) dt + σ dW ∗(t),

2 Cox–Ingersoll–Ross (henceforth CIR) [19]: Z = R+, b ≥ 0,

dr(t) = (b + βr(t)) dt + σ√

r(t) dW ∗(t),

3 Dothan [23]: Z = R+,

dr(t) = βr(t) dt + σr(t) dW ∗(t),

4 Black–Derman–Toy [7]: Z = R+,

dr(t) = β(t)r(t) dt + σ(t)r(t) dW ∗(t),


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Examples continued

5 Black–Karasinski [8]: Z = R+, `(t) = log r(t),

d`(t) = (b(t) + β(t)`(t)) dt + σ(t) dW ∗(t),

6 Ho–Lee [29]: Z = R,

dr(t) = b(t) dt + σ dW ∗(t),

7 Hull–White extended Vasicek [30]: Z = R,

dr(t) = (b(t) + β(t)r(t)) dt + σ(t) dW ∗(t),

8 Hull–White extended CIR [30]: Z = R+, b(t) ≥ 0,

dr(t) = (b(t) + β(t)r(t)) dt + σ(t)√

r(t) dW ∗(t).


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Inverse Problem

• specification of short-rate model parameters fully specifiesinitial term-structure T 7→ P(0,T ) = F (0, r(0); T ) andhence forward curve

• conversely: invert term-structure equation to match giveninitial forward curve

• example Vasicek model: P(0,T ) = F (0, r(0); T , b, β, σ)parameterized curve family with three degrees of freedomb, β, σ (for given (r(0))

• often too restrictive: poor fit of the current data

⇒ time-inhomogeneous short-rate models, such as theHull–White extensions: time-dependent parameters =infinite degree of freedom ⇒ perfect fit of any given curve

• usually functions b(t) etc. are fully determined by theinitial term-structure

• explicit examples in the following . . .


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Affine Term-Structure Models

• recall: diffusion short rate model

dr(t) = b(t0 + t, r(t)) dt + σ(t0 + t, r(t)) dW ∗(t) (5)

• recall: short-rate models admitting closed-form bondprices F (t, r ; T ) favorable

• Definition: r affine term-structure (ATS) model ifF (t, r ; T ) = exp(−A(t,T )− B(t,T )r)

• note: F (T , r ; T ) = 1 implies A(T ,T ) = B(T ,T ) = 0

• nice: short-rate ATS models can be completelycharacterized . . .


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Characterization

Proposition

The short-rate model (5) provides ATS if and only if itsdiffusion and drift terms are of the form

σ2(t, r) = a(t) + α(t)r and b(t, r) = b(t) + β(t)r , (6)

for some continuous functions a, α, b, β, and the functions Aand B satisfy the system of ordinary differential equations, forall t ≤ T ,

∂tA(t,T ) =1

2a(t)B2(t,T )− b(t)B(t,T ), A(T ,T ) = 0,

(7)

∂tB(t,T ) =1

2α(t)B2(t,T )− β(t)B(t,T )− 1, B(T ,T ) = 0.

(8)


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Proof

Proof.Insert F (t, r ; T ) = exp(−A(t,T )− B(t,T )r) in term-structureequation: short-rate model (5) provides an ATS if and only if

1

2σ2(t, r)B2(t,T )−b(t, r)B(t,T ) = ∂tA(t,T )+(∂tB(t,T )+1)r

(9)holds for all t ≤ T and r ∈ Z.By inspection: specification (6)–(8) satisfies (9). This provessufficiency.Necessity of (6)–(8): . . .


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Further Specification of AffineParameters

• Z = R: necessarily α(t) = 0 and a(t) ≥ 0, and b, βarbitrary: Hull–White extension of the Vasicek model

• Z = R+: necessarily a(t) = 0, α(t) ≥ 0 and b(t) ≥ 0(otherwise the process would cross zero), and β arbitrary:Hull–White extension of the CIR model

• general affine processes are discussed later

• recall list: all short-rate models except the Dothan,Black–Derman–Toy and Black–Karasinski models have anATS


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Vasicek: dr = (b +βr) dt +σ dW ∗

• explicit solution:

r(t) = r(0)eβt +b

β

(eβt − 1

)+ σeβt

∫ t

0e−βs dW ∗(s)

⇒ r(t) Gaussian process with mean

EQ [r(t)] = r(0)eβt +b

β

(eβt − 1

)and variance

VarQ[r(t)] = σ2e2βt

∫ t

0e−2βs ds =

σ2

2β

(e2βt − 1

)• Q[r(t) < 0] > 0: not satisfactory (but probability usually

small)

• Vasicek assumed constant market price of risk (on finitetime horizon) ⇒ objective P-dynamics of r(t) also of theabove form


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


If β < 0:

⇒ r(t) mean-reverting with mean reversion level b/|β|⇒ r(t) converges to a Gaussian r.v. with mean b/|β| and

variance σ2/(2|β|), for t →∞

Figure: Vasicek short-rate process for β = −0.86, b/|β| = 0.09(mean reversion level), σ = 0.0148 and r(0) = 0.08.


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


• ATS equations (7)–(8) become

∂tA(t,T ) =σ2

2B2(t,T )− bB(t,T ), A(T ,T ) = 0,

∂tB(t,T ) = −βB(t,T )− 1, B(T ,T ) = 0.


B(t,T ) =1

β

(eβ(T−t) − 1

)A(t,T ) = A(T ,T )−

∫ T

t∂sA(s,T ) ds

= −σ2

2

∫ T

tB2(s,T ) ds + b

∫ T

tB(s,T ) ds

=σ2(4eβ(T−t) − e2β(T−t) − 2β(T − t)− 3

)4β3

+ beβ(T−t) − 1− β(T − t)

β2


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


• recall: closed-form bond pricesP(t,T ) = exp (−A(t,T )− B(t,T )r(t))

• will see below: also closed-form bond option prices


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

CIR: dr = (b + βr) dt + σ√

r dW ∗

• CIR SDE has unique nonnegative solution for r(0) ≥ 0

• if b ≥ σ2/2 then r > 0 whenever r(0) > 0 (e.g.Lamberton and Lapeyre [36, Proposition 6.2.4])

• ATS equation becomes non linear (Riccati equation):

∂tB(t,T ) =σ2

2B2(t,T )− βB(t,T )− 1, B(T ,T ) = 0


B(t,T ) =2(eγ(T−t) − 1

)(γ − β)

(eγ(T−t) − 1

)+ 2γ

where γ =√β2 + 2σ2


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

CIR: dr = (b + βr) dt + σ√

r dW ∗

• integration yields

A(t,T ) = −2b

σ2log

(2γe (γ−β)(T−t)/2

(γ − β)(eγ(T−t) − 1

)+ 2γ

)

⇒ closed-form bond pricesP(t,T ) = exp (−A(t,T )− B(t,T )r(t))

• will see below: also closed-form bond option prices


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Dothan: dr = βr dt + σr dW ∗

• Dothan [23] starts from drift-less geometric Brownianmotion under objective probability measure P:

dr(t) = σr(t) dW (t)

• constant market price of risk gives above Q-dynamics

• easily integrated: for s ≤ t:

r(t) = r(s) exp((β − σ2/2

)(t − s) + σ(W ∗(t)−W ∗(s))

)⇒ Fs -conditional distribution of r(t) is lognormal with mean

and variance

EQ[r(t) | Fs ] = r(s)eβ(t−s)

VarQ[r(t) | Fs ] = r 2(s)e2β(t−s)(eσ

2(t−s) − 1)


DamirFilipovic

Generalities


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SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Lognormal Short-Rate Models

+ lognormal short-rate models (Dothan, Black–Derman–Toy,Black–Karasinski): positive interest rates

− but no closed-form bond (option) prices

− major drawback: explosion of the money-market account:

EQ[B(∆t)] = EQ

[e∫ ∆t

0 r(s) ds]≈ EQ

[e

r(0)+r(∆t)2

∆t]

• fact: EQ

[e eY]

=∞ for Gaussian Y

⇒ EQ[B(∆t)] =∞ for arbitrarily small ∆t

• similarly: Eurodollar future price = ∞ (see later)

• idea of lognormal rates taken up in mid-90s by Sandmannand Sondermann [46] and others → market models withlognormal LIBOR or swap rates (studied below)


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Ho–Lee: dr = b(t) dt + σ dW ∗

• ATS equations (7)–(8) become

∂tA(t,T ) =σ2

2B2(t,T )− b(t)B(t,T ), A(T ,T ) = 0,

∂tB(t,T ) = −1, B(T ,T ) = 0

• explicit solution

B(t,T ) = T − t,

A(t,T ) = −σ2

6(T − t)3 +

∫ T

tb(s)(T − s) ds


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


⇒ forward curve

f (t,T ) = ∂T A(t,T ) + ∂T B(t,T )r(t)

= −σ2

2(T − t)2 +

∫ T

tb(s) ds + r(t)

⇒ b(s) = ∂s f0(s) + σ2s gives a perfect fit of observed initialforward curve f0(T )

• plugging back into ATS:

f (t,T ) = f0(T )− f0(t) + σ2t(T − t) + r(t)

• integrate:

P(t,T ) = e−∫ Tt f0(s) ds+f0(t)(T−t)−σ

2

2t(T−t)2−(T−t)r(t)


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


• interesting:

r(t) = r(0)+

∫ t

0b(s) ds+σW ∗(t) = f0(t)+

σ2t2

2+σW ∗(t)

⇒ r(t) fluctuates along the modified initial forward curve

⇒ forward vs. future rate: f0(t) = EQ[r(t)]− σ2t2

2


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Outline

16 Generalities





DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel

Hull–White:dr = (b(t) + βr) dt + σ dW ∗

• ATS equation for B(t,T ) as in Vasicek model:

∂tB(t,T ) = −βB(t,T )− 1, B(T ,T ) = 0

• explicit solution

B(t,T ) =1

β

(eβ(T−t) − 1

).

• ATS equation for A(t,T ):

A(t,T ) = −σ2

2

∫ T

tB2(s,T ) ds +

∫ T

tb(s)B(s,T ) ds


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


• notice ∂T B(s,T ) = −∂sB(s,T ):

f0(T ) = ∂T A(0,T ) + ∂T B(0,T )r(0)

=σ2

2

∫ T

0∂sB2(s,T ) ds +

∫ T

0b(s)∂T B(s,T ) ds

+ ∂T B(0,T )r(0)

= − σ2

2β2

(eβT − 1

)2

︸︷︷︸=:g(T )

+

∫ T

0b(s)eβ(T−s) ds + eβT r(0)︸︷︷︸

=:φ(T )

.

• φ satisfies

∂Tφ(T ) = βφ(T ) + b(T ), φ(0) = r(0).


DamirFilipovic

Generalities


Examples



SomeStandardModels

Vasicek Model

CIR Model

Dothan Model

Ho–Lee Model

Hull–WhiteModel


• since φ = f0 + g : conclude

b(T ) = ∂Tφ(T )− βφ(T )

= ∂T (f0(T ) + g(T ))− β(f0(T ) + g(T )).

• plugging in (. . . ):

f (t,T ) = f0(T )− eβ(T−t)f0(t)

− σ2

2β2

(eβ(T−t) − 1

)(eβ(T−t) − eβ(T+t)

)+ eβ(T−t)r(t).


DamirFilipovic

Forward CurveMovements

Absence ofArbitrage

ImpliedShort-RateDynamics

HJM Models

ProportionalVolatility

Fubini’sTheorem

Part V

Heath–Jarrow–Morton (HJM) Methodology


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Overview

• Have seen above: short-rate models not always flexibleenough to calibrating them to the observed initialterm-structure

• In late 1980s: Heath, Jarrow and Morton (henceforthHJM) [27] proposed a new framework for modeling theentire forward curve directly

• This chapter: provides essentials of HJM framework


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Recap: Usual Stochastic Setup

• filtered probability space (Ω,F , (Ft)t≥0,P)

• usual conditions:• completeness: F0 contains all of the null sets• right-continuity: Ft = ∩s>tFs for t ≥ 0

• d-dimensional (Ft)-Brownian motionW = (W1, . . . ,Wd)>

• infinite time horizon (w.l.o.g.): F = F∞ = ∨t≥0Ft


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Assumptions

• given R- and Rd -valued stochastic process α = α(ω, t,T )and σ = (σ1(ω, t,T ), . . . , σd(ω, t,T )) s.t.

(HJM.1) α and σ are Prog ⊗ B-measurable

(HJM.2)∫ T

0

∫ T0 |α(s, t)| ds dt <∞ for all T

(HJM.3) sups,t≤T ‖σ(s, t)‖ <∞ for all T (note: thisis a ω-wise boundedness assumption)

• given integrable initial forward curve T 7→ f (0,T )

• for every T : forward rate follows Ito dynamics (t ≤ T )

f (t,T ) = f (0,T ) +

∫ t

0α(s,T ) ds +

∫ t

0σ(s,T ) dW (s) (10)


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Assumptions

⇒ very general setup: only substantive economic restrictionis continuous sample paths assumption (and the finitenumber of random drivers W1, . . . ,Wd)

• integrals in (10) well defined by (HJM.1)–(HJM.3)

• from Fubini corollary below: implied short-rate process

r(t) = f (t, t) = f (0, t) +

∫ t

0α(s, t) ds +

∫ t

0σ(s, t) dW (s)

has progressive modification and satisfies∫ t

0 |r(s)| ds <∞a.s. for all t

• hence money-market account B(t) = e∫ t

0 r(s)ds well defined


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Properties

LemmaFor every maturity T , the zero-coupon bond price

P(t,T ) = e−∫ Tt f (t,u) du follows Ito process (t ≤ T )

P(t,T ) = P(0,T ) +

∫ t

0P(s,T ) (r(s) + b(s,T )) ds

+

∫ t

0P(s,T )v(s,T ) dW (s),

where

v(s,T ) = −∫ T

sσ(s, u) du,

is the T -bond volatility and

b(s,T ) = −∫ T

sα(s, u) du +

1

2‖v(s,T )‖2.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

ProofUsing the classical and stochastic Fubini Theorem below twice,we calculate

logP(t,T ) = −∫ T

tf (t, u) du

= −∫ T

tf (0, u) du −

∫ T

t

∫ t

0α(s, u) ds du −

∫ T

t

∫ t

0σ(s, u) dW (s) du

= −∫ T

tf (0, u) du −

∫ t

0

∫ T

tα(s, u) du ds −

∫ t

0

∫ T

tσ(s, u) du dW (s)

= −∫ T

0f (0, u) du −

∫ t

0

∫ T

sα(s, u) du ds −

∫ t

0

∫ T

sσ(s, u) du dW (s)

+

∫ t

0f (0, u) du +

∫ t

0

∫ t

sα(s, u) du ds +

∫ t

0

∫ t

sσ(s, u) du dW (s)

= −∫ T

0f (0, u) du +

∫ t

0

(b(s,T )− 1

2‖v(s,T )‖2

)ds +

∫ t

0v(s,T ) dW (s)

+

∫ t

0

(f (0, u) +

∫ u

0α(s, u) ds +

∫ u

0σ(s, u) dW (s)

)︸︷︷︸

=r(u)

du

= log P(0,T ) +

∫ t

0

(r(s) + b(s,T )− 1

2‖v(s,T )‖2

)ds +

∫ t

0v(s,T ) dW (s).

Ito’s formula now implies the assertion.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Discounted Price

Corollary

For every maturity T , the discounted bond price processsatisfies

P(t,T )

B(t)= P(0,T ) +

∫ t

0

P(s,T )

B(s)b(s,T ) ds

+

∫ t

0

P(s,T )

B(s)v(s,T ) dW (s).


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Absence of Arbitrage

• investigate restrictions on the dynamics (10) under noarbitrage

• assume given dQ/dP = E∞(γ •W ) for some γ ∈ L• Girsanov transformed Q-Brownian motion:

dW ∗ = dW − γ>dt

• Q an ELMM for the bond market if P(t,T )B(t) is a Q-local

martingale for every T


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

HJM Drift Condition

Theorem (HJM Drift Condition)

Q is an ELMM if and only if

b(t,T ) = −v(t,T ) γ(t)> for all T , dP⊗ dt-a.s.

In this case, the Q-dynamics of the forward rates f (t,T ) are ofthe form

f (t,T ) = f (0,T )+

∫ t

0

(σ(s,T )

∫ T

sσ(s, u)>du

)︸︷︷︸

HJM drift

ds+

∫ t

0σ(s,T ) dW ∗(s),

and the discounted T -bond price satisfies

P(t,T )

B(t)= P(0,T )Et(v(·,T ) •W ∗)

for t ≤ T .


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Proof I

In view of “Discounted Price” corollary we find that

dP(t,T )

B(t)=

P(t,T )

B(t)

(b(t,T ) + v(t,T ) γ(t)>

)dt

+P(t,T )

B(t)v(t,T ) dW ∗(t).

Hence P(t,T )B(t) , t ≤ T , is a Q-local martingale if and only if

b(t,T ) = −v(t,T ) γ(t)> dP⊗ dt-a.s. Since v(t,T ) andb(t,T ) are both continuous in T , we deduce that Q is anELMM if and only if b(t,T ) = −v(t,T ) γ(t)> for all T ,dP⊗ dt-a.s.Differentiating both sides in T yields

−α(t,T ) + σ(t,T )

∫ T

tσ(t, u)>du = σ(t,T ) γ(t)>


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Proof II

for all T , dP⊗ dt-a.s. Insert this in (10). The expression forP(t,T )/B(t) now follows from Stochastic Exponential Lemma.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Market Price of Risk

• follows from Theorem above:

dP(t,T ) = P(t,T )(

r(t)− v(t,T ) γ(t)>)

dt

+ P(t,T )v(t,T ) dW (t)

⇒ −γ = market price of risk for the bond market

• striking feature of HJM framework: Q-law of f (t,T ) onlydepends on volatility σ(t,T ), not on P-drift α(t,T )

⇒ option pricing only depends on σ, similar to Black–Scholesstock price model


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

When Is P(t,T )B(t) Q-martingale?

Corollary

Suppose HJM drift condition holds. Then Q is EMM if

1 Novikov condition EQ

[e

12

∫ T0 ‖v(t,T )‖2 dt

]<∞ for all T ; or

2 forward rates are nonnegative: f (t,T ) ≥ 0 for all t ≤ T .

Proof.Novikov condition is sufficient forP(t,T )B(t) = P(0,T )Et(v(·,T ) •W ∗) to be a Q-martingale.

If f (t,T ) ≥ 0, then 0 ≤ P(t,T )B(t) ≤ 1 is a uniformly bounded

local martingale, and hence a true martingale.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Simple Example

• interplay between short-rate models and HJM framework?

• example (simplest HJM model): constantσ(t,T ) ≡ σ > 0:

f (t,T ) = f (0,T ) + σ2t(

T − t

2

)+ σW ∗(t)

⇒ short rates follow Ho–Lee model:

r(t) = f (t, t) = f (0, t) +σ2t2

2+ σW ∗(t)


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

General Result

Proposition

Suppose f (0,T ), α(t,T ) and σ(t,T ) are differentiable in T

with∫ T

0 |∂uf (0, u)| du <∞ and such that (HJM.1)–(HJM.3)are satisfied for α(t,T ) and σ(t,T ) replaced by ∂Tα(t,T ) and∂Tσ(t,T ).Then the short-rate process is an Ito process of the form

r(t) = r(0) +

∫ t

0ζ(u) du +

∫ t

0σ(u, u) dW (u)

where

ζ(u) = α(u, u)+∂uf (0, u)+

∫ u

0∂uα(s, u) ds+

∫ u

0∂uσ(s, u) dW (s).


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

ProofRecall first that

r(t) = f (t, t) = f (0, t) +

∫ t

0α(s, t) ds +

∫ t

0σ(s, t) dW (s).

Applying the Fubini Theorem 25.1 below to the stochasticintegral gives∫ t

0σ(s, t) dW (s) =

∫ t

0σ(s, s) dW (s) +

∫ t

0(σ(s, t)− σ(s, s)) dW (s)

=

∫ t

0σ(s, s) dW (s) +

∫ t

0

∫ t

s∂uσ(s, u) du dW (s)

=

∫ t

0σ(s, s) dW (s) +

∫ t

0

∫ u

0∂uσ(s, u) dW (s) du.

Moreover, from the classical Fubini Theorem we deduce in asimilar way that∫ t

0α(s, t) ds =

∫ t

0α(s, s) ds +

∫ t

0

∫ u

0∂uα(s, u) ds du,

and finally

f (0, t) = r(0) +

∫ t

0∂uf (0, u) du.

Combining these formulas, we obtain the desired result.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

HJM Models

• HJM model: σ(ω, t,T ) = σ(t,T , f (ω, t,T )) forappropriate function σ

• simplest choice: deterministic σ(t,T ) not depending on ω

⇒ Gaussian distributed forward rates f (t,T ) ⇒ simple bondoption price formulas (see later)

• particular case: Ho-Lee model σ(t,T ) ≡ σ (seen above)


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

HJM Models: Existence andUniqueness

• shown in HJM [27] and Morton [40]:

• assume: σ(t,T , f ) uniformly bounded, jointly continuous,and Lipschitz continuous in f

• assume: continuous initial forward curve f (0,T )

• then ∃ unique jointly continuous solution f (t,T ) of

df (t,T ) =

(σ(t,T , f (t,T ))

∫ T

tσ(t, u, f (t, u)) du

)dt

+ σ(t,T , f (t,T )) dW (t)

• remarkable: boundedness condition on σ cannot besubstantially weakened as following example shows . . .


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Proportional Volatility

• single Brownian motion (d = 1)

• σ(t,T , f (t,T )) = σf (t,T ) for some constant σ > 0:positive and Lipschitz continuous but not bounded

• solution of HJM equation must satisfy

f (t,T ) = f (0,T )eσ2∫ t

0

∫ Ts f (s,u) du dseσW (t)−σ

2

2t (11)

• claim: there is no finite-valued solution to this expression(following Avellaneda and Laurence [2, Section 13.6])


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem


• assume for simplicity: f (0,T ) ≡ 1 and σ = 1

• differentiating both sides of (11) in T :

∂T f (t,T ) = f (t,T )

∫ t

0f (s,T ) ds =

1

2∂t

(∫ t

0f (s,T ) ds

)2

• integrating from t = 0 to t = 1 and interchanging order ofdifferentiation and integration:

∂T

∫ 1

0f (s,T ) ds =

1

2

(∫ 1

0f (s,T ) ds

)2

• solving this differential equation path-wise forX (T ) =

∫ 10 f (s,T ) ds, T ≥ 1, we obtain as unique

solution

X (T ) =X (1)

1− X (1)2 (T − 1)


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem


• since X (1) > 0: X (T ) ↑ ∞ for T ↑ τ where τ = 1 + 2X (1)

is a finite random time

• conclude: f (ω, t, τ(ω)) must become +∞ for some t ≤ 1,for almost all ω.

• nonexistence of HJM models with proportional volatilityencouraged development of LIBOR market models (seelater)


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Outline







DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Fubini’s Theorem

Theorem (Fubini’s theorem for Stochastic Integrals)

Consider the Rd -valued stochastic process φ = φ(ω, t, s) withtwo indices, 0 ≤ t, s ≤ T , satisfying the following properties:

1 φ is ProgT ⊗ B[0,T ]-measurable;

2 supt,s ‖φ(t, s)‖ <∞.2

Then λ(t) =∫ T

0 φ(t, s) ds ∈ L, and there exists aFT ⊗ B[0,T ]-measurable modification ψ(s) of∫ T

0 φ(t, s) dW (t) with∫ T

0 ψ2(s) ds <∞ a.s.

Moreover,∫ T

0 ψ(s) ds =∫ T

0 λ(t) dW (t), that is,∫ T

0

(∫ T

0φ(t, s) dW (t)

)ds =

∫ T

0

(∫ T

0φ(t, s) ds

)dW (t).

2Note that this is a ω-wise boundedness assumption.


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Fubini’s Theorem: Proof

see course book Section 6.5


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Fubini’s Theorem: Corollary

Corollary

Let φ be as in Theorem 25.1. Then the process∫ s

0φ(t, s) dW (t), s ∈ [0,T ],

has a progressive modification π(s) with∫ T

0 π2(s) ds <∞ a.s.

Proof.For φ(ω, t, s) = K (ω, t)f (s), with bounded progressive processK and bounded measurable function f , the process∫ s

0φ(t, s) dW (t) = f (s)

∫ s

0K (t) dW (t)

is clearly progressive and path-wise square integrable. Now usea similar monotone class and localization argument as in theproof of Theorem 25.1 (→ exercise).


DamirFilipovic


Absence ofArbitrage


HJM Models


Fubini’sTheorem

Monotone Class Theorem

Theorem (Monotone Class Theorem)

Suppose the set H consists of real-valued bounded functionsdefined on a set Ω with the following properties:

1 H is a vector space;

2 H contains the constant function 1Ω;

3 if fn ∈ H and fn ↑ f monotone, for some bounded functionf on Ω, then f ∈ H.

If H contains a collection M of real-valued functions, which isclosed under multiplication (that is, f , g ∈M impliesfg ∈M). Then H contains all real-valued bounded functionsthat are measurable with respect to the σ-algebra which isgenerated by M (that is, σf −1(A) | A ∈ B, f ∈M).

Proof.see e.g. Steele [49, Section 12.6].


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing

Example:VasicekShort-RateModel

Black–ScholesModel withGaussianInterest Rates

Example: Black–Scholes–VasicekModel

Part VI

Forward Measures


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Overview

• we replace risk-free numeraire by another traded asset,such as the T -bond

• change of numeraire technique proves most useful foroption pricing and provides the basis for the marketmodels studied below

• we derive explicit option price formulas for Gaussian HJMmodels

• this includes the Vasicek short-rate model and someextension of the Black–Scholes model with stochasticinterest rates


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Outline





DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




T -Forward Measure

• HJM setup from above: ∃ EMM Q for all T -bonds, W ∗

the respective Q-Brownian motion

• recall: T -bond volatility v(t,T ) = −∫ Tt σ(t, u) du

• fix T > 0: dQT

dQ = 1P(0,T )B(T ) defines probability measure

QT ∼ Q on FT (why?) and for t ≤ T :

dQT

dQ|Ft = EQ

[dQT

dQ| Ft

]=

P(t,T )

P(0,T )B(t)

• QT is called the T -forward measure

• from above: dQT

dQ |Ft = Et (v(·,T ) •W ∗)

⇒ Girsanov’s Theorem: W T (t) = W ∗(t)−∫ t

0 v(s,T )>ds,t ≤ T , is a QT -Brownian motion


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Fundamental Property

LemmaFor any S > 0, the T -bond discounted S-bond price process

P(t, S)

P(t,T )=

P(0,S)

P(0,T )Et(σS,T •W T

), t ≤ S ∧ T

is a QT -martingale, where we define

σS ,T (t) = −σT ,S(t) = v(t, S)− v(t,T ) =

∫ T

Sσ(t, u) du.

(12)Moreover, the T - and S-forward measures are related by

dQS

dQT|Ft =

P(t, S) P(0,T )

P(t,T ) P(0, S)= Et

(σS,T •W T

), t ≤ S ∧ T .


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




ProofLet u ≤ t ≤ S ∧ T . Bayes’ rule gives

EQT

[P(t,S)

P(t,T )| Fu

]=

EQ

[P(t,T )

P(0,T )B(t)P(t,S)P(t,T ) | Fu

]P(u,T )

P(0,T )B(u)

=

P(u,S)B(u)

P(u,T )B(u)

=P(u, S)

P(u,T ),

which proves that P(t,S)/P(t,T ) is a martingale. Thestochastic exponential representation follows from StochasticExponential Lemma and the representation of P(t,T )/B(t)(→ exercise). The second claim follows from the identity

dQS

dQT|Ft =

dQS

dQ|Ft

dQdQT

|Ft .


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Pricing with Stochastic InterestRates

⇒ collection of EMMs: each QT corresponds to differentnumeraire: T -bond

• Q is called the risk-neutral (or spot) measure

• simpler pricing formulas: let X be a T -claim s.t.

EQ

[|X |

B(T )

]<∞

• arbitrage price at t ≤ T : π(t) = B(t)EQ

[X

B(T ) | Ft

]• computation: have to know joint distribution of 1/B(T )

and X ⇒ double integral (rather hard work)


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Pricing with Stochastic InterestRates

• if 1/B(T ) and X were independent under Q conditionalon Ft : π(t) = P(t,T )EQ [X | Ft ]

• a much simpler formula, since:• only have to compute single integral EQ[X | Ft ];• P(t,T ) observable at t, and does not have to be

computed within the model

• but: independence of 1/B(T ) and X unrealisticassumption for interest rate sensitive claims X !

• good news: above simple formula holds—under QT . . .


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Forward Measure Pricing

Proposition

Under above assumptions: EQT [|X |] <∞ and

π(t) = P(t,T )EQT [X | Ft ] .


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Application: ExpectationHypothesis

Lemma (Expectation Hypothesis)

If σ(·,T ) ∈ L2, the expectation hypothesis holds under theT -forward measure:

f (t,T ) = EQT [r(T ) | Ft ] for t ≤ T .

Proof.Under QT we have

f (t,T ) = f (0,T ) +

∫ t

0σ(s,T ) dW T (s). (13)

Hence, if σ(·,T ) ∈ L2 then f (t,T ), t ≤ T , is aQT -martingale.


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T -Bond asNumeraire

Bond OptionPricing




A Word of Warning

In view of equation (13) it is tempting to “specify” a forwardrate model by postulating the dynamics of f (·,T ) under QT

for each maturity T separately without reference to someunderlying Q. However, it is far from clear whether a commonrisk-neutral measure Q, tying all QT ’s, exists in this case. Onthe other hand, we note that this is exactly the approach in theLIBOR market model developed below. The importantdifference being that there one considers finitely manymaturities only.


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T -Bond asNumeraire

Bond OptionPricing




Application:Dybvig–Ingersoll–Ross Theorem

• Dybvig–Ingersoll–Ross [24]: long rates can never fall

• recall: zero-coupon yield R(t,T ) = 1T−t

∫ Tt f (t, s) ds

• define: asymptotic long rate R∞(t) = limT→∞ R(t,T )

Lemma (Dybvig–Ingersoll–Ross Theorem)

For all s < t the long rates satisfy R∞(s) ≤ R∞(t) if they exist.


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




ProofLet s < t be such that R∞(s) and R∞(t) exist. Then

p(u) = limT→∞ P(t,T )1T = e−R∞(u) exist for u ∈ s, t, and

it remains to prove that p(s) ≥ p(t).Under the t-forward measure Qt , we have

P(s,T )

P(s, t)= EQt [P(t,T ) | Fs ],

and thusp(s) = lim

T→∞EQt [P(t,T ) | Fs ]

1T .

Now let X ≥ 0 be any bounded random variable withEQt [X ] = 1. Using the Fs -conditional versions of Fatou’slemma, Holder’s inequality and dominated convergence, weobtain

EQt [X p(t)] = EQt

[lim inf

T→∞X P(t,T )

1T

]≤ EQt

[lim inf

T→∞EQt

[X P(t,T )

1T | Fs

]]≤ EQt

[lim inf

T→∞EQt

[X

TT−1 | Fs

]T−1T EQt [ P(t,T ) | Fs ]

1T

]= EQt [X p(s)] .

Since X was arbitrary with the stated properties, we concludethat p(t) ≤ p(s), and the lemma is proved.


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T -Bond asNumeraire

Bond OptionPricing




Outline





DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Bond Option Pricing: General

• consider European call option on S-bond, expiry dateT < S , strike price K

• arbitrage price at t = 0 (for simplicity):

π = EQ

[e−

∫ T0 r(s) ds (P(T ,S)− K )+

]• decompose

π = EQ[B(T )−1P(T , S) 1P(T ,S)≥K

]− KEQ

[B(T )−1 1P(T ,S)≥K

]= P(0, S)QS [P(T , S) ≥ K ]

− KP(0,T )QT [P(T , S) ≥ K ]


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Bond Option Pricing: General

• observe that

QS [P(T , S) ≥ K ] = QS

[P(T ,T )

P(T ,S)≤ 1

K

]QT [P(T , S) ≥ K ] = QT

[P(T ,S)

P(T ,T )≥ K

].

⇒ look for lognormal P(T ,T )P(T ,S) , P(T ,S)

P(T ,T ) under QS , QT . . .


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Bond OptionPricing




Bond Option Pricing: GaussianModels

• assume: σ(t,T ) = (σ1(t,T ), . . . , σd(t,T )) deterministic

⇒ f (t,T ) Gaussian distributed

⇒ P(T ,T )P(T ,S) , P(T ,S)

P(T ,T ) lognormal under QS , QT

⇒ closed-form option price formula . . .


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T -Bond asNumeraire

Bond OptionPricing




Bond Option Pricing: GaussianModels

Proposition

Under the above Gaussian assumption, the bond option price is

π = P(0,S)Φ[d1]− KP(0,T )Φ[d2],

where Φ is the standard Gaussian cumulative distributionfunction,

d1,2 =log[

P(0,S)KP(0,T )

]± 1

2

∫ T0 ‖σT ,S(s)‖2 ds√∫ T

0 ‖σT ,S(s)‖2 ds,

and σT ,S(s) is given in (12).

• similar closed-form expression for put option (. . . )


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T -Bond asNumeraire

Bond OptionPricing




Proof

Proof.It is enough to observe that

log P(T ,T )P(T ,S) − log P(0,T )

P(0,S) + 12

∫ T0 ‖σT ,S(s)‖2 ds√∫ T

0 ‖σT ,S(s)‖2 ds

andlog P(T ,S)

P(T ,T ) − log P(0,S)P(0,T ) + 1

2

∫ T0 ‖σT ,S(s)‖2 ds√∫ T

0 ‖σT ,S(s)‖2 ds

are standard Gaussian distributed under QS and QT ,respectively.


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T -Bond asNumeraire

Bond OptionPricing




Outline





DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Vasicek Model

• Vasicek short-rate model (d = 1):dr = (b + βr) dt + σ dW ∗

⇒ (. . . ) df (t,T ) = α(t,T ) dt + σeβ(T−t) dW ∗

⇒ σ(t,T ) = σeβ(T−t) deterministic


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T -Bond asNumeraire

Bond OptionPricing




Vasicek Model• example: β = −0.86, b/|β| = 0.09 (mean reversion level),σ = 0.0148 and r(0) = 0.08

• ATM cap prices and Black volatilities for: t0 = 0 (today),T0 = 1/4 (first reset date), and Ti − Ti−1 ≡ 1/4,i = 1, . . . , 119 (maturity of the last cap is T119 = 30)

• contrast to market curve: Vasicek model cannot producehumped volatility curves

Figure: Vasicek ATM cap Black volatilities.


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T -Bond asNumeraire

Bond OptionPricing




Vasicek Model

Table: Vasicek ATM cap prices and Black volatilities

Maturity ATM prices ATM vols

1 0.00215686 0.1297342 0.00567477 0.1063483 0.00907115 0.09154554 0.0121906 0.08153585 0.01503 0.07436076 0.017613 0.06896517 0.0199647 0.06475158 0.0221081 0.0613624

10 0.025847 0.056233712 0.028963 0.052529615 0.0326962 0.048575520 0.0370565 0.044396730 0.0416089 0.0402203


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Bond OptionPricing




Outline





DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Generalized Black–Scholes

• Black–Scholes model [9]: stock S , money-market accountB following Q-dynamics

dB = Br dt, B(0) = 1,

dS = Sr dt + SΣ dW ∗, S(0) > 0,

• constant volatility Σ = (Σ1, . . . ,Σd) ∈ Rd

• new: r stochastic within above Gaussian HJM setup


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T -Bond asNumeraire

Bond OptionPricing





• consider European call option on S , maturity T , strikeprice K

• arbitrage price at t = 0 (for simplicity):

π = EQ

[1

B(T )(S(T )− K )+

]= EQ

[S(T )

B(T )1S(T )≥K

]− KEQ

[1

B(T )1S(T )≥K

]• recall T -forward measure QT

• similarly: choose S as numeraire and define EMMQ(S) ∼ Q on FT via

dQ(S)

dQ=

S(T )

S(0) B(T )= ET (Σ •W ∗)

• Girsanov: W (S)(t) = W ∗(t)− Σ>t, t ≤ T ,Q(S)-Brownian motion


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• change of measure (Bayes):π = S(0)Q(S)[S(T ) ≥ K ]− KP(0,T )QT [S(T ) ≥ K ]

• remains: compute probabilities

Q(S)[S(T ) ≥ K ] = Q(S)

[P(T ,T )

S(T )≤ 1

K

],

QT [S(T ) ≥ K ] = QT

[S(T )

P(T ,T )≥ K

]• observe: P(t,T )

S(t) is Q(S)-martingale and S(t)P(t,T ) is QT

martingale for t ≤ T


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing





• Ito’s formula (to practice stochastic calculus):

dS(t)

P(t)=

1

P(t)dS(t)− S(t)

P(t)2dP(t)

− 1

P(t)2d〈S ,P〉t +

S(t)

P(t)3d〈P,P〉t

= (· · · ) dt +S(t)

P(t)(Σ− v(t,T )) dW ∗(t)

(omitted parameter T in P(t,T ))

• recall: T -bond volatility v(t,T ) = −∫ Tt σ(t, u) du

• no need to compute drift term: volatility unaffected bychange of measure ⇒ QT -dynamics:

dS(t)

P(t,T )=

S(t)

P(t,T )(Σ− v(t,T )) dW T (t)


DamirFilipovic

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⇒ stochastic exponential equation:

S(T )

P(T ,T )=

S(0)

P(0,T )ET(

(Σ− v(·,T )) •W T)

is lognormally distributed under QT

• along similar calculations (. . . ):

P(T ,T )

S(T )=

P(0,T )

S(0)ET(−(Σ− v(·,T )) •W (S)

)is lognormally distributed under Q(S)


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Generalized Black–Scholes OptionPrice Formula

Proposition

In the above generalized Black–Scholes model, the option priceis

π = S(0)Φ[d1]− KP(0,T )Φ[d2],

where Φ is the standard Gaussian cumulative distributionfunction and

d1,2 =log[

S(0)KP(0,T )

]± 1

2

∫ T0 ‖Σ− v(t,T )‖2 dt√∫ T

0 ‖Σ− v(t,T )‖2 dt. (14)

Note that v(t,T ) = 0 yields the classical Black–Scholes optionprice formula for constant short rate.


DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Outline





DamirFilipovic

T -Bond asNumeraire

Bond OptionPricing




Black–Scholes–Vasicek

• special case of generalized Black–Scholes formula: Vasicekshort-rate model

• dim W ∗ = d = 2

• Vasicek short-rate dynamics:

dr = (b + βr) dt + σ dW ∗,

where σ = (σ1, σ2)

• note: this corresponds to standard representationdr = (b + βr) dt + ‖σ‖ dW∗ for the one-dimensional

Q-Brownian motion W∗ =σ1 W ∗1 +σ2 W ∗2

‖σ‖

⇒ R2-valued T -bond volatility

v(t,T ) = −σ∫ T

teβ(T−s) ds =

σ

β

(1− eβ(T−t)

)


DamirFilipovic

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• tedious but elementary computation yields∫ T

0‖Σ− v(t,T )‖2 dt

= ‖Σ‖2 T + 2Σσ>eβT − 1− βT

β2

+ ‖σ‖2 e2βT − 4eβT + 2βT + 3

2β3

for aggregate volatility in (14)

• relation between option price π and instantaneouscovariation d〈S , r〉/dt = Σσ> of S and r : π monotone

increasing in∫ T

0 ‖Σ− v(t,T )‖2 dt, which again isincreasing in Σσ> since eβT − 1− βT > 0


DamirFilipovic

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⇒ π increases with increasing covariation between S and r

• for negative covariation, π may be smaller than classicalBlack–Scholes option price with constant short rates(σ = 0)


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ForwardContracts

FuturesContracts

Interest RateFutures

Forwardvs. Futures ina GaussianSetup

Part VII

Forwards and Futures


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ForwardContracts

FuturesContracts



Overview

• we discuss two common types of term contracts:

• forwards: mainly traded over the counter (OTC)

• futures: actively traded on many exchanges

• underlying in both cases: a T -claim Y, e.g. exchange rate,interest rate, commodity such as copper, any traded ornon-traded asset, an index, etc.

• special discussion: interest rate futures, futures rates andforward rates in the Gaussian HJM model


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DamirFilipovic

ForwardContracts

FuturesContracts



Forward Contracts

• assume HJM setup, and let Y denote a T -claim

• forward contract on Y, contracted at t, with time ofdelivery T > t and forward price f (t; T ,Y) defined byfollowing payment scheme:

• at T : holder (long position) pays f (t; T ,Y) and receivesY from underwriter (short position)

• at t: forward price chosen such that present value offorward contract is zero:

EQ

[e−

∫ Tt

r(s) ds (Y − f (t; T ,Y)) | Ft

]= 0

• this is equivalent to

f (t; T ,Y) =1

P(t,T )EQ

[e−

∫ Tt r(s) dsY | Ft

]= EQT [Y | Ft ]


DamirFilipovic

ForwardContracts

FuturesContracts



Forward Contracts

• examples: the forward price at t of:

1 a dollar delivered at T is 12 an S-bond delivered at T ≤ S is P(t,S)

P(t,T )

3 any traded asset S delivered at T is S(t)P(t,T )

• note: forward price f (s; T ,Y) has to be distinguished from(spot) price at time s of the forward contract entered attime t ≤ s, which is

EQ

[e−

∫ Ts r(u) du (Y − f (t; T ,Y)) | Fs

]= P(s,T ) (f (s; T ,Y)− f (t; T ,Y))


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FuturesContracts



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DamirFilipovic

ForwardContracts

FuturesContracts



Futures Contracts

• futures contract on Y with time of delivery T defined as:• at every t ≤ T : there is a market quoted futures price

F (t; T ,Y), which makes the futures contract on Y, ifentered at t, equal to zero

• at T : holder (long position) pays F (T ; T ,Y) and receivesY from underwriter (short position)

• marking to market or resettlement: during any infinitesimaltime interval (t, t + ∆t]: holder receives (or pays, ifnegative) F (t; T ,Y)− F (t + ∆t; T ,Y)

⇒ continuous cash flow between the two parties of a futurescontract, they are required to keep certain amount ofmoney as safety margin


DamirFilipovic

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Futures Contracts

• trading volumes in futures are huge

• one of reasons: often difficult to trade/hedge directly inunderlying object (e.g. an index including illiquidinstruments, or a commodity such as copper, gas orelectricity)

• holding short position in futures: no need to physicallydeliver the underlying object if you exit contract beforedelivery date

• selling short makes it possible to hedge against theunderlying


DamirFilipovic

ForwardContracts

FuturesContracts



Futures Price Process

• Suppose EQ[|Y|] <∞, then futures price process =Q-martingale:

F (t; T ,Y) = EQ [Y | Ft ]

• consequence: forward = futures price if interest ratesdeterministic


DamirFilipovic

ForwardContracts

FuturesContracts



Heuristic Argument

• w.l.o.g F (t) = F (t; T ,Y) is an Ito process

• cumulative discounted cash flow of futures contract in(t,T ]: V = limN VN with

VN =N∑

i=1

1

B(ti )(F (ti )− F (ti−1))

limit over sequence of partitions t = t0 < · · · < tN = Twith maxi |ti − ti−1| → 0 for N →∞

• can rewrite

VN =N∑

i=1

1

B(ti−1)(F (ti )− F (ti−1))

+N∑

i=1

(1

B(ti )− 1

B(ti−1)

)(F (ti )− F (ti−1))


DamirFilipovic

ForwardContracts

FuturesContracts



Heuristic Argument

• B continuous ⇒ 1/B ∈ L(F )

• elementary stochastic calculus: VN → V in probabilitywith

V =

∫ T

t

1

B(s)dF (s) +

∫ T

td

⟨1

B,F

⟩s

=

∫ T

t

1

B(s)dF (s)


DamirFilipovic

ForwardContracts

FuturesContracts



Heuristic Argument

• present value = zero:

EQ

[∫ T

t

1

B(s)dF (s) | Ft

]= 0

• consequence:

M(t) =

∫ t

0

1

B(s)dF (s) = EQ

[∫ T

0

1

B(s)dF (s) | Ft

]is Q-martingale, t ≤ T

• assume: EQ

[∫ T0 B(s)2 d〈M,M〉s

]= EQ [〈F ,F 〉T ] <∞

⇒ B ∈ L2(M) and

F (t) =

∫ t

0B(s) dM(s)

is Q-martingale for t ≤ T , as desired


DamirFilipovic

ForwardContracts

FuturesContracts



Outline





DamirFilipovic

ForwardContracts

FuturesContracts



Eurodollar Futures

• Interest rate futures: can be divided into futures onshort-term instruments and futures on coupon bonds

• here: we only consider an example from the first group

• Eurodollars = deposits of US dollars in institutions outsideof the US

• LIBOR is the interbank rate of interest for Eurodollar loans

• Eurodollar futures contract is tied to LIBOR

• introduced by the International Money Market (IMM) ofthe Chicago Mercantile Exchange (CME) in 1981

• designed to protect its owner from fluctuations in the3-month (=1/4 year) LIBOR

• maturity (delivery) months are March, June, Septemberand December


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ForwardContracts

FuturesContracts



Formal Definition

• fix maturity T

• L(T ) = 3-month spot LIBOR for period [T ,T + 1/4]

• market quote of Eurodollar futures contract on L(T ) att ≤ T is

1− LF (t,T ) [100 per cent]

where LF (t,T ) = corresponding futures rate (comparewith bootstrapping example above)

• futures price, used for the marking to market, defined by

F (t; T , L(T )) = 1− 1

4LF (t,T ) [million dollars]

⇒ change of 1 basis point (0.01%) in futures rate LF (t,T )leads to cash flow of 106 × 10−4 × 1

4 = 25 [dollars]


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ForwardContracts

FuturesContracts



Formal Definition

• definition: LF (T ,T ) = L(T )

⇒ final price F (T ; T , L(T )) = 1− 14 L(T ) = Y: underlying Y

is a synthetic value, no physical delivery at maturity,settlement is made in cash

• since F (t; T , L(T )) = EQ [F (T ; T , L(T )) | Ft ] we obtainexplicit formula for the futures rate

LF (t,T ) = EQ [L(T ) | Ft ]


DamirFilipovic

ForwardContracts

FuturesContracts



A Mathematicians Intuition

• underlying Y = P(T ,T + 1/4)

• futures price = EQ [P(T ,T + 1/4) | Ft ]

• exact: P(T ,T + 1/4) = 1− 14 L(T )P(T ,T + 1/4)

• approximate: P(T ,T + 1/4) ≈ 1− 14 L(T )


DamirFilipovic

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FuturesContracts



Outline





DamirFilipovic

ForwardContracts

FuturesContracts



Forward vs. Futures

• let S be price process of a traded asset with Q-dynamics

dS(t)

S(t)= r(t) dt + ρ(t) dW ∗(t)

• fix delivery date T

• forward price of S for delivery at T :

f (t; T ,S(T )) =S(t)

P(t,T )

• futures price of S for delivery at T

F (t; T ,S(T )) = EQ[S(T ) | Ft ]

• aim: establish relationship between the two prices underGaussian assumption


DamirFilipovic

ForwardContracts

FuturesContracts



Forward vs. Futures

Proposition

Suppose ρ(t) and T -bond volatility v(t,T ) are deterministic.Then

F (t; T , S(T )) = f (t; T ,S(T )) e∫ Tt (v(s,T )−ρ(s)) v(s,T )>ds

for t ≤ T .

Hence, if the instantaneous covariation of S(t) and P(t,T ) isnegative,

d〈S ,P(·,T )〉tdt

= S(t)P(t,T ) ρ(t) v(t,T )> ≤ 0,

then the futures price dominates the forward price.


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FuturesContracts



Proof

Proof.Write µ(s) = v(s,T )− ρ(s). It is clear that

f (t; T ,S(T )) =S(0)

P(0,T )Et(µ•W ∗) exp

(∫ t

0µ(s) v(s,T )>ds

).

Since E(µ •W ∗) is a Q-martingale and ρ(s) and v(s,T ) aredeterministic, we obtain

F (t; T ,S(T )) = EQ[f (T ; T , S(T )) | Ft ]

= f (t; T ,S(T )) e∫ Tt µ(s) v(s,T )>ds ,

as desired.


DamirFilipovic

ForwardContracts

FuturesContracts



Forward vs. Futures Rates

Lemma (Convexity Adjustments)

Assume Gaussian HJM framework. Then relation betweeninstantaneous forward and futures rates:

f (t,T ) = EQ[r(T ) | Ft ]−∫ T

t

(σ(s,T )

∫ T

sσ(s, u)>du

)ds

and simple forward and futures rates:

F (t; T ,S) = EQ[F (T ,S) | Ft ]

− P(t,T )

(S − T )P(t, S)

(e∫ Tt

(∫ ST σ(s,v) dv

∫ Ss σ(s,u)> du

)ds − 1

)


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FuturesContracts



Forward vs. Futures Rates

Hence, if

σ(s, v)σ(s, u)> ≥ 0 for all s ≤ u ∧ v

then futures rates are always greater than the correspondingforward rates.

Proof.Exercise


DamirFilipovic

Multi-factorModels

ConsistencyCondition


PolynomialTerm-Structures

Special Case:m = 1

General Case:m ≥ 1


Nelson–SiegelFamily

Svensson Family

Part VIII

Consistent Term-Structure Parametrizations


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Overview

• practitioners and academics have vital interest inparameterized term-structure models

• in this chapter: take up a point left open at the end ofestimation chapter: exploit whether parameterized curvefamilies φ(·, z), used for estimating the forward curve, gowell with arbitrage-free interest rate models

• recall BIS document [5]: rich source of cross-sectionaldata (daily estimations of the parameter z) for theNelson–Siegel and Svensson families

• suggests that calibrating a diffusion process Z for theparameter z would lead to an accurate factor model forthe forward curve

• conditions for absence of arbitrage can be formulated interms of the drift and diffusion of Z and derivatives of φ

• these conditions turn out to be surprisingly restrictive insome cases


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Short Rate Models Revisited

• seen earlier: every time-homogeneous diffusion short-ratemodel r(t) induces forward rates of the form

f (t,T ) = φ(T − t, r(t))

for some deterministic function φ

+ computational merits (e.g. Jamshidian decomposition)

− unrealistic implications: e.g. family of attainable forwardcurves φ(·, r) | r ∈ R is only one-dimensional

→ term-structure movements explained by single statevariable r(t): conflicts with above principal componentanalysis (2-3 factors needed for statistically accuratedescription of forward curve movements)

− maturity-specific risk: if d = 1 e.g. a bond option withmaturity five years could be perfectly hedged by themoney-market account and a bond of maturity 30 years


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Multi-factor Models

• to gain more flexibility: multiple factors m ≥ 1

• m-factor model := interest rate model of the form

f (t,T ) = φ(T − t,Z (t))

• deterministic φ

• m-dim state space process Z


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Assumptions

• state space Z ⊂ Rm closed with non-empty interior

• φ ∈ C 1,2(R+ ×Z)

• b : Z → Rm continuous

• ρ : Z → Rm×d measurable, s.t. diffusion matrixa(z) = ρ(z)ρ(z)> continuous in z ∈ Z

• W ∗: d-dim Brownian motion on (Ω,F , (Ft),Q)

• ∀z ∈ Z ∃ unique Z-valued solution Z = Z z of

dZ (t) = b(Z (t)) dt + ρ(Z (t)) dW ∗(t)

Z (0) = z

• NA: Q is risk-neutral measure for bond prices

P(t,T ) = exp

(−∫ T−t

0φ(x ,Z z(t)) dx

)for all z ∈ Z


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Remark

Time-inhomogeneous models are included by identifying onecomponent, say Z1, with calendar time. We therefore setdZ1 = dt, which is equivalent to b1 ≡ 1 and ρ1j ≡ 0 forj = 1, . . . , d . Calendar time at inception is now Z1(0) = z1,and t, T , etc. accordingly denote relative time with respect toz1. The NA assumption for all z ∈ Z now means, in particular,that absence of arbitrage holds relative to any initial calendartime z1.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

First Consequences

• short rates given by r(t) = φ(0,Z (t))

⇒ NA holds if and only if

exp(−∫ T−t

0 φ(x ,Z z(t)) dx)

exp(∫ t

0 φ(0,Z z(s)) ds) , t ≤ T ,

is a Q-local martingale, for all z ∈ Z• aim: find consistency condition for a, b and φ such that

NA holds

• possible way: apply Ito’s formula and set drift = 0

• our way: first embed in HJM framework and use HJMdrift condition . . .


DamirFilipovic

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DamirFilipovic

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Special Case:m = 1




Svensson Family

Consistency Condition: Derivation

apply Ito’s formula to f (t,T ) = φ(T − t,Z (t)):

df (t,T ) =(− ∂xφ(T − t,Z (t)) +

m∑i=1

bi (Z (t))∂ziφ(T − t,Z (t))

+1

2

m∑i ,j=1

aij(Z (t))∂zi∂zjφ(T − t,Z (t)))

dt

+m∑

i=1

d∑j=1

∂ziφ(T − t,Z (t))ρij(Z (t)) dW ∗j (t).


DamirFilipovic

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Svensson Family

Consistency Condition: Derivation

⇒ induced forward rate model is of the HJM type with

α(t,T ) = −∂xφ(T − t,Z (t)) +m∑

i=1

bi (Z (t))∂ziφ(T − t,Z (t))

+1

2

m∑i ,j=1

aij(Z (t))∂zi∂zjφ(T − t,Z (t))

σj(t,T ) =m∑

i=1

∂ziφ(T − t,Z (t))ρij(Z (t)), j = 1, . . . , d

satisfying (HJM.1)–(HJM.3)


DamirFilipovic

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Special Case:m = 1




Svensson Family

Consistency Condition: DerivationNA ⇔ HJM drift condition:

− φ(T − t,Z (t)) + φ(0,Z (t)) +m∑

i=1

bi (Z (t))∂zi Φ(T − t,Z (t))

+1

2

m∑i ,j=1

aij(Z (t))∂zi∂zj Φ(T − t,Z (t))

=1

2

d∑j=1

(m∑

i=1

∂zi Φ(T − t,Z (t))ρij(Z (t))

)2

=1

2

m∑k,l=1

akl(Z (t))∂zkΦ(T − t,Z (t))∂zl

Φ(T − t,Z (t))

where we define Φ(x , z) =∫ x

0 φ(u, z) du

• to hold a.s. for all t ≤ T and z = Z (0), now let t → 0 . . .


DamirFilipovic

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Special Case:m = 1




Svensson Family

Consistency Condition

Proposition (Consistency Condition)

NA holds if and only if

∂xΦ(x , z) = φ(0, z) +m∑

i=1

bi (z)∂zi Φ(x , z)

+1

2

m∑i ,j=1

aij(z)(∂zi∂zj Φ(x , z)− ∂zi Φ(x , z)∂zj Φ(x , z)

)(15)

for all (x , z) ∈ R+ ×Z.


DamirFilipovic

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Special Case:m = 1




Svensson Family

Terminology

DefinitionThe pair of characteristics a, b and the forward curveparametrization φ are consistent if NA, or equivalently theabove consistency condition (15), holds.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Interpretations of (15)

• pricing: take φ(0, z), a, b as given and solve PDE (15)with initial condition Φ(0, z) = 0

• inverse problem: given parametric estimation method φgiven, find a and b such that (15) is satisfied for all (x , z)

• it turns out that the latter approach is quite restrictive onpossible choices of a and b . . .


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

First Consequence

Proposition

Suppose that the functions

∂zi Φ(·, z) and1

2

(∂zi∂zj Φ(·, z)− ∂zi Φ(·, z)∂zj Φ(·, z)

),

for 1 ≤ i ≤ j ≤ m, are linearly independent for all z in somedense subset D ⊂ Z. Then there exists one and only oneconsistent pair a, b.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof

Proof.Set M = m + m(m + 1)/2, the number of unknown functionsbk and akl = alk . Let z ∈ D. Then there exists a sequence0 ≤ x1 < · · · < xM such that the M ×M-matrix with kth rowvector built by

∂zi Φ(xk , z) and1

2

(∂zi∂zj Φ(xk , z)− ∂zi Φ(xk , z)∂zj Φ(xk , z)

),

for 1 ≤ i ≤ j ≤ m, is invertible. Thus, b(z) and a(z) areuniquely determined by (15). This holds for each z ∈ D. Bycontinuity of b and a hence for all z ∈ Z.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Practical Implications

• suppose parameterized curve family φ(·, z) | z ∈ Z usedfor daily forward curve estimation in terms of statevariable z .

• above proposition: any consistent Q-diffusion model Z forz is fully determined by φ

• moreover: diffusion matrix a(z) of Z not affected byequivalent measure transformation

⇒ statistical calibration only possible for drift of the model(or equivalently, for the market price of risk), sinceobservations of z are made under objective measureP ∼ Q and dQ/dP is left unspecified by our consistencyconsiderations


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Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Affine Term-Structure (ATS)

• simplest case: time-homogeneous affine term-structure(ATS)

φ(x , z) = g0(x) + g1(x)z1 + · · · gm(x)zm

• second-order z-derivatives vanish, (15) simplifies:

g0(x)− g0(0) +m∑

i=1

zi (gi (x)− gi (0))

=m∑

i=1

bi (z)Gi (x)− 1

2

m∑i ,j=1

aij(z)Gi (x)Gj(x), (16)

where we define

Gi (x) =

∫ x

0gi (u) du


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Svensson Family


• assume: m + m(m + 1)/2 functionsG1, . . . ,Gm, G1G1, G1G2, . . . ,GmGm linearly independent

⇒ can invert and solve the linear equation (16) for a and b(as in proof of above proposition)

• left-hand side of (16) affine in z ⇒ a, b affine:

aij(z) = aij +m∑

k=1

αk;ijzk ,

bi (z) = bi +m∑

j=1

βijzj ,

for parameters aij , αk;ij , bi and βij


DamirFilipovic

Multi-factorModels




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Svensson Family


plugging back into (16) and matching terms ⇒ Riccatiequations

∂xG0(x) = g0(0) +m∑

i=1

biGi (x)− 1

2

m∑i ,j=1

aijGi (x)Gj(x) (17)

∂xGk(x) = gk(0) +m∑

i=1

βkiGi (x)− 1

2

m∑i ,j=1

αk;ijGi (x)Gj(x)

(18)


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Summary

Proposition

If a, b is consistent with above ATS, then a and b are affine,and Gi solve system of Riccati equations (17)–(18) with initialconditions Gi (0) = 0.Conversely, suppose a and b are affine, and let gi (0) be somegiven constants. If the functions Gi solve the system of Riccatiequations (17)–(18) with initial conditions Gi (0) = 0, then theabove ATS is consistent with a, b.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Discussion

• this proposition extends earlier result ontime-homogeneous affine short-rate models with

A(t,T ) = G0(T − t) and B(t,T ) = G1(T − t)

• note 1: we did not have to assume linear independence ofB and B2: this assumption becomes necessary as soon asm ≥ 2 (→ exercise)

• note 2: we have freedom to choose constants gi (0) relatedto short rates by

r(t) = f (t, t) = g0(0) + g1(0)Z1(t) + · · ·+ gm(0)Zm(t)

• typical choice: g1(0) = 1 and all other gi (0) = 0 ⇒ Z1(t)is (non-Markovian) short-rate process


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Polynomial Term-Structure (PTS)

• polynomial term-structure (PTS) φ(x , z) =∑n|i|=0 gi(x) z i

• multi-index notation: i = (i1, . . . , im), |i| = i1 + · · ·+ imand z i = z i1

1 · · · z imm

• n = degree of the PTS: maximal k with gi 6= 0 for some|i| = k


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Maximal Degree Problem

• n = 1: ATS

• n = 2: quadratic term-structure (QTS), intensively studiedin the literature (e.g. Ahn et al. [1])

• question: do we gain something by looking at n = 3 andhigher-degree PTS models?

• surprising answer: no

• we show that there is no consistent PTS for n > 2 . . .

• for simplicity m = 1 only (for general case see course bookSection 9.4.2)


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Maximal Degree Problem I

• special case m = 1: PTS now reads

φ(x , z) =n∑

i=0

gi (x) z i

• define Gi (x) =∫ x

0 gi (u) du

Theorem (Maximal Degree Problem I)

Suppose that Gi and GiGj are linearly independent functions,for 1 ≤ i ≤ j ≤ n, and that ρ 6≡ 0.Then consistency implies n ∈ 1, 2. Moreover, b(z) and a(z)are polynomials in z with deg b(z) ≤ 1 in any case (QTS andATS), and deg a(z) = 0 if n = 2 (QTS) and deg a(z) ≤ 1 ifn = 1 (ATS).


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof

Equation (15) can be rewritten

n∑i=0

(gi (x)− gi (0)) z i =n∑

i=0

Gi (x)Bi (z)−n∑

i ,j=0

Gi (x)Gj(x)Aij(z)

where we define

Bi (z) = b(z)iz i−1 +1

2a(z)i(i − 1)z i−2,

Aij(z) =1

2a(z)ijz i−1z j−1.

By assumption we can solve above linear equation for B and A,and thus Bi (z) and Aij(z) are polynomials in z of order lessthan or equal n.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’d

In particular, this holds for

B1(z) = b(z) and 2A11(z) = a(z).

But then, since a 6≡ 0 by assumption, 2Ann(z) = a(z)n2z2n−2

cannot be a polynomial of order less than or equal n unless2n − 2 ≤ n, which implies n ≤ 2. The theorem is thus provedfor n = 1. For n = 2, we obtain deg a(z) = 0 and thusB2(z) = 2b(z)z + a(z). Hence also in this case deg b(z) ≤ 1,and the theorem is proved for m = 1.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Maximal Degree Problem IIRelax linear independence hypothesis on Gi , GiGj :

Theorem (Maximal Degree Problem II)

Suppose that:

1 supZ =∞;

2 b and ρ satisfy a linear growth condition

|b(z)|+ |ρ(z)| ≤ C (1 + |z |), z ∈ Z,

for some finite constant C;

3 a(z) is asymptotically bounded away from zero:

lim infz→∞

a(z) > 0.

Then consistency implies n ∈ 1, 2.Note: linear growth condition 2 is standard assumption fornon-explosion of Z


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof

Again, we consider equation (15), which reads

n∑i=0

(gi (x)− gi (0)) z i = b(z)n∑

i=0

Gi (x)iz i−1

+1

2a(z)

n∑i ,j=0

Gi (x)i(i − 1)z i−2 −

(n∑

i=0

Gi (x)iz i−1

)2 .

(19)

We argue by contradiction and assume that n > 2, whichimplies 2n − 2 > n. Dividing (19) by z2n−2, for z 6= 0, yields. . .


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’d

1

2a(z)

(∑ni=0 Gi (x)iz i−1

)2

z2n−2=

b(z)

z

∑ni=0 Gi (x)iz i−1

z2n−3

+a(z)

2z2

∑ni ,j=0 Gi (x)i(i − 1)z i−2

z2n−4−∑n

i=0 (gi (x)− gi (0)) z i

z2n−2.

By assumption 1 this holds for all z large enough. Theright-hand side converges to zero, for z →∞, by assumption 2.Taking the lim inf of the left-hand side yields by 3, that

1

2lim infz→∞

a(z)G 2n (x)n2 > 0,

a contradiction. Thus n ≤ 2.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Maximal Degree Problem I

Theorem (Maximal Degree Problem I)

Suppose that Giµ and GiµGiν are linearly independent functions,1 ≤ µ ≤ ν ≤ N, and that ρ 6≡ 0.Then consistency implies n ∈ 1, 2. Moreover, b(z) and a(z)are polynomials in z with deg b(z) ≤ 1 in any case (QTS andATS), and deg a(z) = 0 if n = 2 (QTS) and deg a(z) ≤ 1 ifn = 1 (ATS).


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Maximal Degree Problem II

Theorem (Maximal Degree Problem II)

Suppose that:

1 Z is a cone;

2 b and ρ satisfy a linear growth condition

‖b(z)‖+ ‖ρ(z)‖ ≤ C (1 + ‖z‖), z ∈ Z,

for some finite constant C;3 a(z) becomes uniformly elliptic for ‖z‖ large enough:

〈a(z)v , v〉 ≥ k(z)‖v‖2, v ∈ Rm,

for some function k : Z → R+ with

lim infz∈Z,‖z‖→∞

k(z) > 0.

Then consistency implies n ∈ 1, 2.Note: linear growth condition 2 is standard assumption fornon-explosion of Z


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Nelson–Siegel FamilyRecall Nelson–Siegel: φNS(x , z) = z1 + (z2 + z3x)e−z4x

Proposition

The unique solution to (15) for φNS is

a(z) = 0, b1(z) = b4(z) = 0, b2(z) = z3−z2z4, b3(z) = −z3z4.

The corresponding state process is

Z1(t) ≡ z1,

Z2(t) = (z2 + z3t) e−z4t ,

Z3(t) = z3e−z4t ,

Z4(t) ≡ z4,

where Z (0) = (z1, . . . , z4) denotes the initial point. Hencethere is no non-trivial consistent diffusion process Z for theNelson–Siegel family.

Proof.→ Exercise


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Outline







DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Svensson Family

Recall Svensson: φS(x , z) = z1 + (z2 + z3x)e−z5x + z4xe−z6x

Proposition

The only non-trivial consistent HJM model for the Svenssonfamily is the Hull–White extended Vasicek short-rate model

dr(t) =(z1z5 + z3e−z5t + z4e−2z5t − z5r(t)

)dt+√

z4z5e−z5tdW∗(t),

where (z1, . . . , z5) are given by the initial forward curve

f (0, x) = z1 + (z2 + z3x)e−z5x + z4xe−2z5x

and W∗ is some Q-standard Brownian motion. The form of thecorresponding state process Z is given in the proof below.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Discussion

• Svensson family admits no consistently varying exponents:Z6(t) ≡ z6 = 2z5 ≡ 2Z5(t)

⇒ exponents, z5 and z6 = 2z5, be rather considered as modelparameters than factors

• hypothesis empirically tested on US bond price data inSharef [48]:

• constant exponents hypothesis could not be rejected, whilethe relation z6 = 2z5 does

• statistical properties of z very sensitive on the numericalterm-structure estimation procedure

• Becker and Bouwman [4]: inter-temporal smoothingdevice can substantially improve parameter stability andsmoothness

• open problem: explore consistent inter-temporalsmoothing devices


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

ProofThe consistency equation (15) becomes, after differentiatingboth sides in x ,

q1(x) + q2(x)e−z5x + q3(x)e−z6x

+ q4(x)e−2z5x + q5(x)e−(z5+z6)x + q6(x)e−2z6x = 0, (20)

for some polynomials q1, . . . , q6. Indeed, we assume for themoment that

z5 6= z6, z5 + z6 6= 0 and zi 6= 0 for all i = 1, . . . , 6. (21)


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Svensson Family

Proof cont’dThen the terms involved in (20) are

∂xφS(x , z) = (−z2z5 + z3 − z3z5x)e−z5x + (z4 − z4z6x)e−z6x ,

∇zφS(x , z) =

1e−z5x

xe−z5x

xe−z6x

(−z2x − z3x2)e−z5x

−z4x2e−z6x

,

∂zi∂zjφS(x , z) = 0 for 1 ≤ i , j ≤ 4,


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’d

∇z∂z5φS(x , z) =

0−xe−z5x

−x2e−z5x

0(z2x2 + z3x3)e−z5x

0

,

∇z∂z6φS(x , z) =

000

−x2e−z6x

0z4x3e−z6x

,


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’d

∫ x

0∇zφS(u, z) du

=

x− 1

z5e−z5x + 1

z5(− x

z5− 1

z25

)e−z5x + 1

z25(

− xz6− 1

z26

)e−z6x + 1

z26(

z3z5

x2 +(

z2z5

+ 2z3

z25

)x + z2

z25

+ 2z3

z35

)e−z5x − z2

z25− z3

z35(

z4z6

x2 + 2z4

z26

x + 2z4

z36

)e−z6x − z4

z36

.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’dStraightforward calculations lead to

q1(x) = −a11(z)x + · · · ,

q4(x) = a55(z)z2

3

z5x4 + · · · ,

q6(x) = a66(z)z2

4

z6x4 + · · · ,

deg q2, deg q3, deg q5 ≤ 3,

where · · · stands for lower-order terms in x . Because of (21)we conclude that

a11(z) = a55(z) = a66(z) = 0.

But a is a positive semi-definite symmetric matrix. Hence

a1j(z) = aj1(z) = a5j(z) = aj5(z) = a6j(z) = aj6(z) = 0 j ≤ 6.


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Proof cont’dTaking this into account, expression (20) simplifiesconsiderably. We are left with

q1(x) = b1(z),

deg q2, deg q3 ≤ 1,

q4(x) = a33(z)1

z5x2 + · · · ,

q5(x) = a34(z)

(1

z5+

1

z6

)x2 + · · · ,

q6(x) = a44(z)1

z6x2 + · · · .

Because of (21) we know that the exponents −2z5, −(z5 + z6)and −2z6 are mutually different. Hence

b1(z) = a3j(z) = aj3(z) = a4j(z) = aj4(z) = 0 j ≤ 6.

Only a22(z) is left as positive candidate among the componentsof a(z). The remaining terms are

q2(x) = (b3(z) + z3z5)x + b2(z)− z3 −a22(z)

z5+ z2z5,

q3(x) = (b4(z) + z4z6)x − z4,

q4(x) = a22(z)1

z5,

while q1 = q5 = q6 = 0.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’dIf 2z5 6= z6 then also a22(z) = 0. If 2z5 = z6 then the conditionq3 + q4 = q2 = 0 leads to

a22(z) = z4z5,

b2(z) = z3 + z4 − 25z2,

b3(z) = −z5z3,

b4(z) = −2z5z4.

We derived the above results under the assumption (21). Butthe set of z where (21) holds is dense Z. By continuity of a(z)and b(z) in z , the above results thus extend for all z ∈ Z. Inparticular, all Zi ’s but Z2 are deterministic; Z1, Z5 and Z6 areeven constant.


DamirFilipovic

Multi-factorModels




Special Case:m = 1




Svensson Family

Proof cont’dThus, since

a(z) = 0 if 2z5 6= z6,

we only have a non-trivial process Z if

Z6(t) ≡ 2Z5(t) ≡ 2Z5(0). (22)

In that case we have, writing for short zi = Zi (0),

Z1(t) ≡ z1,

Z3(t) = z3e−z5t ,

Z4(t) = z4e−2z5t

(23)

and

dZ2(t) =(z3e−z5t + z4e−2z5t − z5Z2(t)

)dt+

d∑j=1

ρ2j(t) dW ∗j (t),

(24)where ρ2j(t) (not necessarily deterministic) are such that

d∑j=1

ρ22j(t) = a22(Z (t)) = z4z5e−2z5t .


DamirFilipovic

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Special Case:m = 1




Svensson Family

Proof cont’dBy Levy’s characterization theorem we have that

W∗(t) =d∑

j=1

∫ t

0

ρ2j(s)√

z4z5e−z5sdW ∗

j (s)

is a real-valued standard Brownian motion. Hence thecorresponding short-rate process

r(t) = φS(0,Z (t)) = z1 + Z2(t)

satisfies

dr(t) =(z1z5 + z3e−z5t + z4e−2z5t − z5r(t)

)dt+√

z4z5e−z5t dW∗(t).

Hence the proposition is proved.


DamirFilipovic

Definition andCharacteriza-tion of AffineProcesses

CanonicalState Space

Discountingand Pricing inAffine Models

Examples ofFourierDecompositions

Bond OptionPricing in AffineModels

HestonStochasticVolatility Model

Affine Trans-formationsand CanonicalRepresentation

Existence andUniqueness ofAffineProcesses

On theRegularity ofCharacteristicFunctions

AuxiliaryResults forDifferentialEquations

Part IX

Affine Processes


DamirFilipovic











Overview

• We have seen above: affine diffusion induces affineterm-structure

• In this chapter:

• Study affine processes in detail• Elaborate on their nice analytical properties• Applications in option pricing by Fourier transform


DamirFilipovic











Outline









DamirFilipovic











Standing Assumptions

• fix dimension d ≥ 1

• closed state space X ⊂ Rd with non-empty interior

• b : X → Rd continuous

• ρ : X → Rd×d measurable and s.t. diffusion matrixa(x) = ρ(x)ρ(x)> continuous in x ∈ X

• W : d-dimensional Brownian motion on (Ω,F , (Ft),P)

• for every x ∈ X there exists a unique solution X = X x of

dX (t) = b(X (t)) dt + ρ(X (t)) dW (t)

X (0) = x .


DamirFilipovic











Definition

DefinitionWe call X affine if the Ft-conditional characteristic function ofX (T ) is exponential affine in X (t): there exist C- andCd -valued functions φ(t, u) and ψ(t, u), respectively, withjointly continuous t-derivatives such that X = X x satisfies

E[eu>X (T ) | Ft

]= eφ(T−t, u)+ψ(T−t, u)>X (t) (25)

for all u ∈ iRd , t ≤ T and x ∈ X .

• Re (φ(T − t, u) + ψ(T − t, u)>X (t)) ≤ 0

• φ, ψ uniquely determined by (25) with φ(0, u) = 0,ψ(0, u) = u


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Necessary Conditions

TheoremSuppose X is affine. Then the diffusion matrix a(x) and driftb(x) are affine in x. That is,

a(x) = a +d∑

i=1

xiαi

b(x) = b +d∑

i=1

xiβi = b + Bx

for some d × d-matrices a and αi , and d-vectors b and βi ,where we denote by

B = (β1, . . . , βd)

the d × d-matrix with ith column vector βi , 1 ≤ i ≤ d.


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Necessary Conditions cont’d

Theorem (cont’d)

Moreover, φ and ψ = (ψ1, . . . , ψd)> solve the system of Riccatiequations

∂tφ(t, u) =1

2ψ(t, u)>aψ(t, u) + b>ψ(t, u)

φ(0, u) = 0

∂tψi (t, u) =1

2ψ(t, u)>αi ψ(t, u) + β>i ψ(t, u), 1 ≤ i ≤ d ,

ψ(0, u) = u.(26)

In particular, φ is determined by ψ via simple integration:

φ(t, u) =

∫ t

0

(1

2ψ(s, u)>aψ(s, u) + b>ψ(s, u)

)ds.


DamirFilipovic











Sufficient Conditions

TheoremConversely, suppose the diffusion matrix a(x) and drift b(x) areaffine and suppose there exists a solution (φ, ψ) of the Riccatiequations (26) such that Re (φ(t, u) + ψ(t, u)>x) ≤ 0 for allt ≥ 0, u ∈ iRd and x ∈ X . Then X is affine with conditionalcharacteristic function (25).


DamirFilipovic











Proof

Suppose X is affine. For T > 0 and u ∈ iRd define thecomplex-valued Ito process

M(t) = eφ(T−t,u)+ψ(T−t,u)>X (t).

We can apply Ito’s formula, separately to the real andimaginary parts of M, and obtain

dM(t) = I (t) dt + ψ(T − t, u)>ρ(X (t)) dW (t), t ≤ T ,

with

I (t) = −∂Tφ(T − t, u)− ∂Tψ(T − t, u)>X (t)

+ψ(T − t, u)>b(X (t)) +1

2ψ(T − t, u)>a(X (t))ψ(T − t, u).

Since M is a martingale, we have I (t) = 0 for all t ≤ T a.s.


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Proof cont’d

Letting t → 0, by continuity of the parameters, we thus obtain

∂Tφ(T , u) + ∂Tψ(T , u)>x

= ψ(T , u)>b(x) +1

2ψ(T , u)>a(x)ψ(T , u)

for all x ∈ X , T ≥ 0, u ∈ iRd . Since ψ(0, u) = u, this impliesthat a and b are affine of the stated form. Plugging this backinto the above equation and separating first-order terms in xyields the Riccati equations (26).


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Proof cont’d

Conversely, suppose a and b are affine. Let (φ, ψ) be a solutionof the Riccati equations (26) such that φ(t, u) + ψ(t, u)>x hasa nonpositive real part for all t ≥ 0 , u ∈ iRd and x ∈ X . ThenM, defined as above, is a uniformly bounded (substantial!)local martingale, and hence a martingale, withM(T ) = eu>X (T ). Therefore E[M(T ) | Ft ] = M(t), for allt ≤ T , which is (25), and the theorem is proved.


DamirFilipovic











ODE Facts

K = placeholder for either R or C

LemmaConsider the system of ordinary differential equations

∂t f (t, u) = R(f (t, u))

f (0, u) = u,(27)

where R : Kd → Kd is a locally Lipschitz continuous function.Then the following holds:

1 For every u ∈ Kd , there exists a life time t+(u) ∈ (0,∞]such that there exists a unique solutionf (·, u) : [0, t+(u))→ K × Kd of (27).


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ODE Facts cont’d

Lemma (cont’d)

2 The domain DK = (t, u) ∈ R+ × Kd | t < t+(u) isopen in R+ × Kd and maximal in the sense that eithert+(u) =∞ or limt↑t+(u) ‖f (t, u)‖ =∞ for all u ∈ Kd .

3 The t-section DK (t) = u ∈ Kd | (t, u) ∈ DK is open inKd , and non-expanding in t:

Kd = DK (0) ⊇ DK (t1) ⊇ DK (t2) 0 ≤ t1 ≤ t2.

In fact, we have f (s,DK (t2)) ⊆ DK (t1) for all s ≤ t2 − t1.

4 If R is analytic on Kd then f is an analytic function onDK .


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Outline









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Necessary Parameter Conditions

There is an implicit trade-off between the parametersa, αi , b, βi and the state space X :

• a, αi , b, βi must be such that X does not leave the set X ;

• a, αi must be such that a +∑d

i=1 xiαi is symmetric andpositive semi-definite for all x ∈ X .


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Canonical State Space

• standing assumption: canonical state space

X = Rm+ × Rn

for some integers m, n ≥ 0 with m + n = d

• covers essentially all applications appearing in the financeliterature (except semi-definite matrix-valued)

• notation: I = 1, . . . ,m and J = m + 1, . . . ,m + n• notation: µM = (µi )i∈M and νMN = (νij)i∈M, j∈N


DamirFilipovic











Characterization

TheoremThe process X on the canonical state space Rm

+ × Rn is affineif and only if a(x) and b(x) are affine for admissible parametersa, αi , b, βi in the following sense:

a, αi are symmetric positive semi-definite,

aII = 0 (and thus aIJ = a>JI = 0),

αj = 0 for all j ∈ J

αi ,kl = αi ,lk = 0 for k ∈ I \ i, for all 1 ≤ i , l ≤ d,

b ∈ Rm+ × Rn,

BIJ = 0,

BII has nonnegative off-diagonal elements.

(28)


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Characterization cont’d

Theorem (cont’d)

In this case, the corresponding system of Riccati equations (26)simplifies to

∂tφ(t, u) =1

2ψJ(t, u)>aJJ ψJ(t, u) + b>ψ(t, u)

φ(0, u) = 0

∂tψi (t, u) =1

2ψ(t, u)>αi ψ(t, u) + β>i ψ(t, u), i ∈ I ,

∂tψJ(t, u) = B>JJψJ(t, u),

ψ(0, u) = u,

(29)

and there exists a unique global solution(φ(·, u), ψ(·, u)) : R+ → C− × Cm

− × iRn for all initial valuesu ∈ Cm

− × iRn. In particular, the equation for ψJ forms anautonomous linear system with unique global solutionψJ(t, u) = eB

>JJ t uJ for all uJ ∈ Cn.


DamirFilipovic











Illustration d = 3, m ≤ 1

• m = 0: α(x) ≡ a for some positive semi-definitesymmetric 3× 3-matrix a

• m = 1:

a =

0 0 0+ ∗

+

, α1 =

+ ∗ ∗+ ∗

+


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Illustration d = 3, m = 2

a =

0 0 00 0

+

α1 =

+ 0 ∗0 0

+

, α2 =

0 0 0+ ∗

+


DamirFilipovic











Illustration d = 3, m = 3

a = 0 , α1 =

+ 0 00 0

0

α2 =

0 0 0+ 0

0

, α3 =

0 0 00 0

+


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Sketch of Proof: Necessity

• a(x) symmetric positive semi-definite for all x ∈ Rm+ × Rn

⇔ αj = 0 for all j ∈ J, and a and αi symmetric positivesemi-definite for all i ∈ I .

• stochastic invariance: Rm+ × Rn =

⋂i∈Ix | Rd | e>i x ≥ 0

LemmaX x(t) ∈ Rm

+ × Rn for all x ∈ Rm+ × Rn only if for all boundary

points x with xk = 0 for some k ∈ I :

1 diffusion parallel to boundary: e>k a(x) ek = 0

2 drift inward pointing: e>k b(x) ≥ 0

⇒ implies admissibility conditions


DamirFilipovic











Sketch of Proof: Sufficiency

• solution ψ(t, u) of Riccati equations

• is Cm− × iRn-valued

• has life time t+(u) =∞

for all u ∈ Cm− × iRn

• φ(t, u) is C−-valued

⇒ Re (φ(t, u) + ψ(t, u)>x) ≤ 0 for all t ≥ 0, u ∈ iRd andx ∈ Rm

+ × Rn


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Extension of Affine TransformFormula

notation: S(U) =

z ∈ Ck | Re z ∈ U

TheoremSuppose X is affine. Let DK (K = R or C) denote themaximal domain for the system of Riccati equations, and letτ > 0. Then:

1 S(DR(τ)) ⊂ DC(τ).

2 DR(τ) = M(τ) where

M(τ) =

u ∈ Rd | E[eu>X x (τ)

]<∞ for all x ∈ Rm

+ × Rn.

3 DR(τ) and DR are convex sets.


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Extension of Affine TransformFormula cont’d

Theorem (cont’d)

Moreover, for all 0 ≤ t ≤ T and x ∈ Rm+ × Rn:

4 (25) holds for all u ∈ S(DR(T − t)).

5 (25) holds for all u ∈ Cm− × iRn.

6 M(t) ⊇ M(T ).

Part 6 states: E[eu>X x (T )


+ × Rn, for

some given T and u ∈ Rd , implies E[eu>X x (t)

]for all

x ∈ Rm+ × Rn and t ≤ T .

Proof.Technical. See course book Section 10.7.3.


DamirFilipovic











Key Message

Corollary

Suppose that either side of (25) is well defined for some t ≤ Tand u ∈ Rd . Then (25) holds, implying that both sides are welldefined in particular, for u replaced by u + iv for any v ∈ Rd .


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Outline









DamirFilipovic











Assumptions

• X affine on canonical state space Rm+ × Rn

• pricing: interpret P = Q as risk-neutral measure

• W = W ∗ as Q-Brownian motion

• note: affine property of X not preserved under equivalentchange of measure in general, see e.g. Cheridito, Filipovicand Yor [17].

• affine short-rate model: r(t) = c + γ>X (t), for c ∈ R,γ ∈ Rd

• special cases for d = 1: Vasicek and CIR short-rate models

• recall: affine term-structure model induces an affineshort-rate model


DamirFilipovic











General Problem

• consider T -claim with payoff f (X (T )) s.t.

E[e−

∫ T0 r(s) ds |f (X (T ))|

]<∞

• arbitrage price at t ≤ T :

π(t) = E[e−

∫ Tt r(s) ds f (X (T )) | Ft

]• particular example: T -bond with f ≡ 1

• aim: derive analytic/numerically tractable pricing formula!

• first step: derive formula for Ft-conditional characteristicfunction of X (T ) under the T -forward measure = (up tonormalization with P(t,T ))

E[e−

∫ Tt r(s) ds eu>X (T ) | Ft

]for u ∈ iRd


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T -forward Characteristic Function

TheoremLet τ > 0. The following statements are equivalent:

1 E[e−

∫ τ0 r(s) ds


+ × Rn (e.g.

γ ∈ Rm+ × 0)

2 There exists a unique solution(Φ(·, u),Ψ(·, u)) : [0, τ ]→ C× Cd of

∂tΦ(t, u) =1

2ΨJ(t, u)>aJJ ΨJ(t, u) + b>Ψ(t, u)− c ,

Φ(0, u) = 0,

∂tΨi (t, u) =1

2Ψ(t, u)>αi Ψ(t, u) + β>i Ψ(t, u)− γi , i ∈ I ,

∂tψJ(t, u) = B>JJΨJ(t, u)− γJ ,

Ψ(0, u) = u(30)

for u = 0.


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T -forward Characteristic Functioncont’d

Theorem (cont’d)

Moreover, let DK (K = R or C) denote the maximal domainfor the system of Riccati equations. If either of the aboveconditions holds then DR(S) is a convex open neighborhood of0 in Rd , and S(DR(S)) ⊂ DC(S), for all S ≤ τ . Further:

E[e−

∫ Tt r(s) ds eu>X (T ) | Ft

]= eΦ(T−t,u)+Ψ(T−t,u)>X (t) (31)

for all u ∈ S(DR(S)), t ≤ T ≤ t + S and x ∈ Rm+ × Rn.


DamirFilipovic











Proof

Enlarge the state space: define

Y (t) = y +

∫ t

0

(c + γ>X (s)

)ds, y ∈ R.

Then X ′ = (X ,Y ) is an Rm+ × Rn+1-valued diffusion process

with diffusion matrix a′ +∑

i∈I xiα′i and drift b′ + B′x ′ where

a′ =

(a 00 0

), α′i =

(αi 00 0

),

b′ =

(bc

), B′ =

(B 0γ> 0

)form admissible parameters. We claim that X ′ is an affineprocess.


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Proof cont’dIndeed, the candidate Riccati equations read, for i ∈ I :

∂tφ′(t, u, v) =

1

2ψ′J(t, u, v)>aJJ ψ

′J(t, u, v)

+ b>ψ′1,...,d(t, u, v) + cv ,

φ′(0, u, v) = 0,

∂tψ′i (t, u, v) =

1

2ψ′(t, u, v)>αi ψ

′(t, u, v)

+ β>i ψ′1,...,d(t, u, v) + γiv ,

∂tψ′J(t, u, v) = B>JJψ′J(t, u, v) + γJv ,

∂tψ′d+1(t, u, v) = 0,

ψ′(0, u, v) =

(uv

).

(32)

Here we replaced the constant solution ψ′d+1(·, u, v) ≡ v by vin the boxes.


DamirFilipovic











Proof cont’d

Theorem 38.1 carries over: there exists a unique globalC− × Cm

− × iRn+1-valued solution (φ′(·, u, v), ψ′(·, u, v)) of(32) for all (u, v) ∈ Cm

− × iRn × iR. The second part ofTheorem 37.2 thus asserts that X ′ is affine with conditionalcharacteristic function

E[eu>X (T )+vY (T ) | Ft

]= eφ

′(T−t,u,v)+ψ′1,...,d(T−t,u,v)>X (t)+vY (t)

for all (u, v) ∈ Cm− × iRn × iR and t ≤ T .


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Proof cont’d

The theorem now follows from Theorem 38.3 once we setΦ(t, u) = φ′(t, u,−1) and Ψ(t, u) = ψ′1,...,d(t, u,−1).Indeed, it is clear by inspection thatDK (S) = u ∈ Kd | (u,−1) ∈ D′K (S) where D′K denotes themaximal domain for the system of Riccati equations (32).


DamirFilipovic











Bond Price Formula

Corollary

For any maturity T ≤ τ , the T -bond price at t ≤ T is given as

P(t,T ) = e−A(T−t)−B(T−t)>X (t)

where we define A(t) = −Φ(t, 0), B(t) = −Ψ(t, 0).Moreover, for t ≤ T ≤ S ≤ τ , the Ft-conditional characteristicfunction of X (T ) under the S-forward measure QS is given by

EQS

[eu>X (T ) | Ft

]=

e−A(S−T )+Φ(T−t,u−B(S−T ))+Ψ(T−t,u−B(S−T ))>X (t)

P(t, S)(33)

for all u ∈ S(DR(T ) + B(S − T )), which contains iRd .


DamirFilipovic











Proof

The bond price formula follows from the theorem with u = 0.Now let t ≤ T ≤ S ≤ τ . In view of the flow propertyΨ(T ,−B(S − T )) = −B(S), we know that−B(S − T ) ∈ DR(T ), and thus S(DR(T ) + B(S − T ))contains iRd . Moreover, for u ∈ S(DR(T ) + B(S − T )), weobtain from (31) by nested conditional expectation

E[e−

∫ St r(s) dseu>X (T ) | Ft

]= E

[e−

∫ Tt r(s) ds E

[e−

∫ ST r(s) ds | FT

]eu>X (T ) | Ft

]= e−A(S−T )E

[e−

∫ Tt r(s) dse (u−B(S−T ))>X (T ) | Ft

]= e−A(S−T )+Φ(T−t,u−B(S−T ))+Ψ(T−t,u−B(S−T ))>X (t).

Normalizing by P(t, S) yields the S-forward characteristicfunction.


DamirFilipovic











Affine Pricing

• recall: π(t) = E[e−

∫ Tt r(s) ds f (X (T )) | Ft

]• either: we recognize Ft-conditional distribution

Q(t,T , dx) of X (T ) under T -forward measure QT fromits characteristic function above, then compute by(numerical) integration

π(t) = P(t,T )

∫Rd

f (x) Q(t,T , dx).

• or: employ Fourier transform . . .


DamirFilipovic











Affine Pricing Formula I

TheoremSuppose either condition 1 or 2 of Theorem 39.1 is met forsome τ ≥ T , and let DR denote the maximal domain for thesystem of Riccati equations (30). Assume that f satisfies

f (x) =

∫Rq

e (v+iLλ)>x f (λ) dλ, dx-a.s.

for some v ∈ DR(T ) and d × q-matrix L, and some integrablefunction f : Rq → C, for a positive integer q ≤ d. Then theprice π(t) is well defined and given by the formula

π(t) =

∫Rq

eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) f (λ) dλ. (34)


DamirFilipovic











ProofBy assumption, we have

E[e−

∫ T0 r(s) ds |f (X (T ))|

]≤ E

[∫Rq

e−∫ T

0 r(s) dse v>X (T ) |f (λ)| dλ]<∞.

Hence we may apply Fubini’s theorem to change the order ofintegration, which gives

π(t) = E[e−

∫ Tt r(s) ds

∫Rq

e (v+iLλ)>X (T )f (λ) dλ | Ft

]=

∫Rq

E[e−

∫ Tt r(s) ds e (v+iLλ)>X (T ) | Ft

]f (λ) dλ

=

∫Rq

eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) f (λ) dλ,

as desired.


DamirFilipovic











Affine Pricing Formula II

TheoremSuppose either condition 1 or 2 of Theorem 39.1 is met forsome τ ≥ T , and let DR denote the maximal domain for thesystem of Riccati equations (30). Assume that f is of the form

f (x) = e v>x h(L>x)

for some v ∈ DR(T ) and d × q-matrix L, and some integrablefunction h : Rq → R, for a positive integer q ≤ d. Define thebounded function

f (λ) =1

(2π)q

∫Rq

e−iλ>y h(y) dy , λ ∈ Rq.

1 If f (λ) is an integrable function in λ ∈ Rq then theassumptions of Theorem 39.3 are met.


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Affine Pricing Formula II cont’d

Theorem (cont’d)

2 If v = Lw, for some w ∈ Rq, andeΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) is an integrable functionin λ ∈ Rq then the Ft-conditional distribution of theRq-valued random variable Y = L>X (T ) under theT -forward measure QT admits the continuous densityfunction

q(t,T , y) =1

(2π)q

∫Rq

e−(w+iλ)>y

× eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)

P(t,T )dλ.

In either case, the integral in (34) is well defined and the priceformula (34) holds.


DamirFilipovic











Fourier Transform

For the proof, we recall the fundamental inversion formula fromFourier analysis ([51, Chapter I, Corollary 1.21]):

LemmaLet g : Rq → C be an integrable function with integrableFourier transform

g(λ) =

∫Rq

e−iλ>y g(y) dy .

Then the inversion formula

g(y) =1

(2π)q

∫Rq

e iλ>y g(λ) dλ

holds for dy-almost all y ∈ Rq.


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Proof of 1

Under the assumption of 1, the Fourier inversion formulaapplied to h(y) yields the representation (39.3). HenceTheorem 39.3 applies.


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Proof of 2

We denote by q(t,T , dy) the Ft-conditional distribution ofY = L>X (T ) under the T -forward measure QT . From (33) weinfer the characteristic function of the bounded (why?)

measure ew>y q(t,T , dy):∫Rq

e (w+iλ)>y q(t,T , dy) = E[e (w+iλ)>L>X (T ) | Ft

]=

eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)

P(t,T ), λ ∈ Rq.

By assumption, this is an integrable function in λ on Rq. TheFourier inversion formula thus applies and the injectivity of thecharacteristic function (see e.g. [53, Section 16.6]) yields thatq(t,T , dy) admits the continuous density function as stated.


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Proof of 2 cont’dMoreover, we then obtain

P(t,T )

∫Rq

|ew>y h(y)|q(t,T , y) dy

≤ 1

(2π)q

∫Rq

∫Rq

|h(y)|∣∣∣eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)

∣∣∣ dλ dy <∞.

Hence again we can apply Fubini’s theorem to change the orderof integration, which gives

π(t) = P(t,T )

∫Rq

ew>y h(y) q(t,T , y) dy

=1

(2π)q

∫Rq

∫Rq

ew>y h(y) e−(w+iλ)>y eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) dλ dy

=1

(2π)q

∫Rq

(∫Rq

h(y) e−iλ>y dy

)eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) dλ,

which is (34).


DamirFilipovic











Discussion

• integral in pricing formula

π(t) =

∫Rq

eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) f (λ) dλ

has to be computed numerically in general

• important: integral is over Rq, where possibly q d

• example bond options: f is in closed form and q = 1 (willsee this below . . . )


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Discussion cont’d

reflection on affine pricing formula:

• payoff f (X (T )) =∫

Rq e (v+iLλ)>X (T ) f (λ) dλ isdecomposed into linear combination of (a continuum) of

complex-valued basis “payoffs” e (v+iLλ)>X (T ) with weightsf (λ)

• by nature of affine process X : these basis claims admitclosed-form complex-valued “prices”

πv+iLλ(t) = eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)

• linearity of pricing: price of f (X (T )) is given as linearcombination of πv+iLλ(t) with same weights f (λ)


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Discussion cont’d

→ power of affine diffusion processes!

→ suggests to explore other types of diffusion processes thatadmit closed-form prices for some well specified basis ofpayoff functions (e.g. [15, 10]): open area of research

• examples of non-trivial Fourier decompositions? Yes! . . .


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Outline









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Call Option

LemmaLet K > 0. For any y ∈ R the following identities hold:

1

2π

∫R

e (w+iλ)y K−(w−1+iλ)

(w + iλ)(w − 1 + iλ)dλ

=

(K − e y )+ if w < 0,

(e y − K )+ − e y if 0 < w < 1,

(e y − K )+ if w > 1.

The middle case (0 < w < 1) obviously also equals(K − e y )+ − K .


DamirFilipovic











Proof

Proof.Let w < 0. Then the function h(y) = e−wy (K − e y )+ isintegrable on R. An easy calculation shows that its Fouriertransform

h(λ) =

∫R

e−(w+iλ)y (K − e y )+ dy =K−(w−1+iλ)

(w + iλ)(w − 1 + iλ)

is also integrable on R. Hence the Fourier inversion formulaapplies, and we conclude that the claimed identity holds forw < 0. The other cases follow by similar arguments (→exercise).


DamirFilipovic











Exchange Option

choose K = e z in above lemma:

Corollary

For any y , z ∈ R the following identities hold:

1

2π

∫R

e (w+iλ)y−(w−1+iλ)z

(w + iλ)(w − 1 + iλ)dλ

=

(e y − e z)+ if w > 1,

(e y − e z)+ − e y if 0 < w < 1.


DamirFilipovic











Exchange Option cont’d

• two asset prices Si = eXm+i , i = 1, 2

• exchange option (Magrabe option): option to exchange c2

units of asset S2 against c1 units of asset S1 at T

• payoff at T : f (X (T )) =(c1eXm+1(T ) − c2eXm+2(T )

)+

• meets assumptions of affine pricing formula with q = 1,v = wem+1 + (1− w)em+2, L = em+1 − em+2, and

f (λ) =cw+iλ

1 c−(w−1+iλ)2

2π(w + iλ)(w − 1 + iλ),

for some w > 1

• remains to be checked from case to case: whetherv ∈ DR(T )


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Spread Option

Hurd and Zhou [31]: double Fourier integral representation:

LemmaLet w = (w1,w2)> ∈ R2 be such that w2 < 0 and w1 + w2 > 1.Then for any y = (y1, y2)> ∈ R2 the following identity holds:

(e y1 − e y2 − 1)+ =1

(2π)2

∫R2

e (w+iλ)>y

× Γ(w1 + w2 − 1 + i(λ1 + λ2)) Γ(−w2 − iλ2)

Γ(w1 + 1 + iλ1)dλ1 dλ2,

where the Gamma function Γ(z) =∫∞

0 t−1+z e−t dt is definedfor all complex z with Re (z) > 0.

Proof.Technical. See course book Section 10.3.


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Spread Option

• two asset prices Si = eXm+i , i = 1, 2

• spread option, strike price K > 0, payoff at T :

f (X (T )) =(eXm+1(T ) − eXm+2(T ) − K

)+

• meets assumptions of affine pricing formula with q = 2,v = w1em+1 + w2em+2, L = (em+1, em+2), and

f (λ) =Γ(w1 + w2 − 1 + i(λ1 + λ2)) Γ(−w2 − iλ2)

(2π)2Kw1+w2+i(λ1+λ2)Γ(w1 + 1 + iλ1),

for some w2 < 0 and w1 > 1− w2

• remains to be checked from case to case: whetherv ∈ DR(T )


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Outline









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Assumptions

• assume: either condition 1 or 2 of Theorem 39.1 is met

• European call option on S-bond, expiry date T < S ≤ τ ,strike price K

• call option lemma ⇒ payoff function:(e−A(S−T )−B(S−t)>x − K

)+

=

∫R

e−(w+iλ)B(S−t)>x f (w , λ) dλ

where we define, for any real w > 1,

f (w , λ) =1

2πe−(w+iλ)A(S−T ) K−(w−1+iλ)

(w + iλ)(w − 1 + iλ),

• similar formula for put options (. . . )


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Affine Bond Option PricingFormula

Corollary

There exists some w− < 0 and w+ > 1 such that−B(S − T )w ∈ DR(T ) for all w ∈ (w−,w+), and the lineintegral

Π(w , t) =

∫R

eΦ(T−t,−(w+iλ)B(S−T ))

× eΨ(T−t,−(w+iλ)B(S−T ))>X (t) f (w , λ) dλ

is well defined for all w ∈ (w−,w+) \ 0, 1 and t ≤ T .Moreover, the time t prices of the European call and putoption on the S-bond with expiry date T and strike price K aregiven by any of the following identities: . . .


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Affine Bond Option PricingFormula cont’d

Corollary (cont’d)

πcall(t) =

Π(w , t), if w ∈ (1,w+),

Π(w , t) + P(t, S), if w ∈ (0, 1),

= P(t, S)q(t, S , I)− KP(t,T )q(t,T , I)

πput(t) =

Π(w , t) + KP(t,T ), if w ∈ (0, 1),

Π(w , t), if w ∈ (w−, 0),

= KP(t,T )q(t,T ,R \ I)− P(t, S)q(t, S ,R \ I)

where I = (A(S − T ) + log K ,∞), and q(t,S , dy) andq(t,T , dy) denote the Ft-conditional distributions of thereal-valued random variable Y = −B(S − T )>X (T ) under theS- and T -forward measure, respectively.


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Proof

Proof.See course book Section 10.3.2


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Consequences

• The pricing of European call and put bond options in thepresent d-dimensional affine factor model boils down tothe computation of a line integral Π(w , t), which is asimple numerical task!

• Moreover, in case the distributions q(t,S , dy) andq(t,T , dy) are explicitly known, the pricing is reduced tothe computation of the respective probabilities of theexercise events I and R \ I

• In the following two subsections, we illustrate thisapproach for the Vasicek and CIR short-rate models.


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Example: Vasicek Model• recall: dr = (b + βr) dt + σ dW• state space: R, and we set r = X

⇒ Riccati equations:

Φ(t, u) =1

2σ2

∫ t

0Ψ2(s, u) ds + b

∫ t

0Ψ(s, u) ds

∂tΨ(t, u) = βΨ(t, u)− 1,

Ψ(0, u) = u

• for all u ∈ C ∃ unique global solution:

Φ(t, u) =1

2σ2

(u2

2β(e2βt − 1) +

1

2β3(e2βt − 4eβt + 2βt + 3)

− u

β2(e2βt − 2eβt + 2β)

)+ b

(eβt − 1

βu − eβt − 1− βt

β2

)Ψ(t, u) = eβtu − eβt − 1

β


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Example: Vasicek Model cont’d

⇒ affine S-forward characteristic function formula holds forall u ∈ C and t ≤ T :

EQS

[eur(T ) | Ft

]= exp

(1

2σ2 e2β(T−t) − 1

2βu2 + low order u-terms

)

⇒ under S-forward measure, r(T ) is Ft-conditionally

Gaussian distributed with variance σ2 e2β(T−t)−12β

• in line with earlier findings!

• straightforward calculation: Ft-conditional QS -mean ofr(T ) (. . . )

• re-derive bond option price formula for the Vasicek modelfrom Gaussian HJM formula above (→ exercise)


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Example: CIR Model

• recall: dr = (b + βr) dt + σ√

r dW

• state space: R+, and we set r = X

⇒ Riccati equations:

Φ(t, u) = b

∫ t

0Ψ(s, u) ds,

∂tΨ(t, u) =1

2σ2Ψ2(t, u) + βΨ(t, u)− 1,

Ψ(0, u) = u.


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Example: CIR Model cont’d

• for all u ∈ C− ∃ unique global C−-valued solution:

Φ(t, u) =2b

σ2log

(2θe

(θ−β)t2

L3(t)− L4(t)u

)

Ψ(t, u) = −L1(t)− L2(t)u

L3(t)− L4(t)u

where θ =√β2 + 2σ2 and

L1(t) = 2(e θt − 1

)L2(t) = θ

(e θt + 1

)+ β

(e θt − 1

)L3(t) = θ

(e θt + 1

)− β

(e θt − 1

)L4(t) = σ2

(e θt − 1

).


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• elementary algebraic manipulations (. . . ) ⇒ S-forwardcharacteristic function of r(T ):

EQS

[eur(T ) | Ft

]=

eC2(t,T ,S)r(t)C1(t,T ,S)u

1−C1(t,T ,S)u

(1− C1(t,T , S)u)2bσ2

where

C1(t,T ,S) =L3(S − T )L4(T − t)

2θL3(S − t)

C2(t,T ,S) =L2(T − t)

L4(T − t)− L1(S − t)

L3(S − t)

⇒ Ft-conditional distribution of Z (t,T ) = 2r(T )C1(t,T ,S) under

QS is noncentral χ2 with 4bσ2 degrees of freedom and

parameter of noncentrality 2C2(t,T ,S)r(t)

• see next lemma . . .


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Noncentral χ2-Distribution

Lemma (Noncentral χ2-Distribution)

The noncentral χ2-distribution with δ > 0 degrees of freedomand noncentrality parameter ζ > 0 has density function

fχ2(δ,ζ)(x) =1

2e−

x+ζ2

(x

ζ

) δ4− 1

2

I δ2−1(√ζx), x ≥ 0

and characteristic function∫R+

eux fχ2(δ,ζ)(x) dx =e

ζu1−2u

(1− 2u)δ2

, u ∈ C−.

Here Iν(x) =∑

j≥01

j!Γ(j+ν+1)

(x2

)2j+νdenotes the modified

Bessel function of the first kind of order ν > −1.

Proof.See e.g. [33, Chapter 29].


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Noncentral χ2-Distribution cont’dNoncentral χ2 = generalization of distribution of the sum of thesquares of independent normal distributed random variables:

• fix δ ∈ N, reals ν1, . . . , νδ ∈ R, and define ζ =∑δ

i=1 ν2i

• let N1, . . . ,Nδ be independent standard normal distributedrvs

• define Z =∑δ

i=1(Ni + νi )2

• direct integration shows: characteristic function of Zequals

E[euZ ] =e

ζu1−2u

(1− 2u)δ2

, u ∈ C−

• above lemma ⇒ Z noncentral χ2-distributed with δdegrees of freedom and noncentrality parameter ζ

• good to know: noncentral χ2-distribution hard coded inmost statistical software packages

⇒ explicit bond option price formulas for the CIR model!


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Noncentral χ2-Distribution cont’dx = (0:0.1:10)’;ncx2 = ncx2pdf(x,4,2);chi2 = chi2pdf(x,4);plot(x,ncx2,’b-’,’LineWidth’,2)hold onplot(x,chi2,’g–’,’LineWidth’,2)legend(’ncx2’,’chi2’)


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Example: CIR Model cont’d• numerical example: σ2 = 0.033, b = 0.08, β = −0.9,

r0 = 0.08• set t = 0, T = 1 and S = 2• numerical integration shows: line integral Π(w , 0) in above

bond option pricing corollary behaves numerically stablefor w ranging between (−1, 2) \ 0, 1 (see figure).

• on the other hand, we know that Π(w , 0) diverges forw → −∞ (why?)

Figure: Line integral Π(w , 0) as a function of w .


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Figure: Real part of the integrand of Π(w , 0), for w = −0.5, 0.5, 1.5,as a function of λ.


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• the resulting ATM call and put option strike price isK = 0.9180

• the call and put option price πcall(0) = πput(0) = 0.0078can now be computed by any of the formulas in the bondoption pricing corollary (→ exercise)


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• application: compute ATM cap prices and implied Blackvolatilities (→ exercise)

• tenor: t = 0 (today), T0 = 1/4 (first reset date), andTi − Ti−1 ≡ 1/4, i = 1, . . . , 119 (the maturity of the lastcap is T119 = 30)

• the following table and figure show the ATM cap pricesand implied Black volatilities for a range of maturities

• like the Vasicek model, the CIR model seems incapable ofproducing humped volatility curves


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Table: CIR ATM cap prices and Black volatilities

Maturity ATM prices ATM vols

1 0.0073 0.45062 0.0190 0.37203 0.0302 0.32264 0.0406 0.28905 0.0501 0.26476 0.0588 0.24627 0.0668 0.23168 0.0742 0.2198

10 0.0871 0.201712 0.0979 0.188615 0.1110 0.174420 0.1265 0.159430 0.1430 0.1442


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Figure: CIR ATM cap Black volatilities.


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Outline









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Heston Model

• Heston [28]: generalizes Black–Scholes model by assumingstochastic volatility

• interest rates: constant r(t) ≡ r ≥ 0

• one risky asset (stock) S = eX2

• X = (X1,X2): affine process on R+ × R:

dX1 = (k + κX1) dt + σ√

2X1 dW1

dX2 = (r − X1) dt +√

2X1

(ρ dW1 +

√1− ρ2dW2

)for k, σ ≥ 0, κ ∈ R, ρ ∈ [−1, 1]


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Heston Model cont’d

• implied risk-neutral stock dynamics:

dS = Sr dt + S√

2X1 dW

for Brownian motion W = ρW1 +√

1− ρ2 W2

⇒√

2X1 = stochastic volatility of the price process S

• possibly non-zero covariation d〈S ,X1〉 = 2ρσSX1 dt


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• system of Riccati equations for X equivalent to

φ(t, u) = k

∫ t

0ψ1(s, u) ds + ru2t

∂tψ1(t, u) = σ2ψ21(t, u) + (2ρσu2 + κ)ψ1(t, u) + u2

2 − u2

ψ1(0, u) = u1

ψ2(t, u) = u2,

• Riccati lemma below: ∃ explicit global solution φ, ψ ifu1 ∈ C− and 0 ≤ Re u2 ≤ 1

• in particular u = (0, 1): φ(t, 0, 1) = rt, ψ(t, 0, 1) = (0, 1)>

• Theorem 38.3 ⇒ E[S(T )] <∞ and (EMM):

E[e−rT S(T ) | Ft ] = e−rT E[eX2(T ) | Ft ]

= e−rT e r(T−t)+X2(t) = e−rtS(t)


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Riccati LemmaLemmaConsider the Riccati differential equation

∂tG = AG 2 + BG − C , G (0, u) = u, (35)

where A,B,C ∈ C and u ∈ C, with A 6= 0 andB2 + 4AC ∈ C \ R−. Let

√· denote the analytic extension of

the real square root to C \ R−, and define θ =√

B2 + 4AC .

1 The function

G (t, u) = −2C(eθt − 1

)−(θ(eθt + 1

)+ B

(eθt − 1

))u

θ (eθt + 1)− B (eθt − 1)− 2A (eθt − 1) u

is the unique solution of equation (35) on its maximalinterval of existence [0, t+(u)). Moreover,∫ t

0G (s, u)ds =

1

Alog

(2θe

θ−B2

t

θ(eθt + 1)− B(eθt − 1)− 2A(eθt − 1)u

).

2 If, moreover, A > 0, B ∈ R, Re (C ) ≥ 0 and u ∈ C− thent+(u) =∞ and G (t, u) is C−-valued.


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• European call option, maturity T , strike price K , price:

π(t) = e−r(T−t)E[(S(T )− K )+ | Ft

]• fix w > 1 small enough with (0,w) ∈ DR(T ) (→ exercise):

π(t) = e−r(T−t)

∫R

eφ(T−t,0,w+iλ)+ψ1(T−t,0,w+iλ)X1(t)

× e (w+iλ)X2(t) f (λ) dλ

with f (λ) = 12π

K−(w−1+iλ)

(w+iλ)(w−1+iλ)

• alternatively: fix 0 < w < 1 and then

π(t) = S(t) + e−r(T−t)

∫R

eφ(T−t,0,w+iλ)+ψ1(T−t,0,w+iλ)X1(t)

× e (w+iλ)X2(t) f (λ) dλ


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numerical example: X1(0) = 0.02, X2(0) = 0, σ = 0.1,κ = −2.0, k = 0.02, r = 0.01, ρ = 0.5

Figure: Implied volatility surface for the Heston model.


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Table: Call option prices in the Heston model

T–K 0.8 0.9 1.0 1.1 1.2

0.2 0.2016 0.1049 0.0348 0.0074 0.0012

0.4 0.2037 0.1120 0.0478 0.0168 0.0053

0.6 0.2061 0.1183 0.0571 0.0245 0.0100

0.8 0.2088 0.1239 0.0646 0.0310 0.0144

1.0 0.2115 0.1291 0.0711 0.0368 0.0186


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Table: Black–Scholes implied volatilities for the call option prices inthe Heston model

T–K 0.8 0.9 1.0 1.1 1.2

0.2 0.1715 0.1786 0.1899 0.2017 0.2126

0.4 0.1641 0.1712 0.1818 0.1930 0.2033

0.6 0.1585 0.1656 0.1755 0.1858 0.1954

0.8 0.1544 0.1612 0.1704 0.1799 0.1889

1.0 0.1513 0.1579 0.1664 0.1751 0.1835


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Outline









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Assumption

• let X be affine on the canonical state space Rm+ × Rn with

admissible parameters a, αi , b, βi

⇒ for any x ∈ Rm+ × Rn the process X = X x satisfies

dX = (b + BX ) dt + ρ(X ) dW

X (0) = x ,

with ρ(x)ρ(x)> = a +∑

i∈I xiαi


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Affine Transformations

• let Λ be an invertible d × d-matrix

⇒ the linear transform Y = ΛX satisfies

dY =(Λb + ΛBΛ−1Y

)dt + Λρ

(Λ−1Y

)dW

Y (0) = Λx

• hence Y has affine drift Λb + ΛBΛ−1y and diffusionmatrix Λα(Λ−1y)Λ>


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Affine Transformations cont’d

• affine short-rate model r(t) = c + γ>X (t) can beexpressed in terms of Y (t):

r(t) = c + γ>Λ−1Y (t)

⇒ Y specifies an affine short-rate model producing the sameshort rates (and bond prices etc.) as X

• identification problem: ∃? unique canonical representationfor affine short-rate models with the same observableimplications?

• literature: [20, 18, 34, 16] and others

• will show: diffusion matrix α(x) can always be broughtinto block-diagonal form . . .


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Affine Transformations cont’d

LemmaThere exists some invertible d × d-matrix Λ withΛ(Rm

+ × Rn) = Rm+ × Rn such that Λα(Λ−1y)Λ> is

block-diagonal of the form

Λα(Λ−1y)Λ> =

(diag(y1, . . . , yq, 0, . . . , 0) 0

0 p +∑

i∈I yiπi

)for some integer 0 ≤ q ≤ m and symmetric positivesemi-definite n × n matrices p, π1, . . . , πm. Moreover, Λb andΛBΛ−1 meet the admissibility conditions in lieu of b and B.


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Proof

Λα(x)Λ> is block-diagonal for all x = Λ−1y if and only if ΛaΛ>

and ΛαiΛ> are block-diagonal for all i ∈ I .

Define the invertible d × d-matrix

Λ =

(Im 0D In

)(36)

where the n×m-matrix D = (δ1, . . . , δm) has ith column vector

δi =

−αi,iJ

αi,ii, if αi ,ii > 0

0, else.

Then Λ(Rm+ × Rn) = Rm

+ × Rn and

Dαi ,II = −αi ,JI , i ∈ I .


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Proof cont’d

From here we easily verify that

Λαi =

(αi ,II αi ,IJ

0 Dαi ,IJ + αi ,JJ

),

and thus

ΛαiΛ> =

(αi ,II 0

0 Dαi ,IJ + αi ,JJ

).

Since ΛaΛ> = a, the first assertion is proved (modulopermutation of first m indices).The admissibility conditions for Λb and ΛBΛ−1 can easily bechecked as well.


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Canonical Representation

Combining the preceding findings:

Theorem (Canonical Representation)

Any affine short-rate model r(t) = c + γ>X (t), after somemodification of γ if necessary, admits an Rm

+ ×Rn-valued affinestate process X with block-diagonal diffusion matrix of the form

α(x) =

(diag(x1, . . . , xq, 0, . . . , 0) 0

0 a +∑

i∈I xiαi ,JJ

)for some integer 0 ≤ q ≤ m.


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Outline









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Existence Problem

• Standing assumption always was existence and uniquenessfor

dX = (b + BX ) dt + ρ(X ) dW

X (0) = x

• Problem: ρ(x)ρ(x)> = a +∑

i∈I xiαi . Hence ρ(x) is notLipschitz in general.

• Question: exists a solution at all?

• Recall affine characterization theorem 37.2: admissibilityconditions on a, αi , but not on ρ(x)

• Indeed: can replace ρ(x) by ρ(x)D sinceρ(x)DD>ρ(x)> = ρ(x)ρ(x)> for any orthogonald × d-matrix D


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Law of X

• Theorem 38.1: φ and ψ in affine transform formulauniquely determined by admissible parameters a, αi , b, βi

as solution of the Riccati equations

• φ and ψ uniquely determine the law of the process X byiteration:

E[eu>1 X (t1)+u>2 X (t2)

]= E

[eu>1 X (t1)E

[eu>2 X (t2) | Ft1

]]= E

[eu>1 X (t1)eφ(t2−t1,u2)+ψ(t2−t1,u2)>X (t1)

]= eφ(t2−t1,u2)+φ(t1,u1+ψ(t2−t1,u2))+ψ(t1,u1+ψ(t2−t1,u2))>x

⇒ law of X : uniquely determined by a, αi , b, βi but can berealized by infinitely many variants of the above SDE

• Need canonical choice of ρ(x) . . .


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Canonical Affine SDE

• Recall canonical representation: block-diagonal diffusionafter linear transformation X 7→ ΛX :

ρ(x)ρ(x)> =

(diag(x1, . . . , xq, 0, . . . , 0) 0

0 a +∑

i∈I xiαi ,JJ

)• Obvious: ρ(x) = ρ(xI )

• Set ρIJ(x) ≡ 0, ρJI (x) ≡ 0, and

ρII (xI ) = diag(√

x1, . . . ,√

xq, 0, . . . , 0)

• Determine ρJJ(xI ) via Cholesky factorization ofa +

∑i∈I xiαi ,JJ = unique lower triangular matrix with

positive diagonal elements and

ρJJ(xI )ρJJ(xI )> = a +

∑i∈I

xiαi ,JJ


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Canonical Affine SDE cont’d

• Affine stochastic differential equation now reads

dXI = (bI + BII XI ) dt + ρII (XI ) dWI

dXJ = (bJ + BJI XI + BJJXJ) dt + ρJJ(XI ) dWJ

X (0) = x

(37)

• Yamada–Watanabe (1971): ∃ unique Rm+ × Rn-valued

solution X = X x , for any x ∈ Rm+ × Rn


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Summary

We thus have shown:

TheoremLet a, αi , b, βi be admissible parameters. Then there exists ameasurable function ρ : Rm

+ × Rn → Rd×d withρ(x)ρ(x)> = a +

∑i∈I xiαi , and such that, for any

x ∈ Rm+ × Rn, there exists a unique Rm

+ × Rn-valued solutionX = X x of the above affine SDE.Moreover, the law of X is uniquely determined by a, αi , b, βi ,and does not depend on the particular choice of ρ.


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Outline









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Outline









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DamirFilipovic

HeuristicDerivationFrom HJM

LIBOR MarketModel

LIBORDynamics UnderDifferentMeasures

Implied BondMarket

ImpliedMoney-MarketAccount

SwaptionPricing

Forward SwapMeasure

AnalyticApproximations

Monte CarloSimulation ofthe LIBORMarket Model

VolatilityStructure andCalibration


Calibration toMarket Quotes

Continuous-TenorCase

Part X

Market Models


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LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Overview

• Instantaneous forward rates: not easy to estimate

• Directly model observables such as LIBOR

• Breakthroughs 1997:• Miltersen et al. [39], Brace et al. [11]: HJM-type

lognormal LIBOR rate model• Jamshidian [32]: framework for arbitrage-free LIBOR and

swap rate models not based on HJM

• Principal idea: choose other numeraire than risk-freeaccount

• Both approaches lead to Black’s formula for either caps(LIBOR models) or swaptions (swap rate models)

• Because of this they are usually referred to as “marketmodels”


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LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










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LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Heuristic Derivation

• Point of departure: HJM setup from above

• Recall: forward δ-period LIBOR

L(t,T ) = F (t; T ,T + δ) =1

δ

(P(t,T )

P(t,T + δ)− 1

)• Seen in Chapter ”Forward Measures”:

P(t,T )/P(t,T + δ) is a QT+δ-martingale:

d

(P(t,T )

P(t,T + δ)

)=

P(t,T )

P(t,T + δ)σT ,T+δ(t) dW T+δ(t)

where σT ,T+δ(t) =∫ T+δT σ(t, u) du

• Hence

dL(t,T ) =1

δd

(P(t,T )

P(t,T + δ)

)=

1

δ(δL(t,T ) + 1)σT ,T+δ(t) dW T+δ(t)


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Heuristic Derivation cont’d

• Suppose: ∃ Rd -valued deterministic function λ(t,T ) s.t.

σT ,T+δ(t) =δL(t,T )

δL(t,T ) + 1λ(t,T ) (38)

• Plugging in: dL(t,T ) = L(t,T )λ(t,T ) dW T+δ(t)

• This is equivalent to, for s ≤ t ≤ T ,

L(t,T ) = L(s,T )e∫ ts λ(u,T ) dW T+δ(u)− 1

2

∫ ts ‖λ(u,T )‖2 du

⇒ Ft-conditional QT+δ-distribution of log L(T ,T ) is

Gaussian with mean = log L(t,T )− 12

∫ Tt ‖λ(s,T )‖2 ds

and variance =∫ Tt ‖λ(s,T )‖2 ds


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure








⇒ Caplet (reset date T , settlement date T + δ, strike rate κ)has time t price

EQ

[e−

∫ T+δt r(s) dsδ(L(T ,T )− κ)+ | Ft

]= P(t,T + δ)EQT+δ

[δ(L(T ,T )− κ)+ | Ft

]= δP(t,T + δ) (L(t,T )Φ(d1(t,T ))− κΦ(d2(t,T )))

where Φ = standard Gaussian cdf, and

d1,2(t,T ) =log(

L(t,T )κ

)± 1

2

∫ Tt ‖λ(s,T )‖2 ds(∫ T

t ‖λ(s,T )‖2 ds) 1

2

• This is just Black’s formula withσ(t)2 = 1

T−t

∫ Tt ‖λ(s,T )‖2 ds


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure








• Question: ∃? HJM model satisfying (38)?

• Answer: yes, but the construction and proof are not easy

• Idea: rewrite (38)∫ T+δ

Tσ(t, u) du =

(1− e−

∫ T+δT f (t,u) du

)λ(t,T )

• Differentiating in T gives

σ(t,T + δ) = σ(t,T )

+ (f (t,T + δ)− f (t,T ))e−∫ T+δT f (t,u) duλ(t,T )

+(

1− e−∫ T+δT f (t,u) du

)∂Tλ(t,T )


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure








• This recurrence relation can be solved by forwardinduction, once σ(t, ·) is determined on [0, δ) (typically,σ(t,T ) = 0 for T ∈ [0, δ))

⇒ Complicated dependence of σ on the forward curve

⇒ Existence and uniqueness for HJM equations: carried outby [11], see also [25, Section 5.6]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Direct Approach

• Direct approach to LIBOR models: outside of HJMframework

• Instead of risk-neutral martingale measure: work underforward measures with numeraires = bonds

• Fix finite time horizon TM = Mδ, M ∈ N• (Ω,F , (Ft)t∈[0,TM ],QTM ) carrying a d-dimensional

Brownian motion W TM (t), t ∈ [0,TM ]

• Notation suggests that QTM will play the role of theTM -forward measure

• Write Tm = mδ, m = 0, . . . ,M

• Aim: construct a model for the M forward LIBOR rateswith maturities T0, . . . ,TM−1


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Direct Approach cont’d

Take as given:

• ∀m ≤ M − 1: an Rd -valued deterministic boundedmeasurable function λ(t,Tm) = volatility of L(t,Tm)

• an initial positive and nonincreasing discreteterm-structure P(0,Tm), m ≤ M, and hence nonnegativeinitial forward LIBOR rates

L(0,Tm) =1

δ

(P(0,Tm)

P(0,Tm+1)− 1

), m = 0, . . . ,M − 1


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Direct Approach cont’d• Backward induction: postulate first that

dL(t,TM−1) = L(t,TM−1)λ(t,TM−1) dW TM (t),

t ∈ [0,TM−1], which is equivalent to

L(t,TM−1) = L(0,TM−1)Et(λ(·,TM−1) •W TM

)• Motivated by (38): define Rd -valued bounded progressive

process, t ∈ [0,TM−1],

σTM−1,TM(t) =

δL(t,TM−1)

δL(t,TM−1) + 1λ(t,TM−1)

• Induces equivalent probability measure QTM−1 on FTM−1:

dQTM−1

dQTM= ETM−1

(σTM−1,TM

•W TM

)∝ δL(TM−1,TM−1)+1

• Girsanov: W TM−1(t) = W TM (t)−∫ t

0 σTM−1,TM(s)> ds is a

QTM−1-Brownian motion, t ∈ [0,TM−1]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Intermezzo: Sanity Check

• Recall from part “Forward Measures”: in HJM framework

dQTM−1

dQTM|Ft =

P(t,TM−1) P(0,TM)

P(t,TM) P(0,TM−1)= Et

(σTM−1,TM

•W TM

)• Moreover

P(t,TM−1)

P(t,TM)= δL(t,TM−1) + 1


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Direct Approach cont’d• Hence we can postulate

dL(t,TM−2) = L(t,TM−2)λ(t,TM−2) dW TM−1(t),

t ∈ [0,TM−2], which is equivalent to

L(t,TM−2) = L(0,TM−2)Et(λ(·,TM−2) •W TM−1

)• Define Rd -valued bounded progressive process,

t ∈ [0,TM−2],

σTM−2,TM−1(t) =

δL(t,TM−2)

δL(t,TM−2) + 1λ(t,TM−2)

• Induces probability measure QTM−2 ∼ QTM−1 on FTM−2:

dQTM−2

dQTM−1= ETM−2

(σTM−2,TM−1

•W TM−1

)• Girsanov: W TM−2(t) = W TM−1(t)−

∫ t0 σTM−2,TM−1

(s)> dsis a QTM−2-Brownian motion, t ∈ [0,TM−2]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Direct Approach cont’d

• Repeating this procedure leads to a family of lognormalmartingales (L(t,Tm))t∈[0,Tm] under their respective

measures QTm+1 .


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







LIBOR Dynamics Under DifferentMeasures

Find dynamics of L(t,Tm) under any QTn+1 :

LemmaLet 0 ≤ m, n ≤ M − 1. Then the dynamics of L(t,Tm) underQTn+1 is given according to the three cases, fort ∈ [0,Tm ∧ Tn+1],

dL(t,Tm)

L(t,Tm)= λ(t,Tm)

·

−(∑n

l=m+1 σTl ,Tl+1(t)>dt + dW Tn+1(t)

), m < n,

dW Tn+1(t), m = n,(∑ml=n+1 σTl ,Tl+1

(t)>dt + dW Tn+1(t)), m > n.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Proof

Proof.This follows from the obvious equality

W Ti+1(t) = W Tj+1(t)−j∑

l=i+1

∫ t

0σTl ,Tl+1

(s)> ds, t ∈ [0,Ti+1], 0 ≤ i < j ≤ M−1

(→ exercise ).


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Forward Prices

• What can be said about bond prices?

• Define forward price process (1 ≤ m ≤ M):

P(t,Tm−1)

P(t,Tm)= δL(t,Tm−1) + 1, t ∈ [0,Tm−1]

• Clearly:

d

(P(t,Tm−1)

P(t,Tm)

)= δ dL(t,Tm−1)

=P(t,Tm−1)

P(t,Tm)σTm−1,Tm(t) dW Tm(t)

• Hence

P(t,Tm−1)

P(t,Tm)=

P(0,Tm−1)

P(0,Tm)Et(σTm−1,Tm •W Tm

)is a QTm -martingale, t ∈ [0,Tm−1]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Forward Prices cont’d

• Extension: define Tm-forward price for all Tk -bonds:

P(t,Tk)

P(t,Tm)=

P(t,Tk )

P(t,Tk+1) · · ·P(t,Tm−1)P(t,Tm) , k < m(

P(t,Tm)P(t,Tm+1)

)−1· · ·(

P(t,Tk−1)P(t,Tk )

)−1, k > m,

for t ∈ [0,Tk ∧ Tm]

• Following lemma is in accordance with part “ForwardMeasures” . . .


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Forward Prices cont’d

LemmaFor every 1 ≤ k 6= m ≤ M, the forward price process satisfies

P(t,Tk)

P(t,Tm)=

P(0,Tk)

P(0,Tm)Et(σTk ,Tm •W Tm

), t ∈ [0,Tk ∧ Tm],

for the Rd -valued bounded progressive process

σTk ,Tm =

∑m−1l=k σTl ,Tl+1

, k < m

−∑k−1

l=m σTl ,Tl+1, k > m.

(39)

Hence P(t,Tk )P(t,Tm) , t ∈ [0,Tk ∧ Tm], is a positive QTm -martingale.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







ProofSuppose first k < m. Then (5) and (3) imply

P(t,Tk)

P(t,Tm)=

m−1∏l=k

P(t,Tl)

P(t,Tl+1)=

P(0,Tk)

P(0,Tm)

m−1∏l=k

Et(σTl ,Tl+1

•W Tl+1

)=

P(0,Tk)

P(0,Tm)exp

[∫ t

0

m−1∑l=k

σTl ,Tl+1(s)

(dW Tm(s)−

m−1∑i=l+1

σTi ,Ti+1(s)> ds

)

−1

2

∫ t

0

m−1∑l=k

‖σTl ,Tl+1(s)‖2 ds

]

=P(0,Tk)

P(0,Tm)exp

∫ t

0

m−1∑l=k

σTl ,Tl+1(s) dW Tm(s)− 1

2

∫ t

0

∥∥∥∥∥m−1∑l=k

σTl ,Tl+1(s)

∥∥∥∥∥2

ds

.Hence

P(t,Tk)

P(t,Tm)=

P(0,Tk)

P(0,Tm)Et

((m−1∑l=k

σTl ,Tl+1

)•W Tm

),

as desired. The case k > m follows by similar argumentation(exercise).


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Nominal Bond Prices

• Can derive nominal Tn-bond prices

P(Tm,Tn) =n∏

k=m+1

P(Tm,Tk)

P(Tm,Tk−1)

=n∏

k=m+1

1

δL(Tm,Tk−1) + 1

only at dates t = Tm, for 0 ≤ m < n ≤ M

• However: continuous time bond price dynamics P(t,Tn)not determined

• Knowledge of forward LIBOR rates L(t,T ) for allmaturities T ∈ [0,TM−1] would be necessary: see section“Continuous-Tenor Case” below


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Forward Pricing

• Notwithstanding: can now consistently computeTm-forward price for any Tm-contingent claim X withEQTm [|X |] <∞:

π(t)

P(t,Tm)= EQTm [X | Ft ]

• Change of numeraire:

LemmaThe Tm-forward price price satisfies

π(t)

P(t,Tm)=

P(t,Tn)

P(t,Tm)EQTn

[X

P(Tm,Tn)| Ft

],

for all m < n ≤ M.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Proof

By last lemma: for t ∈ [0,Tk ]

dQTk

dQTk+1|Ft = Et

(σTk ,Tk+1

•W Tk+1

)=

P(0,Tk+1)

P(0,Tk)

P(t,Tk)

P(t,Tk+1).

Hence

dQTm

dQTn|Ft =

n−1∏k=m

dQTk

dQTk+1|Ft =

n−1∏k=m

P(0,Tk+1)

P(0,Tk)

P(t,Tk)

P(t,Tk+1)

=P(0,Tn)

P(0,Tm)

P(t,Tm)

P(t,Tn).

Bayes’ rule now yields the assertion.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Caplet Pricing

From section “Heuristic Derivation” we now obtain:

Corollary

Let m + 1 < n ≤ M. The time Tm price of the nth caplet withreset date Tn−1, settlement date Tn and strike rate κ is

Cpl(Tm; Tn−1,Tn) = δP(Tm,Tn)

× (L(Tm,Tn−1)Φ(d1(n; Tm))− κΦ(d2(n; Tm)))

where Φ = standard Gaussian cdf, and

d1,2(n; Tm) =log(

L(Tm,Tn−1)κ

)± 1

2

∫ Tn−1

Tm‖λ(s,Tn−1)‖2 ds(∫ Tn−1

Tm‖λ(s,Tn−1)‖2 ds

) 12

.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Caplet Pricing cont’d

This is exactly Black’s formula for the caplet price with

σ(Tm)2 =1

Tn−1 − Tm

∫ Tn−1

Tm

‖λ(s,Tn−1)‖2 ds.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Definition

• Define discrete-time implied money-market accountprocess

B∗(0) = 1,

B∗(Tm) = (1 + δL(Tm−1,Tm−1))B∗(Tm−1), 1 ≤ m ≤ M

• Equivalently:

B∗(Tn) = B∗(Tm)n−1∏k=m

1

P(Tk ,Tk+1), m < n ≤ M

• Interpretation: B∗(Tm) = cash amount accumulated up totime Tm by rolling over a series of zero-coupon bonds withthe shortest maturities available

• By construction: B∗ nondecreasing and B∗(Tm) isFTm−1-measurable for all m = 1, . . . ,M


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







First PropertiesDefine function η : [0,TM−1]→ N by Tη(t)−1 ≤ t < Tη(t)

LemmaFor all t ∈ [0,TM−1] we have

EQTM [B∗(TM)P(0,TM) | Ft ] = Et(σT0,TM−1

•W TM

),

where we define (in accordance with σTk ,Tm for k ≥ 1) theRd -valued bounded progressive process

σT0,TM−1(t) =

M−1∑k=η(t)

σTk ,Tk+1(t).

In particular, for all 0 ≤ m ≤ M we have

EQTM [B∗(TM) | FTm ] =B∗(Tm)

P(Tm,TM).


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Proof

Proof.Follows from above lemma for P(t,Tk)/P(t,Tm) (→exercise).


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Risk-Neutral Measure

• In view of preceding lemma: can define Q∗ ∼ QTM onFTM

bydQ∗

dQTM= B∗(TM)P(0,TM)

• We have

dQ∗

dQTM|Ft =

Et(σT0,TM−1

•W TM), t ∈ [0,TM−1],

B∗(Tm) P(0,TM)P(Tm,TM) , if t = Tm, m ≤ M − 1

• Interpretation: Q∗ = risk-neutral martingale measure

• Also called “spot LIBOR measure” (Jamshidian [32])


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Risk-Neutral PricingBayes’ rule implies:

LemmaThe time Tk price of the Tm-contingent claim X from“Forward Pricing” lemma above satisfies

π(Tk) = B∗(Tk)EQ∗

[X

B∗(Tm)| FTk

],

for all k ≤ m.

• Consequence (π(Tk) = P(Tk ,Tm) for X = 1):discrete-time process(

P(Tk ,Tm)

B∗(Tk)

)k=0,...,m

is a Q∗-martingale w.r.t. (FTk)

⇒ We constructed a full discrete-time interest rate model


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Risk-Neutral LIBOR Dynamics

Girsanov: W ∗(t) = W TM (t)−∫ t

0 σT0,TM−1(s) ds is a

Q∗-Brownian motion, t ∈ [0,TM−1]

LemmaLet 0 ≤ m ≤ M − 1. Then the dynamics of L(t,Tm) under Q∗is given according to, t ∈ [0,Tm]

dL(t,Tm)

L(t,Tm)= λ(t,Tm)

m∑k=η(t)

σTk ,Tk+1(t)>dt +λ(t,Tm) dW ∗(t).

Proof.Follows from “LIBOR dynamics” lemma above for n = M − 1and the definition of W ∗.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Problem Formulation

• Consider a payer swaption with nominal 1, strike rate K ,maturity Tµ, underlying tenor Tµ, Tµ+1, . . . ,Tν (Tµ is thefirst reset date and Tν the maturity of the underlyingswap), for some µ < ν ≤ M

• Recall: payoff at maturity Tµ is

Π = δ

ν∑m=µ+1

P(Tµ,Tm)(L(Tµ,Tm−1)− K )

+

• By above pricing lemmas: swaption price at t = 0

π = P(0,Tµ)EQTµ [Π] = EQ∗

[Π

B∗(Tµ)

]⇒ Need joint distribution (under QTµ or Q∗) of

L(Tµ,Tµ), L(Tµ,Tµ+1), . . . , L(Tµ,Tν−1)

• No analytic formula in LIBOR market model . . .


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Definition

• Corresponding forward swap rate at t ≤ Tµ is

Rswap(t) =P(t,Tµ)− P(t,Tν)

δ∑ν

k=µ+1 P(t,Tk)=

1− P(t,Tν)P(t,Tµ)

δ∑ν

k=µ+1P(t,Tk )P(t,Tµ)

.

⇒ Rswap(t) available in our LIBOR market model

• Define positive QTµ-martingale D(t) =∑ν

k=µ+1P(t,Tk )P(t,Tµ) ,

t ∈ [0,Tµ]

• Induces forward swap measure Qswap ∼ QTµ on FTµ by

dQswap

dQTµ=

D(Tµ)

D(0).


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Change of Numeraire

LemmaThe forward swap rate process Rswap(t), t ∈ [0,Tµ], is apositive Qswap-martingale.Moreover, there exists some d-dimensional Qswap-Brownianmotion W swap and an Rd -valued progressive swap volatilityprocess ρswap such that

dRswap(t) = Rswap(t)ρswap(t) dW swap(t), t ∈ [0,Tµ].

• Note: ρswap(t) explicitly available in principle


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Proof

Let 0 ≤ m ≤ M and 0 ≤ s ≤ t ≤ Tm ∧ Tµ. Then

EQswap

[P(t,Tm)

P(t,Tµ)D(t)| Fs

]=

1

D(s)EQTµ

[P(t,Tm)

P(t,Tµ)D(t)D(t) | Fs

]=

1

D(s)

P(s,Tm)

P(s,Tµ).

On the other hand, from above formula for Rswap(t):

Rswap(t) =1

δD(t)− P(t,Tν)

δP(t,Tµ)D(t).

Hence Rswap(t) is a positive Qswap-martingale.The representation of Rswap(t) in terms of W swap and ρswap

follows from above lemma for P(t,Tk)/P(t,Tm) andGirsanov’s theorem.


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Swap Measure Swaption Pricing

• Recall: swaption payoff at maturity can be written as

δD(Tµ) (Rswap(Tµ)− K )+

• Hence the price equals

π = δP(0,Tµ)EQTµ

[D(Tµ) (Rswap(Tµ)− K )+]

= δP(0,Tµ)D(0)EQswap

[(Rswap(Tµ)− K )+]

= δ

ν∑k=µ+1

P(0,Tk)EQswap

[(Rswap(Tµ)− K )+]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Lognormal Hypothesis

• Hypothesis (H): ρswap(t) is deterministic

• Consequence: log Rswap(Tµ) Gaussian distributed under

Qswap with mean = log Rswap(0)− 12

∫ Tµ0 ‖ρswap(t)‖2 dt

and variance =∫ Tµ

0 ‖ρswap(t)‖2 dt

⇒ Swaption price would be

π = δ

ν∑k=µ+1

P(0,Tk) (Rswap(0)Φ(d1)− K Φ(d2))

with

d1,2 =log(

Rswap(0)K

)± 1

2

∫ Tµ0 ‖ρswap(t)‖2 dt(∫ Tµ

0 ‖ρswap(t)‖2 dt) 1

2

• This is Black’s formula with volatilityσ2 = 1

Tµ

∫ Tµ0 ‖ρswap(t)‖2 dt


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Lognormal Hypothesis Valid?

• Fact (without proof): ρswap cannot be deterministic in ourlognormal LIBOR setup

• Alternative for swaption pricing: model forward swap ratesdirectly and postulate that they are lognormal under theforward swap measures (the so-called swap market model)

• Carried out by Jamshidian [32], and computationallyimproved by Pelsser [43]

• But (without proof): then forward LIBOR rate volatilitycannot be deterministic

⇒ Either one gets Black’s formula for caps or for swaptions,but not simultaneously for both

⇒ In lognormal forward LIBOR model swaption prices haveto be approximated

• via Monte Carlo methods• via analytic approximation . . .


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Freezing the Weights

• Recall: forward swap rate can be written as weighted sumof forward LIBOR rates

Rswap(t) =ν∑

m=µ+1

wm(t)L(t,Tm−1)

with weights

wm(t) =P(t,Tm)

D(t)P(t,Tµ)=

11+δL(t,Tµ) · · ·

11+δL(t,Tm−1)∑ν

k=µ+11

1+δL(t,Tµ) · · ·1

1+δL(t,Tk−1)

• Empirical studies show: variability of wm small comparedto variability of L(t,Tm−1)

→ Approximate Rswap(t) ≈∑ν

m=µ+1 wm(0)L(t,Tm−1)


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Freezing the Weights cont’d

⇒ QTµ-dynamics, t ∈ [0,Tµ]

dRswap(t) ≈ (· · · ) dt

+ν∑

m=µ+1

wm(0)L(t,Tm−1)λ(t,Tm−1) dW Tµ

⇒ Forward swap volatility satisfies

‖ρswap(t)‖2 =d 〈log Rswap, log Rswap〉t

dt

≈ν∑

k,l=µ+1

wk(0)L(t,Tk−1)λ(t,Tk−1)

Rswap(t)

× wl(0)L(t,Tl−1)λ(t,Tl−1)>

Rswap(t)


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Further Approximation

• Further approximation: ‖ρswap(t)‖2 ≈ ρswap(t)2 with

ρswap(t)2 =ν∑

k,l=µ+1

wk(0)L(0,Tk−1)λ(t,Tk−1)

Rswap(0)

× wl(0)L(0,Tl−1)λ(t,Tl−1)>

Rswap(0)

• Levy’s characterization theorem:W(t) =

∫ t0ρswap(s) dW swap(s)‖ρswap(s)‖ is a Qswap-Brownian motion,

t ∈ [0,Tµ]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Summary

⇒ ρswap(t) deterministic and

dRswap(t) = Rswap(t)‖ρswap(t)‖ dW(t)

≈ Rswap(t)ρswap(t) dW(t)

⇒ Approximate swaption price by Black’s formula with

σ2 =1

Tµ

∫ Tµ

0ρswap(t)2 dt

• “Rebonato’s formula” (Rebonato [44])

• Goodness of this approximation numerically tested byseveral authors ([12, Chapter 8]): “the approximation issatisfactory in general”


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Towards an Euler Scheme

• Seen in swaption case above: option pricing typicallyrequires Monte Carlo simulation

• Many ways to simulate forward LIBOR rates: e.g.Glasserman [26]

• Here: sketch particular Euler scheme

• Focus on risk-neutral measure Q∗ (albeit the following canbe carried out under any forward measure)

• Aim: simulate entire M-vector (L(t,T0), . . . , L(t,TM−1))>


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Towards an Euler Scheme

• Transform: Hm(t) = log L(t,Tm) satisfies (Ito’s formula)

dHm(t) = αm(t) dt + λ(t,Tm) dW ∗(t), t ≤ Tm

with drift term

αm(t) = λ(t,Tm)m∑

k=η(t)

δeHk (t)

1 + δeHk (t)λ(t,Tk)>−1

2‖λ(t,Tm)‖2

• Advantage of simulating Hm:• keeps L(t,Tm) = exp(Hm(t)) positive• Hm has Gaussian increments ⇒ improves convergence of

Euler scheme


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







General pricing problem

• General pricing problem: Tn-claim with payofff (Hn(Tn), . . . ,HM−1(Tn))

• Price at t = 0: π = EQ∗[

f (Hn(Tn),...,HM−1(Tn))B∗(Tn)

]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Euler Scheme

• Fix time grid ti = i∆t, i = 0, . . . ,N, with ∆t = Tn/N forN large enough

• Z (1), . . . ,Z (N): sequence of independent standard normalrandom vectors in Rd

• Euler approximation for Hm: 1 ≤ i ≤ N

Hm(ti ) = Hm(ti−1) +αm(ti−1) ∆t + λ(ti−1,Tm) Z (i)√

∆t(40)

• Monte Carlo principle:

1 simulate via Euler scheme K independent copies

Π(1), . . . ,Π(K) of Π = f (Hn(Tn),...,HM−1(Tn))B∗(Tn)

2 estimate π via averaging: Π = 1K

∑Kj=1 Π(j)


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Efficiency

Three considerations are important for the efficiency of thissimulation estimator: bias, variance, and computing time

Bias: introduced via Euler approximation (40): EQ∗ [Π]differs from target value π = EQ∗ [Π]. Bias isreduced by increasing number of timediscretization steps N. (In our example: bias isalready negligible for ∆t = 1/12, can assumethat EQ∗ [Π] ≈ π.)


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Efficiency cont’d

Variance: central limit theorem: as number of replicationsK increases, the simulation estimation errorΠ− π approximately normal distributed withmean zero and approximate standard deviation of

sπ =

√∑Kj=1(Π(j) − Π)

K (K − 1).

sπ is called standard error of the Monte Carlosimulation: Π± sπ is an asymptotically (asK →∞) valid 68% confidence interval for thetrue value π.

Computing time: obvious trade-off between bias and variancefor a given computing capacity, which has to becarefully balanced in general.

Thorough treatment of Monte Carlo: e.g. Glasserman [26]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Volatility Estimation

• So far: volatility factors λ(t,Tm) = exogenousdeterministic functions

• In practice: chosen to calibrate to

• market prices of liquidly traded derivatives, such as capsand swaptions

• historical time series

• Note: model is automatically calibrated to initial bondprices

• Volatility estimation: huge topic, see e.g. Brigo andMercurio [12]

• Here: brief discussion of two approaches:

• historical volatility estimation via principal componentanalysis (PCA)

• volatility calibration to market quotes of caps andswaptions . . .


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Sketch of PCA

• Assume: λ(t,Tm) = λ(Tm − t), for all m

• Given: N observations x(1), . . . , x(N) of random vectorsX (i) = (X1(i), . . . ,XM(i))> where (1 ≤ m ≤ M)

Xm(i) = log L(iδ, (i +m−1)δ)− log L((i−1)δ, (i +m−1)δ)

(sliding LIBOR curve increments)

• Euler approximation (40) & neglect drift term (see nextpage) ⇒

Xm(i) ≈ λ(mδ) Z (i)√δ (41)

⇒ X (i) approximately i.i.d. with zero mean


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Sketch of PCA: Drift Term

• Recall Q∗-drift term of log L(iδ, (i + m − 1)δ):

αm((i − 1)δ) = −1

2‖λ(mδ)‖2

+ λ(mδ)m∑

k=1

δL((i − 1)δ, (i + k − 1)δ)

1 + δL((i − 1)δ, (i + k − 1)δ)λ(kδ)>

• For P: to be corrected by

market price of risk× δ ‖λ(mδ)‖

⇒ Drift term of order

δ ‖λ(mδ)‖ ×max‖λ(mδ)‖, market price of risk

⇒ neglected


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Sketch of PCA cont’d• empirical mean µ = 1

N

∑Nt=1 x(t)

• empirical covariance matrix

Qij = Cov[xi , xj ] =1

N

N∑t=1

(xi (t)− µi )(xj(t)− µj)

• PCA decomposition:

x(i) = µ+M∑

j=1

aj yj(i) ≈M∑

j=1

aj yj(i)

with• Q = ALA>, loadings A = (a1 | · · · | an)• empirical principal components y = A>(x − µ): yj

uncorrelated, nonincreasing order Var[y1] ≥ Var[y2] ≥ · · ·• Compare with (41) ⇒ estimate (1 ≤ m ≤ M)

λj(mδ) =

√Var[yj ]

δajm


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







PCA Stylized Facts

• First 2–3 principal components yj enough to explain mostof the variance of x

• First three loadings aj (i.e. volatility curves s 7→ λj(s)) aretypically of the form as seen in Section “PCA of theForward Curve”: flat, upward (or downward) sloping, andhump-shaped

Figure: First three forward curve loadings


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Volatility vs. Correlation

• Recall Black’s caplet price formula: function of ‖λ(t,Tm)‖only

⇒ No gain in flexibility in matching caplet implied volatilitiesby taking number d of Brownian motions greater than one

• Potential value of a multi-factor model lies in capturingcorrelations between forward LIBOR rates of differentmaturities.

• Euler approximation (40) ⇒ instantaneous correlationbetween increments of log L(t,Tm) and log L(t,Tn)approximately

ρmn(t) =λ(t,Tm)λ(t,Tn)>

‖λ(t,Tm)‖ ‖λ(t,Tn)‖

• ρmn(t) chosen to match market quotes of swaptions, orhistorical correlations as indicated in above PCA


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Volatility vs. Correlation cont’d

• Formalize dual aspect volatility vs. correlation:

λ(t,Tm) = σm(t) `m(t)

where• σm(t) = ‖λ(t,Tm)‖ = volatility of L(t,Tm): calibrated to

caplet prices• `m(t) = λ(t,Tm)/‖λ(t,Tm)‖ captures correlation between

different rates: ρmn(t) = `m(t) `n(t)>


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Volatility Specifications

• Assume as given: market quotes of all capletsCpl(Tn−1,Tn) = Cpl(0; Tn−1,Tn) by their impliedvolatilities σCpl(Tn−1,Tn) (1 ≤ n ≤ M − 1):∫ Tn

0σn(t)2 dt = σ2

Cpl(Tn,Tn+1) Tn

• Specification 1: σm(t) = σ(Tm − t) = function of time tomaturity

⇒∫ Tn

Tn−1σ(t)2 dt =

=

σ2

Cpl(T1,T2) T1, n = 1

σ2Cpl(Tn,Tn+1) Tn − σ2

Cpl(Tn−1,Tn) Tn−1, 2 ≤ n ≤ M − 1

• Drawback: requires σ2Cpl(Tn,Tn+1) Tn ≥ σ2

Cpl(Tn−1,Tn) Tn−1,which is not satisfied by caplet data in general !


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Volatility Specifications cont’d

• Specification 2: σm(t) ≡ σm independent of t, consistent

⇒ Calibration easy:σn = σCpl(Tn,Tn+1), 1 ≤ n ≤ M − 1

• Drawback: stipulates that volatility does not change overtime: not plausible for long-matured forward LIBOR rates

• Specification 3: parametric form σm(t) = vm e−β(Tm−t),common exponent β, individual factors vm

• Note: specification under-determined: the system

v 2n

1− e−2βTn

2β= σ2

Cpl(Tn,Tn+1) Tn, 1 ≤ n ≤ M − 1,

leaves one degree of freedom (to be fixed by someadditional data point)

• Note: for β = 0 we obtain back Specification 2

• More systematic classification of admissible volatilityspecifications: see Brigo and Mercurio [12, Section 6.3]


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Correlation Specifications

• Common assumption: `m(t) ≡ `m (e.g. in [12, Section6.3])

⇒ Analytic approximation formula (“Rebonato’s”) forimplied swaption volatility:

σ2swp ≈

1

Tµ

ν−1∑k,l=µ

ρk,l

×wk+1(0)wl+1(0)L(0,Tk)L(0,Tl)

∫ Tµ0 σk(t)σl(t) dt

R2swap(0)

• Estimator for ρ, given initial term-structure L(0,Tk) andσm calibrated to caplet quotes

• Caution (euro zone): δ = 1/2 for caplets, δ = 1 for swaps !

• Alternative to analytic approximation formula: iterativeMonte Carlo simulations


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Correlation Specifications cont’d

• Specification of `m goes together with choice of d

• For illustration: consider two extreme cases d = 1 andd = M − 1

• Remark: recall stylized facts on “Correlation”: ρmn

exponentially decaying ρmn = e−γ |Tm−Tn|. Specifications Iand II then correspond to γ = 0 and γ =∞


DamirFilipovic


LIBOR MarketModel


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Correlation Specification I

• Specification I: d = 1 and `m = 1, for all m

• instantaneous correlation between increments oflog L(t,Tm) and log L(t,Tn) is one

Figure: Perfectly correlated case: trajectories of L(·,T ) forT = 2, 4.5, 7, 9.5.


DamirFilipovic


LIBOR MarketModel


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Correlation Specification II• Specification II: d = M − 1 and `m = e>m , ρ=unit matrix⇒ SDEs decoupled (t ≤ Tm, 1 ≤ m ≤ M − 1):

dHm(t) =

(δeHm(t)

1 + δeHm(t)σm(t)2 − 1

2σm(t)2

)dt

+ σm(t) dWm(t)

⇒ independent forward LIBOR rates

Figure: Independent case: trajectories of L(·,T ) forT = 2, 4.5, 7, 9.5.


DamirFilipovic


LIBOR MarketModel


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Caplet Calibration

• Usual market quotes: prices in bp on caps and floors, seetables (or to be inferred from implied volatilities)

⇒ Have to strip caplet and floorlet volatilities

• Illustration: bootstrapping method analogous to part“Estimating the Term-Structure”

• Tenor: Ti = i/2, i = 0, . . . , 20, where T1 = 1/2 = firstcaplet reset date, T20 = 10 = maturity of last cap

• Prevailing initial forward LIBOR curve given in followingtable


DamirFilipovic


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Forward LIBOR Curve

Table: Forward LIBOR curve (in %) on 18 November 2008

Ti 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5L(0,Ti ) 4.228 2.791 3.067 3.067 3.728 3.728 4.051 4.051 4.199 4.199

Ti 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5L(0,Ti ) 4.450 4.450 4.626 4.626 4.816 4.816 4.960 4.960 5.088 5.088


DamirFilipovic


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Cap Prices

Table: Euro cap prices (in basis points) on 18 November 2008

T–K 3.5% 4% 4.5% 5% 5.5% 6% 6.5% 7% 7.5%

2 25.0 11.0 5.0 2.5 1.5 1.0 0.5 0.0 0.03 77.0 40.5 21.5 12.0 7.0 4.0 2.5 1.5 1.54 148.5 86.0 48.5 27.0 16.0 10.0 6.5 4.5 4.05 230.5 140.5 82.0 47.5 28.5 17.5 11.5 8.0 7.56 325.5 206.0 125.5 74.5 45.5 29.0 19.0 13.5 12.57 431.5 283.5 178.0 109.0 68.0 44.5 29.5 21.0 20.58 545.5 368.5 238.0 149.5 95.0 62.5 42.5 30.0 29.09 664.0 459.0 304.5 196.5 127.0 85.0 58.5 42.0 40.0

10 786.0 554.5 376.5 248.5 164.0 111.0 77.0 56.0 53.0


DamirFilipovic


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Floor Prices

Table: Euro floor prices (in basis points) on 18 November 2008

T–K 3% 2.75% 2.5% 2.25% 2% 1.75% 1.5% 1.25% 1%

2 69.5 50.0 34.0 23.0 14.5 9.0 5.5 3.5 1.53 92.0 66.5 47.0 32.0 21.5 14.0 9.0 5.0 2.54 110.0 80.5 58.0 40.5 28.5 19.5 13.0 8.0 4.05 127.0 94.0 68.5 49.0 35.0 25.0 17.0 11.0 5.56 142.0 107.0 78.5 58.0 42.5 31.0 21.5 13.5 7.57 157.5 119.5 89.5 67.0 50.0 37.0 26.5 16.5 9.08 172.5 132.0 101.0 76.5 58.5 43.5 31.0 20.0 10.59 187.5 145.0 112.0 86.5 66.5 50.0 36.0 23.5 13.0

10 201.5 157.5 122.5 95.5 74.0 56.5 41.0 27.5 15.5


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Forward SwapMeasure







Boostrapping Caplet Volatilities

• Consider: caps at strike rate K = 3.5%

• First cap, maturity T4 = 2 years:

Cp(T4) = Cpl(T1,T2) + Cpl(T2,T3) + Cpl(T3,T4)

• Infer implied caplet volatility by inverting Black’s formula:

σCpl(T1,T2) = σCpl(T2,T3) = σCpl(T3,T4) = 29.3%

• Next cap matures in T6 = 3 years:

Cp(T6)− Cp(T4) = Cpl(T4,T5) + Cpl(T5,T6)

• Infer implied caplet volatilityσCpl(T4,T5) = σCpl(T5,T6) = 20.8%, without altering theprevious ones, etc.


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Boostrapping Caplet Volatilitiescont’d

Table: Implied volatilities (in %) for caplets Cpl(Ti−1,Ti ) at strikerate 3.5%

i 1 2 3 4 5 6 7 8 9 10σCpl(Ti−1,Ti ) n/a 29.3 29.3 29.3 20.8 20.8 18.3 18.3 17.8 17.8

i 11 12 13 14 15 16 17 18 19 20σCpl(Ti−1,Ti ) 16.3 16.3 16.7 16.7 16.1 16.1 15.7 15.7 15.7 15.7


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


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Calibrate LIBOR Market Model

• Parametric specification: σm(t) = vm e−β(Tm−t)

• Recall: v 2n

1−e−2βTn

2β = σ2Cpl(Tn,Tn+1) Tn

⇒ v1, . . . , v19 as functions on β ( = degree of freedom)

• Sanity check: MC algorithm reproduces 3.5% market capprices independently the correlation specification . . .


DamirFilipovic


LIBOR MarketModel


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Forward SwapMeasure







Price a Swaption

• Price the at-the-money 4× 6-swaption: maturity in 4years, underlying swap 6 years long

• Tenor: first reset date T8 = 4, and annual(!) couponpayments at T10 = 5, . . . ,T20 = 10

• Swaption price depends on β and on correlationspecifications I (d = 1, `m = 1) or II (d = M − 1,`m = e>m).

• Compute by MC and analytic approximation formula


DamirFilipovic


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Results

Figure: The swaption price as function of β. The straight horizontalline indicates the real market quote of 248 bp. The upper curves arefor the correlation specification I, the lower curves are for specificationII. The solid lines show the Monte Carlo simulation based prices withstandard errors indicated by the dotted lines. The dashed lines showthe respective prices based on the analytic approximation formula.


DamirFilipovic


LIBOR MarketModel


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Discussion

• Actual market quote for this swaption was 248 bp• For correlation specification I: obtain an estimate β ≈ 0.07

• For correlation specification II: diversification effectbetween underlying independent LIBOR rates

⇒ Lower aggregate volatility and thus lower swaption price

• Cannot calibrate to actual market price: indicates thatspecification II systematically underprices swaptions

• Analytic approximation error: 7–10 pb (specification I),20–35 bp (specification II)

• More systematic tests for quality of analyticalapproximation in [12, Chapter 8]


DamirFilipovic


LIBOR MarketModel


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Forward SwapMeasure







Outlook: Volatility Smile

• Snag: bootstrapped caplet implied volatilities depend onstrike rate in general (volatility smile/skew)

⇒ Lognormal LIBOR market model can only matchterm-structure of caplet volatility for one strike rate at atime (no matter how many driving Brownian motions d)

• Situation similar to Black–Scholes stock market model:incapable of fitting market option prices across all strikes(note: Heston stochastic volatility model can producevolatility skews)

• Possible extensions:• λ(t,Tm) = λ(ω, t,Tm) progressive process (analogous to

Heston model)• replace driving Brownian motion by Levy process

• Much research over last decade to achieve a good fittingof market option data, see e.g. Brigo and Mercurio [12,Part IV] for a detailed overview


DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

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Outline










DamirFilipovic


LIBOR MarketModel


Implied BondMarket


SwaptionPricing

Forward SwapMeasure







Additional Assumptions

• Aim: specify all forward LIBOR rates L(t,T ),T ∈ [0,TM−1]: fill the gaps between the Tjs

• Each forward LIBOR L(t,T ) will follow a lognormalprocess under T + δ-forward measure

Additional assumptions:

• stochastic basis: Ft = FW TMt (for representation theorem)

• ∀T ∈ [0,TM−1]: an Rd -valued deterministic boundedmeasurable function λ(t,Tm) = volatility of L(t,Tm)

• an initial positive and nonincreasing term-structureP(0,T ), T ∈ [0,TM ], and hence nonnegative initialforward LIBOR rates

L(0,T ) =1

δ

(P(0,T )

P(0,T + δ)− 1

), T ∈ [0,TM−1]


DamirFilipovic


LIBOR MarketModel


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Forward SwapMeasure







Construction 1st Step

• Construct discrete-tenor model for L(t,Tm),m = 0, . . . ,M − 1, as in Section “LIBOR Market Model”


DamirFilipovic


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2nd Step: Interpolate forT ∈ [TM−1,TM ]

• No forward LIBOR for T ∈ [TM−1,TM ] (not defined)

• Recall

P(0,Tm) = EQ∗

[1

B∗(Tm)

], m ≤ M

• By monotonicity of P(0,T ): ∃ unique nondecreasingα : [TM−1,TM ]→ [0, 1] with

1 α(TM−1) = 0 and α(TM) = 12 log B∗(T ) := (1−α(T )) log B∗(TM−1) +α(T ) log B∗(TM)

satisfies

P(0,T ) = EQ∗

[1

B∗(T )

], T ∈ [TM−1,TM ]


DamirFilipovic


LIBOR MarketModel


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2nd Step cont’d

• For T ∈ [TM−1,TM ] (B∗(T ) is FT -measurable andpositive): define T -forward measure QT ∼ Q∗ on FT by

dQT

dQ∗=

1

B∗(T )P(0,T )

• Note: dQT

dQTM= dQT

dQ∗dQ∗

dQTM= B∗(TM)P(0,TM)

B∗(T )P(0,T )

• Representation Theorem: ∃ unique σT ,TM∈ L such that

dQT

dQTM|Ft = EQTM

[B∗(TM)P(0,TM)

B∗(T )P(0,T )| Ft

]= Et

(σT ,TM

•W TM

), t ∈ [0,T ]

• Girsanov: W T (t) = W TM (t)−∫ t

0 σT ,TM(s)> ds is a

QT -Brownian motion, t ∈ [0,T ]


DamirFilipovic


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3rd Step: Backward Induction

• Fix T ∈ [TM−2,TM−1]

• Can now define forward LIBOR L(t,T ):

dL(t,T ) = L(t,T )λ(t,T ) dW T+δ(t),

L(0,T ) =1

δ

(P(0,T )

P(0,T + δ)− 1

)• This defines bounded progressiveσT ,T+δ(t) = δL(t,T )

δL(t,T )+1λ(t,T )

• T -Forward measure defined by

dQT

dQT+δ= ET

(σT ,T+δ •W T+δ

)


DamirFilipovic


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3rd Step cont’d

Note: σT ,TM= σT ,T+δ + σT+δ,TM

satisfies (t ∈ [0,T ])

dQT

dQTM|Ft =

dQT

dQT+δ|Ft

dQT+δ

dQTM|Ft

= Et(σT ,T+δ •W T+δ

)Et(σT+δ,TM

•W TM

)= Et

(σT ,TM

•W TM

)• Proceeding by backward induction ⇒ forward measure

QT , QT -Brownian motion W T for all T ∈ [0,TM ]

⇒ forward LIBOR L(t,T ) for all T ∈ [0,TM−1]


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Zero-Coupon Bonds

• Finally: obtain zero-coupon bond prices for all maturities

• Fix 0 ≤ T ≤ S ≤ TM

• Note: σT ,S = σT ,TM− σS,TM

satisfies (t ∈ [0,T ])

dQS

dQT|Ft =

dQS

dQTM|Ft

dQTM

dQT|Ft = Et

(−σT ,S •W T

)• By part “Forward Measures”: consistently define forward

price process

P(t, S)

P(t,T )=

P(0, S)

P(0,T )

dQS

dQT|Ft =

P(0,S)

P(0,T )Et(−σT ,S •W T

)


DamirFilipovic


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Forward SwapMeasure







Zero-Coupon Bonds cont’d

• In particular for t = T :

P(T ,S) =P(0,S)

P(0,T )ET(−σT ,S •W T

)• Exercise: P(t,T )

B∗(t) is Q∗-martingale

• Note: P(T ,S) may be greater than 1, unless S − T = mδfor some m ∈ N

⇒ Even though all δ-period forward LIBOR L(t,T ) arenonnegative, there may be negative interest rates for otherthan δ periods.


DamirFilipovic


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