advanced derivatives pricing

Advanced Derivatives Pricing and Applications

Dr Dale RobertsAustralian National University

Semester 2, 2014FINM3007/MATH3015/MATH6015

Dale Roberts (c) 2011-2014 Last Modified: 14/10/2014 08:19 1 / 561

http://www.maths.anu.edu.au/~robertsd

Week 1

Vectors and Matrices

Functions of Bounded Variation

Stieltjes Integration

Gaussian Random Variables



Vectors and Matrices



Vectors

• A vector (or column) of length n is an n× 1 array

x = (x1, x2, . . . , xn)′ =

x1x2...

xn

.

• The superscript ′means transpose and converts the row

[x1 x2 · · · xn]

to a column.



Vector-Vector Product

• Let x and y be vectors of length n.• We oen write x ∈ Rn and y ∈ Rn.• The dot product or inner product of x and y is given by

x · y := x ′ y =n∑

i

x i yi = x1 y1 + x2 y2 + · · ·+ xn yn.

• The norm of a vector x is given by

‖x‖ :=p

x ′x =

√

√

√

n∑

i=1

x2i



Matrices

• Anm× nmatrix is an array of objects consisting ofm rows and n columns

A= [ai j] =

a11 a12 · · · a1na21 a22 · · · a2n· · · · · · · · · · · ·am1 am2 · · · amn

• The element ai j is the entry in the ith row and jth column.• So a column of length n is simply a n× 1matrix and a row of length n is a

1× nmatrix.



Matrices• A n× n diagonal matrix is a matrix of the form

D =

d1 0 · · · · · · · · · 00 d2 0 · · · · · · 0... · · · · · · · · · · · · ...... · · · · · · 0 dn−1 00 · · · · · · · · · 0 dn

wherein all entries not on the main diagonal are zero.• Also denoted

D = diag(di)n1 = diag(d1, d2, . . . , dn)

• A special case is the identity matrix

I = diag(1, 1, . . . , 1).



Matrix-Vector Product

• The (matrix-vector) product Ax is defined to be the column of lengthmwhose entry in the ith row is

(Ax)i =n∑

j=1

ai j x j = ai1x1 + ai2x2 + · · ·+ ainxn (1)

• If A= [α1 α2 · · · αn] is a single row (i.e.,m= 1), then (1) becomes

Ax = [α1 α2 . . . αn]

x1x2...

xn

=n∑

j=1

α j x j.




• If the ith row of A is denoted by

ai = [ai1 ai2 · · · ain],

then

Ax =

(Ax)1(Ax)2...

(Ax)m

=

a1xa2x...

amx




• If A= [ai j] and B = [bi j] are twomatrices of the same size. Then

A= B⇔ ai j = bi j, ∀i, j

A± B = [ai j ± bi j]

αA= [αai j]

• The following laws of matrix-vector multplication hold:

(αA)x = α(Ax) = A(αx) (Associative law)A(αx + β y) = αAx + βAy (Distributive law)(αA+ βB)x = αAx + βBx (Distributive law)

where A, B are n×mmatrices; α, β real numbers; and x , y vectors oflength n.



Matrix-Matrix Product

• Let Abe anm× nmatrix. A j denotes the jth column of A and we write

A= [A1 A2 · · · An].

• (A)i j denotes the element of A at row i and column j. Hence, if A= [ai j]then (A)i j = ai j.• Let A= [ai j] be anm× nmatrix and B = [bi j] be a p× qmatrix withcolumns B j. The (matrix-matrix) product AB is defined if and only if n= pand in this case

AB = [AB1 AB2 · · · ABq]

is am× qmatrix.



Determinants

• The determinant of n× nmatrix A is denoted by det A= |A|.• When n= 1, det A= |[a]|= a.• When n= 2,

det

a bc d

= ad − bc.

• Determinants of matrices of size n> 2 are computed in terms ofdeterminants of matrices of size n− 1.



Determinants

• Theminor Mi j of ai j in A= [ai j] is the determinant of the submatrix of Aobtained by deleting the ith row and jth column.• The cofactor of ai j is Ci j = (−1)i+ j Mi j.• For each i, j the following hold:

|A|=n∑

k=1

aikCik (Expansion along ith row)

|A|=n∑

k=1

ak jCk j (Expansion along jth row)



Determinants

ExampleExpanding along the first row:

det A=

1 2 34 5 67 8 9

= 1

5 68 9

− 2

4 67 9

+ 3

4 57 8

= 0.

Expanding along the second column:

|A|= −2

4 67 9

+ 5

1 37 9

− 8

1 34 6

= 0.



Inverses of Square Matrices

• Amatrix B is an inverse of A if and only if AB = I and BA= I .• In this case, we write B = A−1.• A is said to be invertible or nonsingularwhen it has an inverse.

Example

A=

cos(θ ) − sin(θ )sin(θ ) cos(θ )

. A−1 =

cos(θ ) sin(θ )− sin(θ ) cos(θ )

.

You can verify that AA−1 = I and A−1A= I .



Nondecreasing Functions

• If f1 and f2 are nondecreasing functions on a closed interval [a, b] thenf1 + f2 is also a non-decreasing function but f1 − f2 need not be

• The function f := f1 − f2 is in the class of functions of bounded variation



Setup

f : [a, b]→ R

P = (x i−1, x i) : i = 1, . . . , k partition of [a, b]

P = set of all partitions of [a, b]



Variation

DefinitionTake f : [a, b]→ R, P = (x i−1, x i) : i = 1, . . . , k a partition of [a, b], andPis the set of all partitions of [a, b]. The variation V b

a ( f ) of f on [a, b] is

V ba ( f ) = sup

¦

k∑

i=1

| f (x i)− f (x i−1)| : P ∈ P©




DefinitionIf V b

a ( f ) is finite then f is of bounded variation on [a, b].

Note: If V ba ( f ) =∞ then we say that f has unbounded variation. For example,

Brownian motion has paths of unbounded variation.



Example 1

ExampleIf function f is non-decreasing on [a, b], then for any P ∈ P :

k∑

i=1

| f (x i)− f (x i−1)|=k∑

i=1

f (x i)− f (x i−1) = f (b)− f (a)

Therefore,V b

a ( f ) = f (b)− f (a).



Example 2

ExampleThe function

f (x) =

¨

0, x = 0

sin( 1x ), x > 0.

f /∈ BV[0, 2π].

0.1 0.2 0.3 0.4 0.5 0.6

-1.0

-0.5

0.5

1.0



Example 3

ExampleThe function

f (x) =

¨

0, x = 0

x sin( 1x ), x > 0.

f /∈ BV[0, 2π].

0.1 0.2 0.3 0.4 0.5 0.6

-0.2

0.2

0.4

0.6



Stieltjes Integration



Stieltjes Integral

• Useful in probability theory as it allows one to handle continuous anddiscrete random variables using the same theory• The Stieltjes generalisation of Riemann (and Lebesgue) integrals involvetwo functions• Integrating one function against the increments of another function



Stieltjes Sum

Let f : [a, b]→ R and g : [a, b]→ R. For any P ∈ P , P := I1, I2, . . . , InwithIk := (xk−1, xk) choose points zk ∈ [xk−1, xk] and form the sum

S( f , g; P) :=n∑

k=1

f (zk)(g(xk)− g(xk−1)).



Riemann-Stieljes Integral

DefinitionIf lim|P|→0 S( f , g; P) exists, then this limit is called the Riemann-Stieljes integralof f with respect to g and is denoted

∫ b

a

f (t) d g(t) or∫ b

a

f d g.

The function f is called the integrand and the function g is called theintegrator.

Note: A Lebesgue-Stieljes integral can also be constructed.



Linearity in the Integrand

LemmaIf∫ b

a f1 d g and∫ b

a f2 d g exist and α,β ∈ R, then

∫ b

a

(α f1 + β f2) d g = α

∫ b

a

f1 d g + β

∫ b

a

f2 d g.



Linearity in the Integrator

LemmaIf∫ b

a f d g1 and∫ b

a f d g2 exist and α,β ∈ R, then

∫ b

a

f d(αg1 + β g2) = α

∫ b

a

f d g1 + β

∫ b

a

f d g2.



Integration by Parts

LemmaIf f integrable with respect to g, then g is integrable with respect to f and

∫ b

a

g d f = f g

b

a−∫ b

a

f d g.



Existence Criteria

TheoremIf f ∈ C[a, b] and g ∈ BV[a, b] then

∫ b

a

f d g exists.

TheoremIf f ∈ BV[a, b] and g ∈ C[a, b], then

∫ b

a

f d g exists.



Outlook to Stochastic IntegrationWewant to be able to integrate a function f against the paths of a Brownianmotion (Wt)t≥0.

• However, each path t 7→Wt(ω) is of unbounded variation but continuous• We could potentially write

∫ b

a

f (t) dWt

if f ∈ BV[a, b] and use Stieltjes integration. This is called path-by-pathintegration.• What about

∫ b

a

sin1

t

dWt or∫ b

a

Wt dWt ?




A random variable X on a probability space (Ω,F ,P) is normally distributed orGaussianwith parameters µ andσ2, written as X ∼ N(µ,σ2), if

PX < x= Φ x −µσ

, x ∈ R

where Φ is the normal distribution function

Φ(x) :=

∫ x

−∞

1p2π

e−u2/2 du.



Characterisation

Themoment generating function characterises the class of Gaussian randomvariables.

LemmaA random variable X ∼ N(µ,σ2) under a measure P if and only if

E[eθX ] = eθµ+12θ

2σ2, for all real θ .



Gaussian Shis

Lemma (Gaussian shi)Let X ∼ N(0,1), let h be ameasurable function, and c ∈ R. Then

E[ecX h(X )] = ec2/2E[h(X + c)]



Gaussian Shis

ExampleTake h(x) = 1 then

E[eσZ] = eσ2/2

Note: σZ ∼ N(0,σ2)



Gaussian Shis

ExampleIf Z ∼ N(0,1) then X := µ+σZ ∼ N(µ,σ2) so

E[eX ] = E[eµ+σZ]

We identify h(x) = eµ so h(Z) = eµ and

E[eX ] = E[eµ+σZ] = E[eσZh(Z)] = eσ2/2eµ = eµ+σ

2/2.



Multivariate Random Variables

• Consider vector X whose elements are random variables on the sameprobability space (Ω,F ,P):

X = (X1, X2, . . . , Xn)′

• E[X i] = µi and Cov[X i, X j] = qi j for i, j = 1, . . . , n.• We call the vector µ := (µ1, . . . ,µn)′ themean and the matrix

Q := (qi j)ni, j=1 the covariance matrix of X .



Multivariate Gaussians

DefinitionAssume detQ > 0, then the (random) vector X has amultivariate normaldistribution if it has the density

φ(x) =1

p

(2π)n detQexp

−12(x −µ)′Q−1(x −µ)

, x ∈ Rn.

Wewrite X ∼ N(µ,Q) if this is the case.

For example: x ∈ Rn,

PX > x=∫ ∞

x1

∫ ∞

x2

· · ·∫ ∞

xn

φ(x1, x2, . . . , xn) d x1d x2 · · · d xn.



Multivariate Gaussians

• If Z ∼ N(0,Q) and c = (c1, c2, . . . , cn)′ ∈ Rn then X := cT Z is Gaussian with

X ∼ N(0, c′Qc).

• If C ∈ Rm×n, i.e. am× nmatrix, then we have

X = C Z ∼ N(0, CQC ′),

and CQC ′ is am×m covariance matrix.• Given a θ ∈ Rn, X is alternatively characterized by its moment generatingfunction

E[eθ′X ] = exp

θ ′µ−12θ ′Qθ

.



Multivariate Gaussian Shis

LemmaLet X ∼ N(0,Q), let h be ameasurable function of x ∈ Rn, and c ∈ Rn. Then

E[ec′X h(X )] = e12 c′QcE[h(X + c)]



Week 2

Brownian Motion

Monte Carlo Method

Simulating Stochastic Processes



Brownian Motion



Probability Space

A probability space (Ω,F ,P) consists of three parts:• the sample spaceΩ (the set of all possible outcomes),• the set of eventsF (each event is a set containing zero or more outcomes),

Example (Coin flip)Ω= H, T.F contains 4 events: H, T, (neither head nor tails) andH, T (either heads or tails).F = , H, T, H, T. Then

P() = 0, P(H) = P(T) =12

P(H, T) = 1



The Set of Events

The set of events F (the set of all subsets ofΩ) is aσ-algebra as:• F contains the sample spaceΩ ∈ F• F is closed under complements: if A∈ F then (Ω \ A) ∈ F• F is closed under countable unions: if A1, A2, . . . ∈ F then ∪iAi ∈ F



Stochastic Processes

DefinitionA stochastic process X is a family (X t)0≤t≤T of random variables defined on thesame probability space (Ω,F ,P) taking values inRn (the state space).

The process X can be viewed in terms of:• a (random) path t 7→ X t(ω)• the probability distribution of X at some fixed time t , i.e., PX t < x.



Finite-dimensional Distributions

DefinitionIf X := (X t)0≤t≤T is aRn-valued stochastic process, then for every finitesequence of times t1, t2, . . . , tk the probability distribution of the randomvariable

X := (X t1, X t2

, . . . , X tk)′

is called the finite-dimensional distribution of X .

Note: Since X isRn valued and we are evaluating at k times t1, . . . , tk thenX isRn×k-dimensional.



Filtrations and Adaptedness

DefinitionGiven a probability space (Ω,F ,P), a filtration is an increasing collection ofevents (Ft)t≥0 and bounded above byF , i.e.,

Fs ⊆Ft ⊆F , s < t

We say that a stochastic process X = (X t)t≥0 is adapted to the filtration if, forevery time point t , the random variable X t “only relies on the information upto time t , i.e.,Ft”.Examples: 2X1 is adapted toF1 but X1 is not adapted toF1/2.



Brownian Motion

DefinitionA stochastic processW := (Wt)t≥0 is a called a Brownianmotion (under P) if ithas all of the following properties:1. the paths t 7→Wt(ω) are continuous for P-almost allω ∈ Ω;2. W (0) = 0, P-almost surely;3. for 0≤ s < t <∞,Wt −Ws ∼ N(0, t − s);4. for k ∈ N and 0≤ t1 < t2 < . . .< tk, the incrementsWt i+1

−Wt iwith i < k,

are independent.

We shall denote byFt = (Ft)t≥0 the natural filtration generated by theprocessW : the event setF is generated by all values ofWs up to time s = t .



Existence

Theorem (Wiener 1923)Brownianmotion exists.

In homage to Norbert Wiener, a Brownian motion is also known as aWienerprocess.

Trivia:Wiener (1894 - 1964) was rejected for a job at University of Melbourne.Ended up getting a job at MIT where he stayed the rest of his life.



Simple Properties

The following properties of a Brownian motionW can easily be proved:• Wt ∼ N(0, t);• Cov[Ws, Wt] =min(s, t);• E[eθWt ] = e−

12θ

2 t ;• E[W 4

t ] = 3t2.



Invariances

Brownian motion (BM) also has a number of useful invariance properties. Inparticular, ifW is a BM then so are the following processes:• the reflected process X t := −Wt ;• the scaled process X t := 1

c Wc2 t for c > 0;• the stationary process X t :=Wt+u −Wu for u> 0;• the inverted process X t := tW1/t for t > 0 and X0 =W0 = 0.



Donsker’s Theorem

Theorem (Donsker 1951)Let (ξn)n∈N be a sequence of independent and identically distributed randomvariables with E[ξi] = 0 and 0< Var[ξi] = σ2 <∞. Let

S0 = 0, Sn =n∑

i=1

ξi.

We construct a sequence of Xn of stochastic processes by

Xn(t,ω) =1σp

nS[nt](ω) + (nt − [nt])

1σp

nξ[nt]+1(ω)

for t ∈ [0,1], n ∈ N. Then this sequence converges (weakly) towards theBrownianmotion B := (Bt)0≤t≤1, i.e., we have Xn→ B in probability dist.



Monte Carlo Method



Main Idea

The main idea of the Monte Carlo method is to approximate an expected valueE[X ] by an arithmetic average of the results of a big number of independentexperiments which all have the same distribution as X .

Trivia: In 1946, physicists needed to perform complicated calculations for theManhattan project. They couldn’t solve the problem using standardmathematical methods. The Mathematician Stanislaw Ulam (1909 - 1984)suggested using random experiments. Being a secret project, this required acode name. Von Neumann chose the nameMonte Carlo.



Strong Law of Large Numbers

• One of the most powerful theorems of probability theory• Has been a central object of research during the history of the subject• It is the basis of the Monte Carlo method



Strong Law of Large Numbers

Theorem (SLLN)If (Xn)n∈N be a sequence of integrable,R-valued random variables that areindependent, identically distributed and defined on the probability space(Ω,F ,P). Let µ= E[X1]. Then, we have for P-almost allω ∈ Ω

1n

n∑

i=1

X i(ω)→ µ n→∞

The arithmetic mean of the (realisations of) X i tends to the theoretical mean µof every X i.



Crude Monte Carlo Method

Approximate E[X ] by the arithmetic mean

1N

N∑

i=1

X i(ω)

for some finite N ∈ N.

Here, the X i(ω) are the results of N independent experiments that the sameprobability distribution as X .



Unbiased Estimator

TheoremLet (Xn)n∈N be a sequence of integrableR-valued random varaibles that are i.i.d.as X . Then the Monte Carlo estimator

X N :=1N

N∑

i=1

X i, N ∈ N,

is an unbiased estimator for µ= E[X ], i.e., we have

E[X N] = µ.

Note: This ensures that the MC Estimator is correct in the mean, it does nothelp us get a feel for the absolute value of the error.



Increasing Accuracy

• We look at the standard deviation of the dierence between X N and µ• As

Var[X N −µ] = Var[X N] =1

N2

N∑

i=1

Var[X i] =σ2

N

the standard deviation is of orderO(1/p

N)• This means that to increase the accuracy of the crude MC estimate by onedigit (i.e., reduce the std. dev by a factor of 0.1) requires increasing N by afactor of 100.• To acheive higher accuracy, we need a significant eort.



Central Limit Theorem

Theorem (CLT i.i.d case)Let (Xn)n∈N be a sequence of i.i.d. R-valued random variables on (Ω,F ,P).Assume that they all have finite varianceσ2 = Var[X ]. Then, the normalised andcentralised sum of these random variables converges in distribution to thestandard normal distribution:

1p

Nσ

N∑

i=1

X i − Nµ

D→ N(0, 1) as N →∞.

We can infer from the CLT that for large values of N the crude Monte Carloestimator is approximatelyN(µ,σ2/N)-distributed.



Approximate Confidence Interval

An approximate (1−α)-confidence interval for the expectation µ is

1N

N∑

i=1

X i − z1−α/2σp

N,

1N

N∑

i=1

X i + z1−α/2σp

N

where z1−α/2 is the 1−α/2-quantile of the standard Normal distribution.

As 97.5%-quantile is about 1.96, an approximate 95% confidence is

1N

N∑

i=1

X i − 2σp

N,

1N

N∑

i=1

X i + 2σp

N



Approximate Confidence Interval

Typically, the standard deviationσ is unknown. We need to estimateσ2 by thesample variance

σN =

√

√

√ 1N − 1

N∑

i=1

(X i − X N )2 =

√

√

√ NN − 1

1N

N∑

i=1

X 2i − X 2

N

Giving our approximation of 95% confidence

1N

N∑

i=1

X i − 2σNp

N,

1N

N∑

i=1

X i + 2σNp

N



CalculatingπExample (π)The classic example of the Monte Carlo method is calculating the value ofπ.Consider the part of the unit circle C with center at the origin (0,0) thatintersects with the positive unit square [0, 1]2. Our experiment is to randomlychoose points p1, p2, . . . , pN of the unit square (i.e., uniformly distributed) andconsider

X i = 1pi∈C.

We have then P(pi ∈ C) = π/4 as the probability of hitting C is it’s areaπr2/4and r = 1. Since

E[1pi∈C] = P(pi ∈ C) = π/4

we can estimateπ by

π(ω) =4N

N∑

i=1

1pi∈C(ω).



Monte Carlo Integration

Example (MC Integration)Suppose you wanted to calculate the value of the deterministic n-dimensionalintegral of the form

∫

[0,1]ng(x) d x .

Introduce the density f (x) of the n-dimensional Uniform distribution on[0,1]n via

f (x) = 1x∈[0,1]n(x), x ∈ Rn

then

I =

∫

[0,1]ng(x) d x =

∫

g(x) f (x) d x = E[g(X )]



Monte Carlo Integration

We approximate by

IN (ω) =1N

N∑

i=1

g(X i(ω))

where X1, . . . , XN are all i.i.d. uniformly distributed random variables on [0, 1]n

Remark: By the SLLN the rate of convergence isO(1/p

N) but is independentof the dimension n. As opposed to deterministic quadrature formulas thathave a rate of convergenceO(N−2/n). The MC approximation outperformswhen n> 4.




• In practice, we oen need to calculate expected values by the Monte Carlomethod• If X = (X t)0≤t≤T is a stochastic process, we oen want to calculate

µ= E[g(X )] = E[g(X t , 0≤ t ≤ T )]

where g is a function of the whole path of X .



Algorithm for Stochastic Processes

Approximate E[g(X )] by the arithmetic mean

1N

N∑

i=1

g(X i(ω)).

where X1, X2, . . . are independent replications of paths of the stochasticprocess X .



Special Cases

• If the function g only depends on the value of the stochastic process X at aparticular time, i.e., if we have

g(X ) = h(XT )

for a fixed time T andR-valued function h, then we only have to know thedistribution of XT . Either the distribution of XT is known, or we have tosimulate the whole path to get XT .• If g only depends on X at a finite number of times t1, t2, . . . , tn, i.e.,

g(X ) = h(X t1, . . . , X tn

)

for some function h then we are in the random variable case.



General Case

If g cannot be reduced to one of the special cases then we need to simulatethe true path of X . Unfortunately, this is not possible and we need to findapproximation methods. There is no general approach andmethods have tobe tailored to the specific problem.



Simulating Paths: Basics

• The fact that the random variables X t for all t ’s are related to each otherhas the consequence that we cannot simply simulate independentrandom variables that have the same distribution as the dierent X t

• We have to take care of the relation among the X t which can be extremelystrong• Since the times t ∈ [0, T] are uncountable, we cannot simulate the truepath of the process X . We need to approximate the path at finite times0= t1 < t2 < · · ·< tn = T



Algorithm

Let 0= t0 < t1 < · · ·< tn = T be a partition of [0, T]. Let Pk denote theconditional distribution of X tk

given X tk−1. We obtain the approximate path

X (ω) by:1. Set X0(ω) = 0

2. For k = 1 to n do:(a) Simulate a random number Yk(ω)with Yk ∼ Pk(b) Set X tk

(ω) = X tk−1(ω) + Yk(ω)

(c) Between tk+1 and tk obtain X t via linear interpolation

X t(ω) = X tk−1(ω) +

t − tk−1

tk − tk−1Yk(ω), t ∈ (tk−1, tk).



Brownian Motion

Let 0= t0 < t1 < · · ·< tn = T be a partition of [0, T]. We obtain theapproximate path t 7→ Bt(ω) by:1. Set B0(ω) = 0

2. For k = 1 to n do:(a) Simulate a random number Zk ∼ N(0,1)(b) Set Btk

(ω) = Btk−1(ω) +

p

tk − tk−1Zk(c) Between tk=1 and tk obtain X t via linear interpolation

Bt(ω) = Btk−1(ω) +

t − tk−1

tk − tk−1(Btk(ω)− Btk−1

(ω)), t ∈ (tk−1, tk).



Week 3

Stochastic Integration

Stochastic Dierential Equations

Simulating Solutions of SDE



Stochastic Integration



Simple Functions

Let F= (Ft)t≥0 and h an F-adapted process onΩ× [0, T] taking values inR.

DefinitionWe call h a simple process if for a partitionπ := 0= t0 < t1 < · · ·< tn = TandFt i

-measurable random variables (hi)ni=1, the process h(t) := h(ω, t)satisfies

h(ω, t) =n−1∑

i=0

hi(ω)1(t i ,t i+1](t) for 0≤ t ≤ T andω ∈ Ω

Note: In the easy case, h is a simple function given by h(t) =∑n−1

i=0 hi1(t i ,t i+1](t)for 0≤ t ≤ T where hi ∈ R for i = 1, . . . , n.



Properties

• Simple process and functions allow us to approximate arbitrary processesand functions• We approximate the “area under the curve” by rectangles• We build up the concept of stochastic integrals by starting with the casefor simple processes



Stochastic Integral for Simple Processes

DefinitionThe stochastic integral (aka. Itô integral) of a simple process h is defined as

∫ T

0

h(t) dWt =n−1∑

i=0

hi(ω)(Wt i+1−Wt i

)

whenever E[h2i ]<∞ for all i = 0, . . . , n− 1.



Square-integrable Simple Processes

• We required E[h2i ]<∞ for all i = 0, . . . , n− 1 for the stochastic integral

to exist.• WewriteH 2

[0,T] for the set of all F-adapted simple processes h that satisfy

∫ T

0

E[h2] d t <∞.



Properties of Stochastic Integral

TheoremFor simple processes h ∈H 2

[0,T] we have

1. E

∫ T

0

h(t) dWt

= 0

2. E

∫ T

0

h(t) dWt

2

=

∫ T

0

E[h(t)2] d t



Proof of 1

Since,∫ T

0

h(t) dWt =n−1∑

i=0


)

taking expectations of both sides and from the fact that

E[Wt i+1−Wt i

] = 0

gives the result.



Proof of 2From the definition, we have

∫ T

0

h(t) dWt =n−1∑

i=0


).

So

E

∫ T

0

h(s) dWs

2

= E

n−1∑

i=0

hi (Wt i+1−Wt i

)2

= E

n−1∑

i=0

h2i (Wt i+1

−Wt i)2

+ 2E

n−1∑

i, j=0i 6= j

hih j (Wt i+1−Wt i

)(Wt j+1−Wt j

)



Proof of 2

For i < j, by the tower property of conditional expectations

E[hih j(Wt i+1−Wt i

)(Wt j+1−Wt j

)] = E[hih j(Wt i+1−Wt i

)E[(Wt j+1−Wt j

)|Ft j]] = 0

Moreover,

E[h2i (Wt i+1

−Wt i)2] = E[h2

i E[(Wt i+1−Wt i

)2|Ft i]] = E[h2

i ](t i+1 − t i)

Substituting, we obtain

E

∫ T

0

h(t) dWt

2

=n−1∑

i=0

E[h2i ](t i+1 − t i) =

∫ T

0

E[h(t)2] d t



Martingales

DefinitionA F-adapted stochastic process X := (X t)t≥0 is amartingale if E[|X t |]<∞ foreach t > 0 and E[X t |Fs] = Xs for all s ≤ t .

The following processes are examples of martingales:• the Brownian motionW ;• X t :=W 2

t − t ;• Et := exp(cWt −

12 c2 t), where c ∈ R.



Stochastic Integrals as Martingales

LemmaIf h is a simple process inH 2

[0,T], then the process

X t :=

∫ t

0

h(s) dWs, t ≤ T

is a martingale.



Construction

• We have seen how to construct stochastic integrals of simple processes• This construction can now be extended to the full stochastic integralunder some conditions on the integrand h• The construction uses the idea that we can approximate the integrand hby simple processes and then take the limit• The full mathematical proof is about taking care of all the technicalaspects of convergence to the limiting object and existence of this object



Square-integrable Processes

Let h := (ht)0≤t≤T be a stochastic process. We say that h is in the set of H2[0,T]

processes if, for all t ∈ [0, T], we have

E

∫ t

0

|h(s)|2 ds

<∞.

Note: Processes in H2[0,T] can be approximated arbitrarily close by simple

processes inH 2[0,T].



Construction

TheoremLetW be a BM. For every h ∈ H2

[0,T] the stochastic integral

It(h) :=

∫ t

0

h(s) dWs, t ≤ T (2)

exists.



Properties

LetW be a BM. Then the stochastic integral given by (2) has the followingproperties:• linearity: It(αh+ β g) = αIt(h) + β It(g) for h, g ∈ H2

[0,T] and α,β ∈ R;

• If h ∈ H2[0,T] then I is a continuous martingale;

• The Itô isometry holds:

E

∫ T

0

h(t) dWt

2

=

∫ T

0

E[|h(t)|2] d t.



Itô Processes

An Itô process is the sum of an absolutely continuous dri plus a continuouslocal martingale of the form

X t = X0 +

∫ t

0

µ(s) ds+

∫ t

0

σ(s) dWs, (3)

whereσ ∈ H2[0,T] and µ is a progressively measurable process satisfying

∫ t

0 |µ(s)| ds <∞ for all t > 0, such that the above integrals are defined.We also write

dX t = µ(t) d t +σ(t) dWt .



Identifying Martingales from Itô Processes

If X is an Itô process with representation dX t = µ(t) d t +σ(t)dWt whichsatisfies E[(

∫ T

0 σ2s ds)1/2]<∞, then

X is a martingale ⇐⇒ X is driless (µt ≡ 0).



Novikov’s Condition

If dX t = σt X t dWt for someF -previsible processσt , then

E

exp

12

∫ T

0

σ2s ds

<∞ ⇒ X is a martingale.



Quadratic covariation

In addition to (3), let

Yt = Y0 +

∫ t

0

ν(s) ds+

∫ t

0

ρ(s) dWs,

be another Itô process defined in terms of the same Brownian motionW . Thequadratic covariation of the processes X and Y is given by

⟨X , Y ⟩t =∫ t

0

σ(s)ρ(s) ds,

and ⟨X , X ⟩ is called the quadratic variation of the process X .



Product RuleGiven X t and Yt adapted to the same Brownian motionW ,

dX t = σt dWt +µt d t,dYt = ρt dWt + νt d t.

Thend(X t Yt) = X t dYt + Yt dX t +σtρt d t

In the other case, if X t and Yt are adapted to two dierent and independentBrownian motionsW and W ,

dX t = σt dWt +µt d t

dYt = ρt dWt + νt d t

Thend(X t Yt) = X t dYt + Yt dX t




Let µ(t, x) andσ(t, x) bemeasurable functions of t ∈ [a, b] and x ∈ R.Consider the stochastic dierential equation (SDE)

dX t = µ(t, X t) d t +σ(t, X t) dWt , Xa = ξ ∈ R, (4)

which should be interpreted as the stochastic integral equation

X t = ξ+

∫ t

a

µ(s, Xs) ds+

∫ t

a

σ(s, Xs) dWs, a ≤ t ≤ b. (5)



Solutions

A stochastic process X := (X t)t≥0 is a (strong) solution to (4) if it satisfies:• The stochastic processσ(t, X t) ∈ H2 so that

∫ t

a

σ(s, Xs) dWs

is a well-defined Itô integral for each t ∈ [a, b];• Almost all sample paths of µ(t, X t) satisfy

∫ b

a

|µ(s, Xs)| ds <∞;

• For each t ∈ [a, b], (5) holds almost surely.



Existence and Uniqueness

To obtain solutions of (4), one needs to impose conditions on the functionsµ(t, x) andσ(t, x). Depending on the conditions imposed one can obtain:• existence of (potentially multiple) solutions;• existence of a unique solution;



Lipschitz and Linear Growth

We say that a measurable function f (t, x) on [a, b]×R satisfies the Lipschitzcondition in x if there exists a constant K > 0 such that

| f (t, x)− f (t, y)| ≤ K |x − y|, ∀a ≤ t ≤ b, x , y ∈ R.

We say that a measurable function f (t, x) on [a, b]×R satisfies the Lineargrowth condition in x if there exists a constant K > 0 such that

| f (t, x)| ≤ K(1+ |x |), ∀a ≤ t ≤ b, x ∈ R.



Existence of Unique Solution

TheoremSuppose that µ(t, x) andσ(t, x) satisfy the Lipschitz and linear growthconditions in x for all t ≥ 0. Then for every initial time a ∈ R+ and point ξ ∈ R,there exists a unique solution X of the SDE (4).



Simulating Solutions of SDE



Overview

• As with ODE, most SDE do not admit explicit solutions• Therefore, we need to consider numerical methods to solve them• However, there are fundamental dierences between ODE and SDE



Dierences between ODE and SDE

The twomain dierences between ODE and SDE are:• A solution to an SDE is a (function-valued) random variable, and thus weobtain dierent solutions for dierentω ∈ Ω.• A (strong) solution to an SDE is not smooth, as the underlying Brownianmotion is not smooth at all.

The second dierence means that numerical schemes for ODEs that rely onthis property are not automatically good when adapted to the SDE situation.



Goal of Approximation

Numerical schemes for SDE depend on our specific goal. Are we interested in• obtaining a path t 7→ eX (t,ω) that is as close as possible to the (unknown)solution path t 7→ X (t,ω),• or in computing an expectation of a functional E[g(X )] of the SDE?

The first is called “strong approximation” and the second is “weakapproximation”.



Exact Simulation

In some rare cases, we can find an explicit solution to the SDE. For example,the SDE

dX t = aX t d t + bX t dWt , X0 = x ,

has the explicit solution

X t = x exp

(a−12

b2)t + bWt

= f (t, Wt)

Then calculating E[g(XT )] for some function g : R→ R is no dierent fromsimulating a random variable.



Approximating Paths

• The task becomes a lot more diicult when we want to calculate theexpectation of the function of the whole path t 7→ X t

• Even in the case of Brownian motion, we need to find a way toapproximate the path• We follow the same approximation idea: approximate the paths on anpartition of the interval [0, T] by simulating the Brownian motion on thatpartition and then constructing the values of X t on the partition



Approximating Paths

TheoremAssume that the SDE

dX t = a(t, X t) d t +σ(t, X t) dWt , X0 = x

has an explicit solution of the form X (t) = f (t, Wt)with f a continuous,real-valued function. Let eXn be an approximation of X that is constructed by

eXn(t) = f (t, Wt), if t = iT/n for some i = 0, 1, . . . , n

and extended to all t ∈ [0, T] by linear interpolation. Then, for each boundedand continuous function g of the path of X , we have

E[g(eXn)]→ E[g(X )], n→∞



Strong Approximation

Consider the SDE

dX t = a(t, X t) d t +σ(t, X t) dWt , X0 = x

and consider a partitionπ= 0= t0 < t1 < · · ·< tn = T of the interval[0, T]. We approximate the SDE by X0 = x and for i = 0, . . . , n− 1 by

X t i+1= X t i

+ a(t i, X t i)(t i+1 − t i) +σ(t i, X t i

)(Wt i+1−Wt i

).

This is called the Euler-Maruyama scheme.



Euler-Maruyama Scheme

Let∆t := T/N for a given N so that t i = i∆t . Then approximate the SDE by:1. Set eX (0) = X (0) = x2. For i = 0 to N − 1 do

(a) Simulate a Zi ∼ N(0, 1)(b) Set

eX t i+1= eX t i

+ a(t i, eX t i)∆t +σ(t i, eX t i

)p∆tZi



Week 4

Itô’s Formula

Multidimensional Stochastic Calculus

Stochastic Exponential

Black-Scholes Market



Itô’s Formula



Chain Rule

• Remember the (calculus) chain rule states that

dd t

f (g(t)) = f ′(g(t))g ′(t)

• Or for a function f (t, x t) in terms of total derivatives

d fd t=∂ f∂ t

d td t+∂ f∂ x

d x t

d t

andmultiplying throught by d t gives

d f (t, x t) =∂ f∂ t(t, x t)d t +

∂ f∂ x(t, x t)d x t



Itô’s Formula

• Itô’s formula is one of the most useful tools in stochastic calculus• It is a stochastic version of the chain rule• In the stochastic version, we get an extra “correction” term



Itô’s Formula for Brownian Motion

Let (Wt)t≥0 be a BM. What are the dynamics of f (t, Wt)?

TheoremFor a function f : [0, T]×R such that the partial derivatives ∂ f

∂ t ,∂ f∂ x and

∂ 2 f∂ x2

exist and are continuous and for which ∂ f∂ x ∈ H2

[0,T] we have

f (t, Wt)− f (0,0) =

∫ t

0

∂ f∂ x(s, Ws) dWs+

∫ t

0

∂ f∂ s(s, Ws) ds+

12

∫ t

0

∂ 2 f∂ x2(s, Ws) ds

Alternatively written:

d f (t, Wt) =∂ f∂ x(t, Wt) dWt +

∂ f∂ t(t, Wt) d t +

12∂ 2 f∂ x2(t, Wt) d t



Applications

• E[W nt ]

• Solving SDEs



Itô’s Formula for ProcessesLet (X t)t≥0 satisfy dX t = µ(t, X t) d t +σ(t, X t) dWt . What are the dynamics off (t, X t)?

TheoremFor a function f (t, x) that is twice dierentiable in x and once in t . Then forYt = f (t, X t), we have

dYt =∂ f∂ t(t, X t) d t +

∂ f∂ x(t, X t) dX t +

12σ(t, X t)

2∂2 f∂ x2(t, X t)d t

Expanding out the dX t term,

∂ f∂ t(t, X t) +µ(t, X t)

∂ f∂ x(t, X t) +

12σ(t, X t)

2∂2 f∂ x2(t, X t)

d t+σ(t)∂ f∂ x(t, X t) dWt



Multidimensional Stochastic Calculus



Multidimensional Brownian Motion

DefinitionA standard n-dimensional Brownian motionW := (W (t))t≥0 is defined as

W (t) =

W1(t)W2(t)...

Wn(t)

and satisfies the following two properties:1. Each of the processes (Wj(t))t≥0 with j = 1, . . . , n, is a one-dimensionalBrownian motion.

2. The Brownian motionsWi andWj are independent for i 6= j and indexvalues i, j = 1, . . . , n.



Multidimensional Brownian Motion

Correlation of Brownian motions can be modeled using weighted sums ofindependent Brownian motions or multiplication by a covariance matrix.

ExampleLet (W1(t))t≥0 and (W2(t))t≥0 be two independent Brownian motions. For−1≤ ρ ≤ +1, let

Wρ(t) = ρW1(t) +Æ

1−ρ2W2(t).

Then (Wρ(t))t≥0 is a Brownian motion and

E(W1(t)Wρ(t)) = ρ t.



Multidimensional Stochastic Integration

• Wewould like to extend the idea of stochastic integration to multipledimensions.• To handle things like:

∫ T

0

W1(t) dW2(t) or∫ T

0

sin(W 21 (t) +W 2

2 (t)) dW2

• Wewill use vectors andmatrices.



Multidimensional Stochastic Integration

• Let H2[0,T] be the set ofR

n-valued adapted processes

h(t) = [h1(t) h2(t) · · · hn(t)]

such that

E

∫ ∞

0

‖h(t)‖2 d t

<∞.

Recall: ‖x‖=q

x21 + x2

2 + · · ·+ x2n.



Multidimensional Stochastic Integration• Let h beRn-valued, (W (t))t≥0 be a n-dimensional Brownian motion, andtimes S < T .• For every h ∈ H2

[0,T] one can define the stochastic integral

IT :=

∫ T

S

h(t) dW (t) =n∑

i=1

∫ T

S

h j(t) dWj(t).

• If h ∈ H2[0,T] then the process (It)t≥S is a martingale and

E

∫ T

S

h(t) dW (t)

2

= E

∫ T

S

‖h(t)‖2 d t

.

• I is one-dimensional in this case (i.e., h a vector).



Multidimensional Stochastic Integration• Let H be am× nmatrix of progressive processes

H(t) =

h11(t) · · · h1n(t)...

...hm1(t) · · · hmn(t)

.

• If hi j ∈ H2[0,T] then

I =∫ T

S

H(t) dW (t) =

∫ T

S

h11(t) · · · h1n(t)...

...hm1(t) · · · hmn(t)

dW1(t)...

dWn(t)

• In other words,I is am×1matrix (i.e., column vector) whose ith elementis given by

(I )i =n∑

j=1

∫ T

S

hi j(t) dW (t).



Multidimensional Diusion Processes

• Since we have defined multidimensional stochastic integration, we cannow consider multidimensional diusion processes or Itô processes.• Of the form

dX t = a(t) d t + b(t) dW (t), X (0) = x ∈ R

where a(t) ∈ R, b(t) = [b1(t) b2(t) · · · bn(t)]withW an n-dim BM.• Or of the form

dX t = a(t) d t + B(t) dW (t), X (0) = x ∈ Rn

where a(t) ∈ Rn, B(t) ∈ Rm×n, andW anm-dim BM.



Multidimensional Diusion Processes• A stochastic dierential with dri and diusion coeicient a and b for X incomponent form is given by

dX i(t) = ai(t) d t +m∑

j=1

bi j(t)dW j(t),

for t in [0, T] and i = 1, . . . , n.• The corresponding integral form for the components is

X i(t) = X i(0) +

∫ t

0

ai(s)ds+m∑

j=1

∫ t

0

bi j(s)dW js ,

for t in [0, T] and i = 1, . . . , n.• The stochastic dierential is just an abbreviation for the correspondingstochastic integral equation.



Multidimensional Diusion Processes• The SDE in vector/matrix form is given by

dX (t) = a(t)d t + b(t)dW (t), X (0) = x ∈ Rn

• The column vectors X (t), a(t) andW (t) are given by

X (t) =

X1(t)...

Xn(t)

, a(t) =

a1(t)...

an(t)

andW (t) = [W1(t), . . . , Wm(t)]T .

• Thematrix b(t) is given by

b(t) =

b11(t) . . . b1m(t)... . . . ...

bn1(t) . . . bnm(t)

.



Quadratic Variation

• Consider the n-dimensional Itô process

X (t) = X (0) +

∫ t

0

a(s) ds+

∫ t

0

B(s) dW (s).

• The quadratic variation of the component process X i(t), for i = 1, . . . , n, isthe random integral given by

⟨X i⟩t =∫ t

0

Bi(s)′Bi(s)ds

where Bi(t) is the ith column of B(t).



Quadratic Covariation• Consider the n-dimensional Itô process

X (t) = X (0) +

∫ t

0

a(s) ds+

∫ t

0

B(s) dW (s).

• The quadratic covariation of the component processes X i and X j, fori, j = 1, . . . , n, is the random integral given by

⟨X i, X j⟩(t) =∫ t

0

Bi(s)′B j(s)ds.

• The stochastic dierential for ⟨X i, X j⟩(t) is denoted by d⟨X i, X j⟩t and

d⟨X i, X j⟩t = Bi(t)′B j(t) d t.



Multidimensional Itô Formula• The multi-dimensional Itô formula gives the stochastic dierential for theprocess Y (t) = f (t, X (t)) for some transformation function

f (t, x) = ( f1(t, x), . . . , fm(t, x))′

• The function f is assumed to map from [0,∞)×Rn→ Rm and be of classC1,2(R,Rn): i.e., the functions (t, x)→ ∂ f (t,x)

∂ t and (t, x)→ ∂ 2 f (t,x)∂ x i∂ x j

arecontinuous on (0,∞)×Rn for i, j = 1, . . . , n.• Itô’s formula states that the stochastic dierential for

Y (t) = (Y1(t), Y2(t), . . . , Ym(t))′ for t in [0, T] is given by

dYk(t) =∂ fk(t, X (t))

∂ td t +

n∑

i=1

∂ fk(t, X (t))∂ x i

dX i(t)

+12

n∑

i, j=1

∂ 2 fk(t, X (t))∂ x i∂ x j

d⟨X i, X j⟩(t).



Multidimensional Itô Formula

• Themulti-dimensional Itô formula using dri and diusion coeicients isgiven by

dY (t) = d f (t, X (t)) =

∂ f (t, X (t))∂ t

+n∑

i=1

ai(t)∂ f (t, X (t))

∂ x i

+12

n∑

i, j=1

bi(t)T b j(t)

∂ 2 f (t, X (t))∂ x i∂ x j

!

d t

+m∑

j=1

n∑

i=1

bi j(t)∂ f (t, X (t))

∂ x i

dWj(t).

• This is obtained by substitution of the dierentials for dX i(t) andd

X i, X j

(t) using the covariation form of Itô’s formula.



Change of Measure

• Girsanov’s theorem explains how the dynamics change for ann-dimensional Brownian motion under a change in measure.• The change in measure from P to some other measureQ is related to whatis called the Radon-Nikodym derivative for the twomeasures.



Change of Measure• Let (W (t))t≥0 be anm-dimensional Brownian motion under P.• Let (θ (t))t≥0 be an F-adapted n-dimensional diusion process thatsatisfies the Novikov condition.• Then there is a measureQ with the properties:

1. Q is equivalent to P.2. The processfW with

fW (t) =W (t) +

∫ t

0

γ(s) ds

is anm-dimensional BM underQ.• In dierential form we have

dfW (t) = dW (t) + γ(t)d t.



Girsanov’s Theorem

Theorem (Girsanov)Let γ ∈ H2

[0,T] and X satisfy dX t = γ(t) dWt . If Et(X ) is a uniformly integrablemartingale with E∞(X )> 0 then

dQdP= ET (X ) = exp

−n∑

i=1

∫ T

0

γi(s) dW is +

12

∫ T

0

γ2i (s) ds

defines an equivalent probability measureQ∼ P and the process

fW (t) :=W (t) +

∫ t

0

γ(u) du

is aQ-Brownianmotion.



Martingale Representation Theorem

• Suppose we have two n-dimensional diusion processes X (t) and Y (t)given by

dX (t) = H(t) dW (t) dY (t) = F(t) dW (t).

• H and F are n× n progressive processes.• Themartingale representation theorem states that there exists a uniquematrix process Φ(t) = [φi j(t)] such that

dY (t) = Φ(t) dX (t) or dYi(t) =n∑

j=1

φi j(t)dX j(t).




The stochastic exponential of an Itô process X is defined as

Et(X ) = eX t−12 ⟨X ,X ⟩t

The stochastic exponential Et(X ) has the following properties:• Et := Et(X ) is a positive Itô process and the unique solution to the SDE

dEt = Et dX t , E0 = eX0;

• E (X ) is a continuous local martingale if X is a local martingale;• E (0) = 1;• E (X )E (Y ) = E (X + Y )e⟨X ,Y ⟩;• E (X )−1 = E (−X )e⟨X ,X ⟩.



Black-Scholes Market



Black-Scholes MarketThe Black-Scholes market typically consists of a risky asset S whose pricedynamics is given by the solution of the SDE

dSt = St(µ d t +σ dWt), S0 = 1

and a risk-free asset B with value Bt satisfying

dBt = Bt r d t B0 = 1.

Writing dSt = St dX t with

dX t = µ d t +σ dWt , X0 = 0,

it follows from the stochastic exponential that

St = S0E (X ) = S0 exp(µt − 12σ

2 t).



Self-financing PortfoliosA portfolio or trading strategy is a pair of processes (φ,ψ) that give the amountof holding at time t of the asset pair (S, B). The value process of this portfolio is

V (t) = φ(t)S(t) +ψ(t)B(t)

The portfolio is called self-financing for (S, B) if∫ t

0

|φ(u)| dSu <∞ for all t > 0,

and there is no inflow or outflow of capital during the trading. In other words,the change in value of the portfolio over any time period is only due to thechanges in the value of B and S:

dV (t) = φ(t)S(t) +ψ(t)B(t).



Arbitrage Portfolio

A arbitrage portfolio is a self-financing portfolio (φ,ψ)with value process Vsatisfying

V (0) = 0, V (T )≥ 0, PV (T )> 0> 0,

for some T > 0.

If no arbitrage portfolio exists then we say that the model is arbitrage-free.



Martingale Measures

We call a measureQ an equivalent martingale measure to P (EMM), denotedQ∼ P, if the discounted price process

Z(t) :=S(t)B(t)

is aQ-martingale.



Fundamental Theorem of Asset Pricing

TheoremLetQ be the EMM. Suppose we have a contingent claim XT ∈ FT . IfEQ[X 2

T ]<∞ then the claim is replicable and the value at time t of anyreplicating portfolio is given by

Vt = BtEQ[B−1T XT |Ft].

In particular, under the Black-Scholes model, the fair price at time zero for theoption is

V0 = EQ[e−rT XT ]

where XT = f (ST ) and ST = S0 exp((r − 12σ

2)T +σfWT )wherefW is aQ-BM.



Week 5

Black-Scholes Model

Definition of Implied Volatility

Empirical Observations

Arbitrage Bounds

Asymptotics

Approximating Implied Volatility



Black-Scholes Model



Black-Scholes Model

Black-Scholes ModelUnder real-world measure P, we assume there exists deterministic r, µ, andσsuch that

Bt = exp(r t)St = S exp(σWt +µt),

where r is the risk-free interest rate,σ is the stock volatility and µ is the stockdri, and S := S0 is the current stock price.

There are no transaction costs and both instruments are freely andinstantaneously tradable either long or short at the price quoted.



Black-Scholes Formula

Explicit solution was obtained by Black and Scholes (1973) relying heavily onthe notion of no-arbitrage in Merton (1973).

Black-Scholes-Merton (BSM) Formula

C = SΦ(d1)− e−rT KΦ(d2)

where

d2 :=log( S

K ) + (r −12σ

2)T

σp

T, d1 := d2 +σ

pT

Jointly awarded Nobel prize in economics in 1997.



Benchmark Model

• Widely acknowledged that assumptions underlying the BSMmodel are farfrom realistic• BSMmodel enjoys unrivalled popularity in practice due to its role as abenchmark model• Provides a convenient mapping device from the space of option prices to asingle real number called implied volatility (IV)



Definition of Implied Volatility



Implied Volatility

• Unknown parameter in the BSM formula isσ: the volatility• “Backed out” of market option prices, IV gives a straight-forwardcomparison of the relative expensiveness of options across various strikes,expiries, and underlying assets.• It is interesting to plot IV against relative strikes and time to expiry



IV Surface

Electronic copy available at: http://ssrn.com/abstract=1922441 Electronic copy available at: http://ssrn.com/abstract=1922441

2 Matthias R. Fengler

0.5

1

1.5

2 0 1 2 3 4 5

20 %

40 %

60 %

80 %

100 %

Time to maturity [years]Moneyness [X/S]

Fig. 1. IV surface of DAX index options from 28 Oct. 2008, traded at the EUREX.IV given in percent across a spot moneyness metric, time to expiry in years.

for each expiry is more finely spaced. The function resulting for a fixed expiryis frequently called the ‘IV smile’ due to its U-shaped pattern. For a fixed(relative) strike across several expiries one speaks of the term structure ofIV. Understandably, the non-flat surface, which also fluctuates from day today, is in strong violation to the assumption of a Geometric Brownian motionunderlying the BSM model.

Although IV observations are observed on this degenerate design, practi-tioners think of them as stemming from a smooth and well-behaved surface.This view is due to the following objectives in option portfolio management:(i) market makers quote options for strike-expiry pairs which are illiquid ornot listed; (ii) pricing engines, which are used to price exotic options andwhich are based on far more realistic assumptions than the BSM model, arecalibrated against an observed IV surface; (iii) the IV surface given by a listedmarket serves as the market of primary hedging instruments against volatil-ity and gamma risk (second-order sensitivity with respect to the spot); (iv)risk managers use stress scenarios defined on the IV surface to visualize andquantify the risk inherent to option portfolios.

Each of these applications requires suitably chosen interpolation and ex-trapolation techniques or a fully specified model of the IV surface. This sug-gests the following structure of this contribution: Section 2 introduces the



IV Smile and Term Structure

• The function resulting for a fixed expiry is frequently called the impliedvolatility smile due to the U-shaped pattern• For a fixed strike across several expiries one speaks of the term structure ofimplied volatility• Practioners like to think of them as stemming from a smooth andwell-behaved surface



Applications

• Market makers use the IV surface to quote prices for strike-expiry pairswhich are illiquid or not listed• Pricing engines used to price exotic options are calibrated against the IVsurface• Risk managers use stress scenarios defined on IV surface to visualise andquantify the risk inherent to option portfolios



Formal Definition

Concept of IV was first introduced by Latané and Rendelman (1976).

DefinitionGiven an observed market price of European option eC with strike K and timeto maturity T , the implied volatility of this option is

eσ : C (K , T, eσ)− eC = 0

Bymonotonicity of the BSM price inσ, there exists a unique solution eσ ∈ R+.Put options can also be used (Put-Call Parity).



IV Surface

The IV eσ is just a constant. However, given the same underlying asset, it variesacross strikes K and expiry T .

DefinitionAn implied volatility surface is the mapping

eσ : (t, K , T ) 7→ eσt(K , T ).




• IV surfaces can be observed for indices, stocks, etc.• Dierent asset classes may display dierent features: smiles may be moreshallow, symmetric or even upward-sloping• Smiles are very pronounced for short expiries and become flattish forlonger dated options• This has not always been the case. Strong asymmetry appeared aer the1987 market turmoil• For equity options (indices and stocks), the smile is negatively skewed



Implied Volatility Skew

We callm := log(K/S) the log moneyness.

DefinitionWe define the implied volatility skew as

∂ eσ2

∂m

m=0

The skew oen increases during times of crisis (e.g., dot-com crash 2001-2003,Sept. 11 2001, GFC 2008).



Further Empirical Observations

• When expiry is small, the fluctuations of the skew are larger• The IV term structure is typically upward sloping (i.e., increasing levels ofIV for longer dated options) in calm times, while in times of crisis it isdownward sloping with short-dated options having higher levels of IVthan longer dated ones• Returns of IV and returns of underlying asset are negatively correlated• IV appears to be mean-reverting (though this is hard to show statistically)



Time Series of ATM-IVBSM implied volatility 5

2001 2003 2005 2007 2009

15 %

20 %

25 %

30 %

35 %

40 %

45 %

3000

4000

5000

6000

7000

8000

1Y ATM−IV DAX

Fig. 2. Time series of 1Y ATM IV (left axis, black line) and DAX index closingprices (right axis, gray line) from 2000 to 2008.

as defined in Section 5. Figure 3 depicts the time series of the DAX indexskew (left axis) for 1M and 1Y options. The skew is negative throughoutand – particularly the short-term skew – increases during times of crisis.For instance, skews increase in the aftermath of the dot-com boom 2001to 2003, or spike at 11 Sep. 2001 and during the heights of the financialcrisis 2008. As theory predicts, see Section 5, the 1Y IV skew has most ofthe time been flatter than the 1M IV skew.

4. Fluctuations of the short-term skew are much larger. Figure 4 gives thequantiles of the skew as a function of time to expiry. Similar patterns alsoapply to IV levels and returns.

5. The IV surface term structure is typically upward sloping (i.e. has increas-ing levels of IV for longer dated options) in calm times, while in times ofcrisis it is downward sloping with short dated options having higher levelsof IV then longer dated ones. This is seen in Figure 3 giving the differ-ence of 1M ATM IV minus 1Y ATM in terms of percentage points on theright axis. A positive value therefore indicates a downward sloping term



Time Series of IV Skew6 Matthias R. Fengler

2001 2003 2005 2007 2009

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

Time

Skew

0

10

20

30

40

50

IV te

rm s

truct

ure

IV term structure

1M skew1Y skew

Fig. 3. Time series of 1M and 1Y IV skew (left axis, gray line and black linerespectively) and time series of the IV term structure (right axis, black dotted line).

Skew is defined as ∂!σ2

∂m

!!!m=0

, where m is log-forward moneyness. The derivative is

approximated by a finite difference quotient. IV term structure is the differencebetween 1M ATM and 1Y ATM in terms of percentage points. Negative valuesindicate an upward sloping term structure.

structure. During the financial crisis the term structure slope achievedunprecedented levels. Humped profiles can be observed as well.

6. Returns of the underlying asset and returns of IV are negatively correlated.For the present data set we find a correlation between 1M ATM IV andDAX returns of ρ = −0.69.

7. IV appears to be mean-reverting, see Figure 2, but it is usually difficultto confirm mean reversion statistically, since IV data is often found to benearly integrated, see Fengler et al. (2007) for a discussion.

8. Shocks cross the IV surface are highly correlated, as can be observedfrom the comovements of IV levels in Figure 2 and the skew and theterm structure in Figure 3. In consequence IV surface dynamics can bedecomposed into a small number of driving factors, see Chapter ???Setlink to Domininik’s Contribution??? of this handbook.



Empirical Quantiles of ATM-IVBSM implied volatility 7

1M 3M 1Y 2Y−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Time to expiry

Skew

95%75%Median25%5%

Fig. 4. Empirical quantiles of the ATM IV skew as a function of time to expiry.

Skew is defined as ∂!σ2

∂m

!!!m=0

, where m is log-forward moneyness.

4 Arbitrage bounds on the implied volatility surface

Despite the rich empirical behavior, IV cannot simply assume any functionalform. This is due to constraints imposed by no-arbitrage principles. For IV,these constraints are very involved, but are easily stated indirectly in theoption price domain. From now on, we set t = 0 and suppress dependence ont for sake of clarity.

We state the bounds using a (European) call option; deriving the corre-sponding bounds for a put is straightforward. The IV function must be suchthat the call price is bounded by

max!S − e−rT X, 0

"≤ C(X, T ) ≤ S . (8)

Moreover, the call price must be a decreasing and convex function in X, i.e.

−e−rT ≤ ∂C

∂X≤ 0 and

∂2C

∂X2≥ 0 . (9)

To preclude calendar arbitrage, prices of American calls for the samestrikes must be nondecreasing across increasing expiries. This statement doesnot hold for European style calls because their theta can change sign. No-arbitrage implies, however, that there exists a monotonicity relationship along



Arbitrage Bounds



Arbitrage Bounds on IV Surface

IV cannot simply take any value due to constraints imposed by theno-arbitrage principles. For IV, these are very diicult to write down, but easilystated indirectly in the option price domain.

Arbitrage BoundsThe IV eσmust be such that the call price is bounded above and below:

(S − e−rT K)+ ≤C (K , T, eσ)≤ S

The call price must be decreasing and convex function in K :

−e−rT ≤∂C∂ K(K , T, eσ)≤ 0 and

∂ 2C∂ K2

(K , T, eσ)≥ 0



Arbitrage Bounds on IV Surface

The quantity x := K/FT is called forward-moneynesswhere FT is the price ofthe forward with expiry T .

Calendar Arbitrage BoundsLet T1 < T2 be the expiry dates of two call options whose strike prices K1 andK2 are related by forward-moneyness, i.e. x1 = x2. Then

C (K2, T2, eσ2)≥C (K1, T1, eσ1)

must hold.



Asymptotics



Asymptotics

• Asymptotics are mathematical results obtained when you take one of thevariables to infinity or to zero• These types of results provide interesting theoretical information aboutthe model• They are oen easier to establish and help reduce the complexity



Asymptotics for the IV Surface

• Many results were conjectures until recently• Understanding the behaviour of IV for far expiries and far strikes is of utterimportance for extrapolation problems oen arising in practice• These results are model independent• m= log(K/S) is the (simple) log-moneyness



IV Surface Flattens

The first theorem shows that the IV surface flattens for infinitely large expiries.

Theorem (Rogers and Tehranchi 2009)For any M > 0we have

limT→∞

supm1,m2∈[−M ,M]

|eσ(m2, T )− eσ(m1, T )|= 0.

Note: “sup” means supremumwhich is equivalent to themaximum if there arefinitely many elements in the set.



Flattening RateThe rate of flattening of the IV skew can bemademore precise by the followingresult.

Theorem (Rogers and Tehranchi 2009)

1. For any 0≤ m1 < m2, we have

eσ(m2, T )2 − eσ(m1, T )2

m2 −m1≤

4T

2. For anym1 < m2 ≤ 0,

eσ(m2, T )2 − eσ(m1, T )2

m2 −m1≥ −

4T



Representation Formula

The level of the IV for far expiry can be written in terms of the expected shareprice.

Theorem (Tehranchi 2009)For any M > 0, we have

limT→∞

supm∈[−M ,M]

eσ(m, T )−

√

√

−8T

logE[ST ∧ 1]

= 0

Note: a ∧ b :=min(a, b).



At Short Expiry

Theorem (Roper and Rutkowski 2009)IfC (K ,ε) = (S − K)+ for some ε > 0 then

limT→0+

eσ(K , T ) = 0.

Otherwise,

limT→0+

eσ(K , T ) =

p2πC (K , T )

Sp

T, S = K

| log(S/K)|p

−2T log(C (K , T )− (S − K)+), S 6= K



At Short Expiry

• The asymptotics are dierent when S = K and S 6= K• The ATM case is similar to Brenner and Subrahmanyam approximationformula• The limit does not need to exist



At Far StrikeLee has established a one-to-one correspondance between the large-strike tailand the number of moments of ST and the small-strike tail and the number ofmoments of S−1

T .

Theorem (Lee 2004)Define ep := supp : E[S1+p

T ]<∞ and

βR = lim supm→∞

eσ2

|m|/T

Then βR ∈ [0,2] and

ep =1

2βR+βR

8−

12

.

Equivalently,βR = 2− 4(

Æ

ep2 + ep− ep).



At Far Strike

Theorem (Lee 2004)Define eq := supp : E[S−q

T ]<∞ and

βL = limsupm→−∞

eσ2

|m|/T

Then βL ∈ [0,2] and

eq =1

2βL+βL

8−

12

.

Equivalently,βL = 2− 4(

Æ

eq2 + eq− eq).



Approximating Implied Volatility



Approximation Formulas

• There is no closed-form formula for IV, even for European options• A numerical solver has to be used to find eσ (e.g., Goal Seek in Excel)• Approximation formulas can be useful when you don’t have a solver(Spreadsheet)



BSF Approximation

The simplest approximation to IV is due to Brenner and Subrahmanyam (1988)and Feinstein (1988).

BSF Approximation

eσ ≈

√

√2πTCS

This formula is only accurate when S = Ke−rT .



Li Approximation

Amore accurate formula was proposed by Li (2005) and hold for ITM (S K)and OTM (S K) options.

Li Approximation

eσ ≈

2zq

2T −

1pT

Ç

8z2 − 6αp2z

if ρ ≤ 1.4

1

2p

T

α+Ç

α2 − 4(K−S)2S(S+K)

if ρ > 1.4

where z = cos(13 arccos( 3αp

32)), α :=

p2π

S+K (2C + Ke−rT − S) andρ := |Ke−rT − S|SC −2.



Corrado-Miller Approximation

Other approximation formulas oen lack a rigourous mathematicalfoundation. The most prominent amongst these are those suggested byCorrado and Miller (1996) and Bharadia et al. (1996)

Corrado-Miller Approximation

eσ ≈1p

T

p2π

S + X

C −S − X

2+

√

√

√

C −S − X

2

2

−(S − X )2

π

where X = Ke−rT .



Bharadia et al. Approximation

Bharadia et al. Approximation

eσ ≈

√

√2πTC − (S − X )/2S − (S − K)/2

.



Accuracy of Approximations

• Isengildina-Massa et al. (2007) investigate the accuracy of sixapproximation formulas• According to their criteria, Corrado and Miller (1996) is the best, followedby Li (2005) and Bharadia et al. (1996)



Week 5b

Bonds



Bonds



Types of Bonds

• A bond is a securitized form of a loan• The buyer of the bond lends the issuer an initial price P in return for apredetermined sequence of payments• The payments may be fixed in nominal terms or linked to some index (e.g.,CPI)



Types of Bonds

• In UK, government bonds are called gilt-edged securities (gilts)• Most other countries: treasury bills or treasury notes• In Australia: Treasury Fixed Coupon Bonds and Treasury Capital IndexedBonds



RBA Treasury Fixed Coupon Bonds

• Pays interest on a semi-annual basis at the prescribed coupon rate,applied to the face value• At maturity, the face value amount is repaid• More details:http://www.aofm.gov.au/content/_download/Treasury_Bond_Information_Memo_18_December_2009.pdf


http://www.aofm.gov.au/content/_download/Treasury_Bond_Information_Memo_18_December_2009.pdf



Treasury Capital Indexed Bonds

• Pays interest on a quarterly basis at the prescribed coupon rate, applied tothe face value• However, the face value is adjusted by indexing the principal to inflation• At maturity, investors receive the adjusted capital value of the security• More details:http://www.aofm.gov.au/content/_download/Treasury_Indexed_Bond_Information_Memo_18_December_2009.pdf


http://www.aofm.gov.au/content/_download/Treasury_Indexed_Bond_Information_Memo_18_December_2009.pdf

http://www.aofm.gov.au/content/_download/Treasury_Indexed_Bond_Information_Memo_18_December_2009.pdf


Australian Bonds (small investors)

• Subject to availability, a range of series across dierent maturity dates willbe on sale at any one time.• Applications may be made for amounts of $1,000 face value and inmultiples of $1,000 up to $250,000 per investor per day (all seriescombined).• Indicative prices:http://www.rba.gov.au/fin-services/bond-facility/xls/otc-prices.xls


http://www.rba.gov.au/fin-services/bond-facility/xls/otc-prices.xls

http://www.rba.gov.au/fin-services/bond-facility/xls/otc-prices.xls


Types of Bonds

• Bonds are also issued by state government (NSW, QLD, etc), banks, andcompanies• Bonds issued by companies are called corporate bonds



Types of Bonds

• Some corporate bonds trade on the ASX (here)• Some are OTC like BHP bonds (here)• Australia’s corporate bondmarket is very small compared to the US orEurope


http://www.asx.com.au/asx/markets/interestRateSecurityPrices.do?type=CORPORATE_BOND

http://quicktake.morningstar.com/stocknet/bonds.aspx?symbol=bhp


Types of Bonds

• Bonds with identical characteristics but sold by dierent issuers may havedierent prices• Example: 2 bonds of 20Y with 6% coupon, one issued by a company andone issued by RBA• Bond issued by company will probably trade at a lower price becausemarket participants take into account the possibility of default on thepayments or the redemption process



Types of Bonds

• Wewill not consider or model credit risk in this course• We only model “risk-free” government bonds• Government bonds are not always riskless: think about Greece and US atthe moment



Fixed Coupon Bonds

• Wewill only concentrate on fixed coupon (fixed-interest) governmentbonds that have no probability of default• We pay a price P for a bond in return for a stream of payments c1, c2, . . . , cnat times t1, t2, . . . , tn from now• The amount of the payments are fixed at the time of issue




• Pays interest on a semi-annual basis at the prescribed coupon rate,applied to the face value• At maturity, the face value amount is repaid• More details:http://www.aofm.gov.au/content/_download/Treasury_Bond_Information_Memo_18_December_2009.pdf





Notation

• g: coupon rate per 100 nominal• n: number of coupon payments• ∆t (or δ): fixed time between payments• t1: time of first payment (t1 ≤∆t)• t j = t j−1 +∆t for j = 2, . . . , n• tn: time to redemption




• Interest is paid on semi-annual basis:∆t = δ = 0.5

• We assume Australian Government will not default: P(T, T ) = 1 for all T• We assume that RBA oers all maturities T



Fixed Coupon Bonds

• c1 =

¨

g∆t first coupon0 if bond has gone ex-dividend

• c j = g∆t for j = 2, . . . , n− 1

• cn = 100+ g∆t



Clean and Dirty Prices

• Bond prices are oen quoted in two dierent forms• The dirty price is the actual amount paid in return for the right to the fullamount of each future coupon payment and the redemption proceeds• The clean price is an artificial price which is typically quoted in the marketplace



Dirty price

• Assume semi-annual (twice a year) coupon payments• If the bond has gone ex-div, then the dirty price will give the buyer theright to the full coupon payable in just over six months• But not the coupon due in a few days• The dirty price of a bond will drop by an amount approximately equal to acoupon payment at the time it goes ex-div• The dirty price of a bond typically rises steadily in between ex-div dates



RBA Prices

• Bank’s selling price includes coupon interest accrued since the lastcoupon payment and up to the date of purchase• Purchasers of Bonds subsequently receive a full coupon payment when itnext falls due, irrespective of when the purchase takes place in the couponperiod.



RBA Prices

• The only exception to this is when Bonds are purchased during theex-interest period, which commences seven days prior to a couponpayment date. Investors who purchase Bonds during the ex-interestperiod do not receive the impending coupon payment.• As a result, Bond prices in the ex-interest period do not include accruedinterest and are commensurately lower.• Question: Are RBA prices clean or dirty?



Zero-Coupon Bonds

• A ZCB is a bond with a coupon rate of zero and a nominal value of 1• We denote the price at time t of a ZCB that matures at time T by P(t, T )• Sometimes we call this a T -ZCB or T -bond• Note that the value of $1 due immediately is P(t, t) = 1 for all t• Arbitrage arguments give P(t, T )≤ 1 for all T



ZCB Cash Flow

P(t,T ) 1

t T

P(t,T ) 1

t T



Modelling Assumptions

• There exists a frictionless market for T -bonds for every T > 0

• P(T, T ) = 1 for all T• P(t, T ) is dierentiable in T



Term structure

1

0Years

Price P(t,T )

T 7→ P(t,T )



T -Bond Price Process

1

0Years

t 7→ P(t,T )



Modelling Assumptions

• In reality these assumptions are not always satisfied• Zero-coupon bonds are not traded for all maturities• It might happen that P(T, T )≤ 1 if the issuer defaults• The condition that P(t, T ) is dierentiable in T is purely technical: itensures that T 7→ P(t, T ) is a smooth curve



Simple vs. Continuous Compounding

Compounding Rate t = 0 t = 11 R 1 1+R2 R/2 1 (1+ R/2)2

m R/m 1 (1+ R/m)m...

......

...∞ eR

eR = 1+ R+ o(R) for small R

e0.04 = 1.04081 but e0.04 − 1.04081= 8.1bp



Week 6

Interest Rates

Arbitrage-Free Pricing


The Interest Rate Setting

Black’s model



Interest Rates



Spot Rates

• The spot rate at time t for maturity at time T is defined as the yield tomaturity of the T -bond:

R(t, T ) = −log P(t, T )

T − t

• In other words,P(t, T ) = exp

−(T − t)R(t, T )

• If we invest $1 at time t in the T -bond for T − t years, then this willaccumulate at an average rate of R(t, T ) over the whole period



Forward Rates

• The forward rate at time t which applies between times T and S(t ≤ T < S) is defined as

F(t, T, S) =1

S − Tlog

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

• The forward rate arises from a contract: agree at time t that we will invest$1 at time T in return for e(S−T )F(t,T,S) at time S



No-arbitrage argument

t T S

e(S−T )F (t,T ,S)−1

By definition, contract has value zero at time t provided F(t, T, S) is the fairforward rate between T and S

We argue that:

F(t, T, S) =1

S − Tlog

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




• Suppose this is not true and that

F(t, T, S)>1

S − Tlog

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

• Then setup a portfolio at time t :

Units ValueForward +1 0T -Bond +1 P(t, T )S-Bond −P(t, T )/P(t, S) −P(t, T )

0




• Hold the contracts until maturity of the Forward• At time T the portfolio is worth 0• At time S, the portfolio is worth

e(S−T )F(t,T,S) −P(t, T )P(t, S)

> 0

• We have an arbitrage: we started with a portfolio with value 0 at time tand have a sure profit at S




• Hence, we cannot have

F(t, T, S)>1

S − Tlog

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

• By reversing the argument, we cannot have

F(t, T, S)<1

S − Tlog

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

• We conclude that

F(t, T, S) =1

S − Tlog

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



Forward-rate curve

• The instantaneous forward rate curve (or forward-rate curve) at time t isfor t < T given by

f (t, T ) = limS→T

F(t, T, S)

...

t T S



Forward-rate curve

limS→T

F(t, T, S) = −∂

∂ Tlog P(t, T )

= −∂∂ T P(t, T )

P(t, T )Therefore,

P(t, T ) = exp

−∫ T

t

f (t, u) du



Forward-rate curve

f (t, u)

Tt A =∫ T

tf (t, u) du



Forward-rate curve

u u+δ Tt

F (t, u, u + δ)

f (t, u) = limδ↓0

F (t, u, u + δ)



Forward Rates

• It is a lot easier to work with the instantaneous forward-rate f (t, T ) thanthe forward F(t, T, S) between T and S• The forward curve T 7→ f (t, T ) is positive for all T ≥ t• This implies that P(t, T ) is a decreasing function of T



Short Rate

• R(t, T ) is the risk-free rate of interest over the fixed period from t to T• When we talk about the risk-free rate of interest wemean theinstantaneous risk-free rate:

r(t) = limT→t

R(t, T ) = R(t, t) = f (t, t)

• Think of r(t) as the rate of interest on a bank account: this can change ona daily basis• r(t) is oen called the short rate



Short Rate

r(t) = limT→t R(t,T )

t T t T t t

r(t)

R(t,T )



Proxies for Short Rate

• The short rate r(t) is fundamental to no-arbitrage pricing• However, it can not be directly observed• The overnight interest rate is not usually considered a good proxy for r(t)• Some people use a slightly longer rate as a proxy: the seven-dayEurodollar rate• Ideally, you want to use a liquid security as a proxy



Par Yields

• The par-yield curve ρ(t, T ) specifies the coupon rates,

100ρ(t, T ),

at which new bonds should be priced if they are to be issued at par• Price of 100 per 100 nominal



Par Yields

• Par yield for a maturity T (annual coupons, δ = 1), solve followingequation for coupon rates ρ(t, T ):

100= 100ρ(t, T )T∑

s=t+1

P(t, s) + 100P(t, T )

• Solution given by

ρ(t, T ) =1− P(t, T )∑T

s=t+1 P(t, s)



Yield to Maturity

• Yield to maturity usually applies to coupon bonds• It is the “internal rate of interest” (IRR) of the bond• Given the price of a bond P, coupons c1, c2, . . ., then

find κ: P(t, T ) =n∑

j=1

c je−κ(t j−t)

• κ is the YTM



Relationships

• For a given t , each of the curves P(t, T ), f (t, T ) and R(t, T ) uniquelydetermines the other two• Recall that we have

P(t, T ) = exp (−R(t, T )(T − t))

= exp

−∫ T

t

f (t, u) du



Relationships

f (t, u)

R(t,T )

Tt

A = R(t,T )(T − t) =∫ T

tf (t, u) du

P(t,T ) = exp(−A)



Example

T 0 1 2 3F(0, T, T + 1) 0.0420 0.0500 0.0550 0.0560

P(0, T ) = exp

−T−1∑

t=0

F(0, t, t + 1)

T 0 1 2 3P(0, T ) 0.95887 0.91211 0.86239 0.81628



Example

R(0, T ) = −log P(0, T )

T + 1

T 0 1 2 3F(0, T, T + 1) 0.0420 0.0500 0.0550 0.0560

P(0, T ) 0.95887 0.91211 0.86239 0.81628R(0, T ) 0.0420 0.0460 0.0490 0.05075



Arbitrage-Free Pricing



Arbitrage

• We are able to construct at time 0 some portfolio which has a NPV of zero• At some fixed time T > 0, the portfolio gives us a sure profit• This is sometimes called a free lunch



Arbitrage

• The concept of arbitrage becomes clear when we consider static portfolios• Also known as ‘buy-and-hold’ strategies



Arbitrage with Static Portfolios

• Suppose we can invest in n assets• Asset i has price Pi(t) at time t• Asset i pays no dividend or coupons• We have x i units of asset i in our portfolio• Portfolio value is

V (t) =n∑

i=1

x i Pi(t)



Arbitrage with Static Portfolios

• At time t = 0:

V (0) =n∑

i=1

x i Pi(0) = 0

• At time T > 0:P(V (T )≥ 0) = 1

P(V (T )> 0)> 0



Principle of No Arbitrage

• The Principle of No Arbitrage states simply that such arbitrageopportunities do not exist• If they did exist, smart investors could make infinite amounts of money



Equivalent Forms for PNA

• We cannot construct a riskless portfolio which returns more than therisk-free rate of return• If two portfolios A and B give rise to identical (but possibly random) futurecashflows with certainty, then A and B must have the same value at thepresent time (aka. law of one price)



Example: Parallel Yield Curve Shis

• It is easy to find a model which admits arbitrage• Suppose that

P(0, T ) = exp

−∫ T

0

f (0, u) du

for some initial forward-rate curve f (0, T )




• Probability space (Ω,F ,P)• Our model is defined so that at time 1 the forward rate curve will be

f (1, u) = f (0, u) + ε, for u> 1

• ε : Ω→ R is a random variable




0 T

u 7→ f (0, u)

11




0

u 7→ f (0, u)

u 7→ f (1, u)

ε

1 T




0

1 T

ε




• Suppose exists: T1-Bond, T2-Bond, T3-Bond with 1< T1 < T2 < T3

0 1 T1 T2 T3

$1 $1 $1

• Let x i be the units held at time 0 of the Ti-Bond




• At time t = 1we have

P(1, T ) = exp

−∫ T

1

f (1, u) du

= exp

−∫ T

1

( f (0, u) + ε) du

= exp

−∫ T

1

f (0, u) du

e−ε(T−1)

=P(0, T )P(0,1)

e−ε(T−1)




• Remember this for later:

P(1, T ) =P(0, T )P(0,1)

e−ε(T−1) (6)




• For an arbitrage, we need:

3∑

i=1

x i P(0, Ti) = 0

3∑

i=1

x i P(1, Ti)≥ 0 with prob. 1

3∑

i=1

x i P(1, Ti)> 0 with positive prob.




• The value of the portfolio at time 1 is

V1(ε) =3∑

i=1

x i P(1, Ti)

=3∑

i=1

x iP(0, Ti)P(0,1)

e−ε(Ti−1) by (6)

=e−ε(T2−1)

P(0, 1)g(ε)




• Where

g(ε) =3∑

i=1

x i P(0, Ti)e−ε(Ti−T2)

• The trick is to choose time T2, add/substract T2, to get

e−ε(Ti−1) = e−ε(Ti−T2)e−ε(T2−1)




• If value of initial portfolio is zero,

3∑

i=1

x i P(0, Ti) = 0

then g(0) = 0

• Also, at time 1 and for a shi ε,

V1(ε)< 0 ⇐⇒ g(ε)< 0




• As

g(ε) =3∑

i=1

x i P(0, Ti)e−ε(Ti−T2)

• g(ε) is continuous and twice dierentiable in ε• We are looking for the situation where V1(ε)> 0 for all ε 6= 0

• As g(0) = 0, we must also have g ′(0) = 0




• The condition g ′(0) = 0 implies

3∑

i=1

x i(T2 − Ti)P(0, Ti) = 0

• Which, in turn, implies

3∑

i=1

x i Ti P(0, Ti) = 0

since T2

3∑

i=1

x i P(0, Ti) = 0




• Finally, it is suicient that g ′′(ε)> 0 for all ε to ensure that g(ε)> 0 for allε 6= 0

• As

g ′′(ε) =3∑

i=1

x i(T2 − Ti)2P(0, Ti)e

−ε(Ti−T2)

• g ′′(ε)> 0 for all ε if and only if x1 and x3 are both greater than or equal to0




• Take x2 = −1

• Eq (6) and x2 = −1 imply that at least one of x1 and x3 must be greaterthan zero• But

3∑

i=1

x i Ti P(0, Ti) = 0

implies that x1 and x3 must be both positive or both negative• It follows that g ′′(ε)> 0 for all ε




• Hence, g(ε)> 0 for all ε 6= 0

• We can conclude that

V1(ε) =3∑

i=1

x i P(1, Ti)> 0

for all ε 6= 0

• Arbitrage!



Example 2

• Suppose that P(0, t) = e−0.08t for all t > 0

• And

P(1, t + 1) =

¨

e−0.1t if ξ= 1

e−0.06t if ξ= 0

for the random variable ξ : Ω→ 0,1• Spot and forward rate curves will randomly shi up or down by 2%



Example 2

• Suppose we hold x1, x2, x3 units of T1-Bond, T2-Bond, and T3-Bond suchthat

x2P(0,2) = −1

x1P(0,1) + x2P(0, 2) + x3P(0,3) = 0

x1P(0, 1) + 2x2P(0, 2) + 3x3P(0,3) = 0



Example 2

• Solving the system of equations:

x2 =−1

P(0,2)= −1.173511

x3 =1

2P(0,3)= 0.635624

x1 =1

2P(0,1)= 0.541644

• At time 1, the value of this portfolio is 0.00021 if ξ= 1 or 0.00022 if ξ= 0



Example 2

• Themodel is not arbitrage-free• It violates the law of one price



Conclusion

• Parallel shis of the yield curve cannot occur at any time in the future




• Suppose that the risk-free rate r(t) is stochastic• Randomness is underpinned by the probability space (Ω,F ,P)• P is the real-world probability measure• The cash account is given by

B(t) = B(0)ex p

∫ t

0

r(s) ds

• And satisfies dB(t) = r(t)B(t) d t




Theorem (FTAP)

(i) Bond prices evolve in arbitrage-free manner if and only if exists measureQequivalent to P under which P(t,T )

B(t) is a martingale for 0< t < T(ii) If (i) holds, then the market is complete if and only ifQ is the unique

measure under which P(t,T )B(t) are martingales




• ThemeasureQ is oen called the equivalent martingale measure• It is also called risk-neutral measure




CorollaryHence

P(t, T ) = EQ

exp

−∫ T

t

r(s) ds

Ft

whereFt is theσ-algebra generated by the price histories up to time t , and EQis the expectation with respect toQ




• X is someFT -measurable claim payable at T• V (t) is fair value of X at time t• Then V (t)/B(t) is also a martingale underQ

• Hence,

V (t) = EQ

exp

−∫ T

t

r(u) du

X

Ft



Example: Forward Pricing

• Forward contract: K will be paid at time T in return for repayment of 1 attime S where T < S• Equivalently, K is paid at T in return for delivery of an S-bond with value

P(T, S)• Howmuch is this contract worth at time t < T?• X = P(T, S)− K at time T




V (t) = EQ

exp

−∫ T

t

r(u) du

X

Ft

= EQ

h

e−∫ T

t r(u) du EQ

h

e−∫ S

T r(u) du

FT

i

Ft

i

− KEQ

h

e−∫ T

t r(u) du

Ft

i

= EQ

h

e−∫ S

t r(u) du

Ft

i

− KEQ

h

e−∫ T

t r(u) du

Ft

i

= P(t, S)− KP(t, T )




• We now choose K so that V (t) = 0

• This gives

K =P(t, S)P(t, T )

• The contract can be hedged at no cost at time t by buying one unit ofP(t, S) and selling P(t, S)/P(t, T ) units of P(t, T )




• What is special about the interest rate setting?• There are a few new things to think about...



Factors

• A one-factormodel is one under which there is only one source ofrandomness aecting the bond prices• In a one-factor model all price changes are perfectly (but non-linearly)correlated• If we know one change in quantity (e.g. risk-free rate r(t)) we know thechange in all asset prices



Factors

• Amultifactormodel has multiple sources of randomness• Price changes are not perfectly correlated• Example: m bonds, each bond driven by a dierent Brownian motion



A Bond is a Derivative

• A bond is a derivative itself• In a one-factor model, the price of any bond is derived from ourknowledge of the short-rate r(t)• The short-rate is the underlying



Types Of Model

• There are twomain types of models: Equilibrium and No-arbitrage• Equilibriummodels are built on assumptions about how the economyworks• Take into account the varying risk preferences of dierent investors• Relationship between the economy and the term-structure of interestrates



Types Of Model

• This means constructing a simple stochastic model for the evolution ofthe risk-free rate• Invoke the FTAP to derive theoretical bond prices• theoretical prices evolve in a way that is arbitrage-free• It may happen that theoretical prices is dierent from the observed pricesin the market, giving rise to arbitrage opportunities (cheap/dear analysis)



Types Of Model

• Oen short-rate models are seen as equilibriummodels• This is not always true• It is diicult to prove that a short-rate model has an equilibrium derivation



Types Of Model

• No-Arbitrage Models use the observed term structure at the current timeas the starting point• Future prices evolve in a way which is consistent with this initial pricestructure and which is arbitrage free• These models are used to price short-term derivatives



Types Of Model

• Actual vs. theoretical price dierences in equilibriummodels getmagnified when pricing derivatives• A 1% error could lead to a 10% error in the price of an option• On the other hand, no-arbitrage models have strange dynamics forquantities like r(t) in the long-term which are hard to justify



Black’s model



Types Of Model

• Black, F. (1976). The pricing of commodity contracts. Journal of FinancialEconomics.• Assumed that futures prices have the same lognormal property assumedfor stock prices• Gives us “Black-Scholes style” formula for valuing futures options



Black’s formula is given by

C = e−r t (F0Φ(d1)− KΦ(d2))

P = e−r t (KΦ(−d2)− F0Φ(−d1))

where

d1 =log(F0/K) +σ2T/2

σp

T, d2 =

log(F0/K)−σ2T/2

σp

T= d1 −σ

pT .



ExampleEuropean put future option on crude oil. TTM is 4 months, current futuresprice is $20, exercise price is $20, risk-free rate is 5% per annum, and thevolatility of the futures price is 25% per annum.

F0 = 20, K = 20, r = 0.05, T = 4/12, σ = 0.25.

log(F0/K) = 0, d1 =σp

T2= 0.07216, d2 = −

σp

T2= −0.07216.

Φ(−d1) = 0.4712, Φ(−d2) = 0.5288

V = e−0.05×4/12(20× 0.5288− 20× 0.4712) = 1.12



Black’s model for a Bond option

• Traders became comfortable with the lognormal assumption andvolatility to describe uncertainty• This lead to the formula being applied to interest rate derivatives (Bondoptions, interest rate caps, and swap options)• During 1980s and 1990s, the volume of trading in interest rate derivativesin both over-the-counter and exchange-tradedmarkets increased veryquickly




Consider a European call option on a variable whose value is V . Define:

T : Maturity of the optionF : Forward price of V for a contract with maturity T

F0: Value of F at time zeroK : Strike price of the option

P(t, T ): Price at time t of a zero-coupon bond paying $1 at time TVT : Value of V at time Tσ: Volatility of F




• Black’s model assumes:1. VT has a lognormal distribution with standard deviation of log VTequal toσ

pT

2. The expected value of VT is F0

• Discounts the expected payo by multiplying by P(0, T )• European call payo ismax(VT − K , 0) at time T




• Lognormal assumption implies that the expected payo is

E(VT )Φ(d1)− KΦ(d2)

where E(VT ) is the expected value of VT and

d1 =log(E(VT )/K) +σ2T/2

σp

T, d2 = d1 −σ

pT .

• This gives

C = P(0, T ) (F0Φ(d1)− KΦ(d2)) , P = P(0, T ) (KΦ(−d2)− F0Φ(−d1))




• Black’s model can be extended to allow for the situation where the payois calculated at time T but the payo is actually made at some time S > T• This basically changes the discounting:

C = P(0, S) (F0Φ(d1)− KΦ(d2)) , P = P(0, S) (KΦ(−d2)− F0Φ(−d1))




• Black’s model does not assume that the evolution of V or F is driven by a(geometric) Brownian motion• All that is required is that VT is lognormally distributed at time T• σ is referred to as the volatility of F or the forward volatility of V• σ doesn’t say anything about the standard deviation of log V at timesother than T



Validity of Black’s model

• Black’s model is appropriate when interest rates are constant ordeterministic• In this situation, the forward price of V equal its futures price




When interest rates are stochastic Black’s model involves two approximations:• E(VT ) = F0. In a risk-neutral world, E(VT ) is equal to its futures price. Theforward price and futures price are not the same when interest rates arestochastic• The stochastic behaviour of interest rates is not taken into account for thediscounting




• It is inappropriate to price interest rate derivatives by just modelling thedynamics of underlying security• Consistent pricing must be based on the evolution of the entire termstructure of interest rates• Using Black’s model naïvely can lead to arbitrage opportunities



Week 7

Continuous-Time Interest Rate Models

The Martingale Approach

The PDE Approach

Additional Comments



Continuous-Time Interest Rate Models



One-Factor Models for the Risk-Free Rate

• Continuous-time setting• One-factor models for the term structure of interest rates• Bond pricing given a one-factor diusion model for the risk-free rate r(t)




• We assume that r(t) is modelled by

dr(t) = a(t)d t + b(t)dW (t)

• (W (t))t≥0 is a standard Brownian motion under the real-world probabilitymeasure P

• a(t) and b(t) are previsible processes• Ft = σ(W (s) : s ≤ t) is theσ-algebra generated by the history ofW (s)up to time t




• a(t) = a(r(t)) and b(t) = b(r(t))• Ensures that r(t) is Markov and time homogeneous




Model a(r) b(r)Merton (1973) µ σDothan (1978) µr σrVasicek (1977) α(µ− r) σCIR (1985) α(µ− r) σ

pr

Pearson-Sun (1994) α(µ− r) σp

r − βBrenann-Schwartz (1979) α(µ− r) σrBlack-Karasinski (1991) αr − γr log r σr




Model r(t)≥ 0? AR? Simple Formula?M N N YD Y N NV N Y YCIR Y Y YPS Y if β > 0 Y NBS Y Y NBK Y Y (γ > 0) N




• All models are approximation to reality but some are better than others• Onemust consider:

• Desired characteristics• Ease of implementation• Howwell does model approximate reality• Fast vs. slow calculation




• Sometimes simple formula doesn’t mean fast calculation• Ex: analytical formulae exist for CIR however involve using the law of thenon-central chi-squared distribution• In R: dchisq,pchisq,rchisq• Ex: Vasicek and other models only require Normal distribution



Other Characteristics

• Are bond and derivative prices straightforward and simple to calculatenumerically?• Is the model flexible enough to cope with new andmore complexderivative products?• Does the model produce dynamics which are realistic?




• Does the model fit historical data well? (e.g., mean and variance)• If the model keeps rates positive, does it allow forward rates, spot ratesand par yields to take values close to zero?• Does the model have equilibrium derivation?




• One factor models generally fail many of these criteria• This is due to their dependence on a single-factor (e.g., dynamics of r(t))• This is why we will consider multi-factor models in the second half ofsemester




• Over the next couple of weeks we will consider one-factor models• Wewill consider a number of concepts that will carry over to themulti-factor setting• Wewill consider a general approach to pricing whereby we obtain acoherent set of price dynamics that do not admit arbitrage




• Wewill now discuss theMartingale approach to pricing




• We consider an abstract model with only one source of randomness• Risk-free rate r(t)

dr(t) = a(t)d t + b(t)dW (t)

• P(t, T ) is price of T -bond at time t

dP(t, T ) = P(t, T )[m(t, T )d t + S(t, T )dW (t)]

• a(t), b(t),m(t, T ) and S(t, T ) are previsible functions (possiblystochastic)




• We also have the risk-free cash account B(t)• B(t) satisfies the SDE

dB(t) = r(t)B(t)d t

• The solution is

B(t) = B(0)exp

∫ t

0

r(u) du




• The risk premium at time t on the risky bond P(t, T ) is defined as theexcess expected rate of return on the bondm(t, T ) over the risk-free rateof interest r(t)• The risk premium represents the extra reward we get for investing in therisky asset rather than the risk-free cash account




• Themarket price of risk is the previsible process

γ(t) =m(t, T )− r(t)

S(t, T )

• Themarket price of risk represents the excess expected return per unit ofvolatility




• Consider a claim XS at some time S < T• What is the no-arbitrage price, V (t), at time t < S for this claim?




TheoremThere exists a measureQ∼ Pwith

V (t) = EQ

exp

−∫ S

t

r(u) du

X

Ft

where dr(t) = (a(t)− γ(t)b(t))d t + b(t)dfW (t) andfW (t) is a BM underQ




• In Baxter and Rennie, we had a 3 step approach• Cairns expands this to 5 steps



Fundamental Theorem of Asset PricingProof

Define the discounted price process

Z(t, T ) =P(t, T )

B(t)= P(t, T )exp

−∫ t

0

r(u) du

where B(t) is the risk-free cash account B(0) = 1



Fundamental Theorem of Asset PricingProof

1. Establish the probability measureQ equiv. to P s.t. Z(t, T ) is a martingale2. For t < S < T define D(t) = EQ[B(S)−1X |Ft]3. Apply MRT to get holdingsφ(t)4. Setψ(t) = D(t)−φ(t)Z(t, T ), show that portfolio is self-financing5. Show that portfolio replicates claim X



Fundamental Theorem of Asset PricingProof: Step 1

Apply product rule to

Z(t, T ) =P(t, T )

B(t)= B(t)−1P(t, T )

to get

dZ(t, T ) = B(t)−1dP(t, T ) + P(t, T )d(B(t)−1) + d⟨B−1, P⟩(t)




Apply Itô’s formula:

d(B(t)−1) = −1

B(t)2dB(t) +

12

2B(t)3

d⟨B⟩(t)

= −r(t)B(t)

d t

as dB(t) = r(t)B(t)d t and d⟨B⟩(t) = 0d t




Substituting dP(t, T ) = P(t, T )[m(t, T )d t + S(t, T )dW (t)] and d(B(t)−1)into product formula gives

dZ(t, T ) =P(t, T )

B(t)(m(t, T )d t + S(t, T )dW (t))−

r(t)P(t, T )d tB(t)

+ 0d t

= Z(t, T )[(m(t, T )− r(t))d t + S(t, T )dW (t)]

asd⟨B−1, P⟩(t) = 0 d t




Recall that the market price of risk is

γ(t) =m(t, T )− r(t)

S(t, T )

so defining

fW (t) =W (t) +

∫ t

0

γ(u) du ⇒ dfW (t) = dW (t) + γ(t) d t

dZ(t, T ) = Z(t, T )[(m(t, T )− r(t))d t + S(t, T )dW (t)]= Z(t, T )[(m(t, T )−r(t)−γ(t)S(t, T ))d t+S(t, T )(dW (t)+γ(t)d t)]

= Z(t, T )[(m(t, T )− r(t)− γ(t)S(t, T ))d t + S(t, T )dfW (t)]

= Z(t, T )S(t, T )dfW (t)




• Notice that Z(t, T ) is an exponential process• We need the Novikov condition to hold, i.e.,

EP

exp

12

∫ T

0

γ(u)2 du

<∞

so we can apply Girsanov’s theorem




• Girsanov’s theorem allows us to state that there exists a measureQequivalent to P

• With Radon-Nikodým derivative given by

dQdP= exp

−∫ T

0

γ(u)dW (u)−12

∫ T

0

γ(u)2 du

• Under whichfW (t) =W (t) +

∫ t

0

γ(u) du

is a standard Brownian motion




• Note that under the same change of measure from P toQwe have that

dP(t, T ) = P(t, T )[m(t, T ) d t + S(t, T ) dW (t)], under P

becomes

dP(t, T ) = P(t, T )[r(t) d t + S(t, T ) dfW (t)], underQ

• UnderQ, all T -bond prices have dri equal to the risk-free rate r(t)




• The process Z(t, T ) underQ satisfies

dZ(t, T ) = Z(t, T )S(t, T ) dfW (t)

• The SDE has no dri• The SDE is a martingale underQ if

EQ

exp

12

∫ T

0

S(t, T )2 d t

<∞




• We have the claim XS = X that is a random variable• We need to make it into a process• For t < S < T define

D(t) = EQ[B(S)−1XS|Ft]

• This is a martingale underQ by the tower property for conditionalexpectations




• Z(t, T ) and D(t) are bothQ-Martingales• We can apply the Martingale representation theorem (MRT)• By MRT, there exists a previsible processφ(t) such that

D(t) = D(0) +

∫ t

0

φ(u)dZ(u, T )

• This requires S(t, T ) to be non-zero for all t < S almost surely




• We setup a (P, B) portfolio with value

V (t) = φ(t)P(t, T ) +ψ(t)B(t)

• We takeψ(t) = D(t)−φ(t)Z(t, T )• We need to show that the portfolio is self-financing




• Consider a short time interval [t, t + d t)• Over the period, holdφ(t) of P(s, t) andψ(t) of B(s)• The change of value over [t, t + d t) is

φ(t)dP(t, T ) +ψ(t)dB(t)

• Wewill now compute the change of value of V (t) but

V (t) = φ(t)P(t, T )+ψ(t)B(t) = B(t)[φ(t)Z(t, T )+ψ(t)] = B(t)D(t)

asψ(t) = D(t)−φ(t)Z(t, T )




dV (t) = d[B(t)D(t)]= B(t)dD(t) + D(t)dB(t) + dB(t)dD(t)= B(t)φ(t)dZ(t, T ) + D(t)r(t)B(t)d t + 0d t

= φ(t)B(t)S(t, T )Z(t, T )dfW (t) + (φ(t)Z(t, T ) +ψ(t))r(t)B(t)d t

= φ(t)P(t, T )(r(t)d t + S(t, T )dfW (t)) +ψ(t)r(t)B(t) d t= φ(t)dP(t, T ) +ψ(t)dB(t)

• The portfolio process is self-financing




• We also haveV (S) = B(S)EQ[B(S)

−1XS|FS] = XS

• So the portfolio replicates the claim XS

• It follows that for t < S, V (t) is the unique no-arbitrage price at time t forXS payable at S




V (t) = B(t)D(t)

= EQ

B(t)B(S)

XS

Ft

= EQ

exp

−∫ S

t

r(u) du

XS

Ft




Uniqueness of the price V (t) is due to:• We are considering a one-factor model• We specified the dynamics under P for P(t, T )which allows us to establisha market price of risk• Randomness is driven by a Brownian motionW (t). Other processes (e.g.,jumps) may mean that V (t) is not uniquely determined




CorollaryFor all S such that 0< S < T ,

P(t, S) = EQ

exp

−∫ S

t

r(u) du

Ft

Proof.Take XS = 1 in previous theorem.




• UnderQ the prices of all tradable assets have the risk-free rate of interestas the expected growth rate• Under P,

dV (t) = V (t)[(r(t) + γ(t)σV (t)) d t +σV (t)dW (t)]

• γ(t)σV (t) is called themarket risk premium or risk premium




• Notice that γ(t)σV (t) depends on the market price of risk γ(t)• This implies that the risk premiums on dierent assets are closely linked• They can dier only through the volatility in the tradable asset (σV (t) or

S(t, T ) for a ZCB)




• We anticipate that ZCB will have a positive risk premium (γ(t)S(t, T )> 0for all T > t) to reward for the extra risk• Derivatives V (t) for whichσV (t) has the same sign as S(t, T ) also have apositive risk premium (e.g., call option on P(t, T ))• Opposite sign forσV (t) leads to a negative risk premium (e.g., Put optionon P(t, T ))



The PDE Approach



The PDE Approach

• We shall now discuss the Partial Dierential Equation (PDE) approachgiven in Vasicek (1977)• Previously, we have used the martingale approach to pricing• The PDE approach is also useful especially for numerical methods• Black and Scholes used a PDE approach in their classic paper



The PDE Approach

• r(t) is Markov, i.e., the future dynamics of r(t) given its current value isindependent of past behaviour• Themarket is eicient: no transactions costs and investors are rational



The PDE Approach

• Our abstract model for r(t) is of the form

dr(t) = a(t, r(t))d t + b(t, r(t))dW (t)

• Prices changes for all bonds with dierent maturities are perfectly (butnonlinearly) correlated



The PDE Approach

• By assumption P(t, T ) is only a function of the current time t and r(t), i.e.,

P(t, T ) = f (t, r(t))

for some unknown f (t, x)• By Itô’s formula,

d f (t, r(t)) =∂ f∂ t

d t +∂ f∂ x

dr(t) +12

b2∂2 f∂ x2

d t

=∂ f∂ t

d t +∂ f∂ x(ad t + bdW (t)) +

12

b2∂2 f∂ x2

d t

=

∂ f∂ t+ a∂ f∂ x+

12

b2∂2 f∂ x2

d t + b∂ f∂ x

dW (t)



The PDE Approach

• As f (t, r(t)) = P(t, T ), this implies that the price dynamics are given byan exponential SDE

dP(t, T ) = P(t, T )[m(t, T )d t + S(t, T )dW (t)]

• With average price returns

m(t, T ) =1P

∂ P∂ t+ a∂ P∂ r+

12

b2∂2P∂ r2

• And price returns volatility

S(t, T ) =1P

b∂ P∂ r



The PDE Approach

• We can now consider the market price of risk and use it to derive anequation for price of a T -bond• Construct a portfolio of:

• −V1(t) of a T1-bond (i.e., short)• V2(t) of a T2-bond (i.e., long) with T2 > T1

• Total portfolio worth is V (t) = V2(t)− V1(t)• We now vary V1(t) and V2(t) so that V (t) remains risk-free



The PDE Approach

• The instantaneous investment gain from t to t + d t is

− V1(t)dP(t, T1)P(t, T1)

+ V2(t)dP(t, T2)P(t, T2)

= −V1(t)(m1d t + S1dW ) + V2(t)(m2d t + S2dW )= (V2m2 − V1m1)d t + (V2S2 − V1S1)dW

• Where for i = 1,2:

mi = m(t, Ti), Si = S(t, Ti)

• We now simplify that expression under some assumptions



The PDE Approach

• If we assume thatV1(t)V2(t)

=S(t, T2)S(t, T1)

=S2

S1

• Then we getV1(t)S1 − V2(t)S2 = 0

andV2m2 − V1m1 =

S1VS1 − S2

m2 −S2V

S1 − S2m1

• So the investment gain over [t, t + d t] is

V (t)

m2S1 −m1S2

S1 − S2

d t



The PDE Approach

• Therefore, through our choice of portfolio strategy we have obtained arisk-free investment strategy• Further, by varying V1(t) appropriately we can ensure that this portfolio isalso self-financing:

dV (t) = V (t)

m2S1 −m1S2

S1 − S2

d t



The PDE Approach

• Since the portfolio is risk-free, the principle of no-arbitrage states that thegrowth rate must equal r(t):

m2S1 −m1S2

S1 − S2

= r(t) orm1 − r

S1=

m2 − rS2

• This must be true for all maturities. Thus for all T > t

m(t, T )− r(t)S(t, T )

= γ(t)

• γ(t) is the market price of risk and cannot depend on the maturity date T



The PDE Approach

• On one hand, we have

m(t, T ) = r(t) + γ(t)S(t, T )

• On the other hand,

m(t, T ) =1P

∂ P∂ t+ a∂ P∂ r+

12

b2∂2P∂ r2

S(t, T ) =1P

b∂ P∂ r



The PDE Approach

• We equate the two expressions form(t, T ) to find

∂ P∂ t+ (a− bγ)

∂ P∂ r+

12

b2∂2P∂ r2− rP = 0



The PDE ApproachTheorem (Feynman-Kac Formula)Let T > 0 andΦ be a continuous function. Suppose that a function F(t, r)satisfies

∂t F(t, r) + f (t, r)∂r F(t, r) +12ρ2(t, r)∂ 2

r F(t, r)− rF(t, r) = 0

F(T, r) = Φ(r)

Then1 M(t) = exp(−∫ t

0 r(u) du)F(t, r(t)) is a martingale and

F(t, r(t)) = EQ

exp

−∫ T

t

r(u) du

Φ(r(T ))

Ft

where r(t) = r(t) and underQ satisfies d r(t) = f (t, r)d t +ρ(t, r)dfW1under some technical conditions, see appendix



The PDE Approach

• We now apply the Feynmann-Kac formula with F(t, r) = P(t, T )

f (t, r) = a(t, r)− γ(t)b(t, r), ρ(t, r) = b(t, r)

• And terminal condition P(T, T ) = Φ(r(t)) = 1 as we are pricing a T -bond• Hence,

P(t, T ) = EQ

exp

−∫ T

t

r(u) du

Ft



The PDE Approach

• Under Pwe havedr(t) = a(t)d t + b(t)dW (t)

• UnderQwe have

d r(t) = f (t, r(t))d t +ρ(t, r(t))dfW (t)

• We need to satisfy ourselves that these measures P andQ are equivalent



The PDE Approach

• Assume that γ(t) satisfies Novikov condition• DefinefW (t) =W (t) +

∫ t

0 γ(s) ds

• Then by Girsanov’s theorem, there existsQ equiv. to P under whichfW is aBM with Radon-Nikodym derivative

dQdP= exp

−∫ T

0

γ(t)dW (t)−12

∫ T

0

γ(t)2d t



The PDE Approach

• We then have

dP(t, T ) = P(t, T )(m(t, T )d t + S(t, T )dW )

= P(t, T )[m(t, T )d t + S(t, T )(dfW − γ(t)d t)]

= P(t, T )[(m(t, T )− γ(t)S(t, T ))d t + S(t, T )dfW ]

= P(t, T )[r(t)d t + S(t, T )dfW ]

• Thus underQ, the expected return on any bond is the risk-free rate• This implies thatQ is the EMM



The PDE Approach

• This Feymann-Kac approach can be applied to other derivative payos• Simply choose the function Φ appropriately• This is the basis of finite dierence methods for pricing options as wedetermine P(t, T ) by solving a PDE with a specific terminal condition



Additional Comments



Market Price of Risk

• Note that we started by specifying P dynamics and thenmoved toQdynamics using the market price of risk γ(t)• The market price of risk links the dynamics of the cash bond B(t) and the

T -bond prices• We can’t replicate a payo X only using the cash bond B(t) therefore oneneeds to specify γ• A short-rate model is not fully determined under P unless one specifies themarket price of risk γ



Start under Risk-Neutral Measure

• It is customary from amodelling point-of-view to simply specify theQdynamics of r directly• This implies theQ dynamics of all bond prices by the formula

P(t, T ) = EQ

exp

−∫ T

t

r(u) du

• Some people “back out” the market price of risk γ by statistical methodsfrom historical observations of price movements• This is actually wrong... statistical observations are under P



Week 8

Particular Models

Aine Short-Rate Models



Particular Models



The Vasicek Model

• The Vasicek model is specified underQ by

dr(t) = α(µ− r(t)) d t +σdfW (t)

wherefW (t) is BM under EMMQ and α,µ,σ are all positive constants• µ is the risk-neutral long-termmean risk-free rate• α is the rate at which r(t) reverts back to µ• σ is the local volatility of the short-term interest rates



The Vasicek Model

• For s > 0, r(t + s) given r(t) is Normally distributed underQwith mean

µ+ (r(t)−µ)e−αs

and variance

σ2 1− e−2αs

2α• This implies that the long-term standard deviation of r(t) is

σp

2α



Bond Prices in Vasicek Model

TheoremPrices for T -bonds in the Vasicek model are given by

P(t, T ) = exp[A(t, T )− B(t, T )r(t)]

where

B(t, T ) =1− e−α(T−t)

α

A(t, T ) = (B(t, T )− (T − t))

µ−σ2

2α2

−σ2

4αB(t, T )2



European Options in Vasicek Model

TheoremThe price of a European call option on a S-bond with strike price K and exercisedate T (T < S) is

V (t) = P(t, S)Φ(d1)− KP(t, T )Φ(d2)

whered1 =

1σp

logP(t, S)

KP(t, T )+σp

2, d2 = d1 −σp

σp =σ

α(1− e−α(S−T ))

√

√1− e−2α(T−t)

2α



The Cox-Ingersoll-Ross Model

• The Vasicek model has a major problem: the risk-free rate can becomenegative• Further, empirical evidence shows that the volatility of r(t) is not constantbut is an increasing function of r(t)• The first tractable model that fixed these problems was the CIR model:

dr(t) = α(µ− r(t)) d t +σÆ

r(t)dfW (t)

where α,µ,σ > 0 andfW is BM under EMMQ



Bond Prices in CIR Model

TheoremPrices for T -bonds in the CIR model are given by

P(t, T ) = exp[A(t, T )− B(t, T )r(t)]

where

B(t, T ) =2(eγt − 1)

(γ+α)(eγt − 1) + 2γ

A(t, T ) =2αµσ2

log

2γe(γ+α)t/2

(γ+α)(eγt − 1) + 2γ

γ=p

α2 + 2σ2



European Option Prices in CIR Model

• Prices of European options are given in terms of the non-centralchi-squared distribution underQ

• They involve a pretty nasty formula• Exercise: Implement the pricing formula in R



Calibration of Vasicek and CIR Model

• In Vasicek and CIR, T -bond prices are given by

P(t, T ) = exp(A(t, T )− B(t, T )r(t))

where A and B depend on the model parameter vector θ• For Vasicek model: θ = (µ,α,σ)• For CIR model: θ = (µ,α,σ)




• We have the current market YTMs denoted R(0, Ti) for the Ti-bond withi = 1, . . . , n• We can plot the yield curve (i.e., the term structure)




Time To Maturity (Log Scale)

Yield

2.0%

2.4%

2.8%

3.2%

3M 6M 1Y 2Y 5Y 10Y 20Y




• Recall that the yield to maturity R(t, T ) satisfies

P(t, T ) = exp(−(T − t)R(t, T ))

• We also haveP(t, T ) = exp(A(t, T )− B(t, T )r(t))

• Equating the two terms and setting t = 0we get

R(0, T ) = −1T(A(0, T )− B(0, T )r(0))




• We now choose the parameter vector θ to minimize the squared distancebetween the market yields R(0, Ti) and the theoretical yields R(0, Ti):

minθ

n∑

i=1

R(0, Ti)− R(0, Ti)

2

• We are fitting the model underQ

• We do not fit to historical data! Historical data gives the past dynamicsunder P




Time To Maturity (Log Scale)

Yield

2.0%

2.4%

2.8%

3.2%

3M 6M 1Y 2Y 5Y 10Y 20Y




• The Vasicek and CIR models are equilibriummodels• They are nice and simple: we can obtain formulas for prices• The disadvantage of equilibriummodels is that they do no automaticallyfit today’s term structure• Can you trust the price of a bond option when the model does not pricethe underlying bond correctly?• A 1% error in the price of the underlying bond could lead to a 25% error inan option price




• A hedge fundmight view the dierence between the model prices and themarket prices as “the market is mispricing certain assets” and build atrading strategy to capture the mispricings




• There exist models that fit the current market perfectly: they are calledno-arbitragemodels (e.g., Ho & Lee, Hull & White)• In an equilibriummodel, today’s term structure of interest rates is anoutput• In a no-arbitrage model, today’s term structure of interest is an input



Aine Short-Rate Models



Aine Models

• We have seen that both the Vasicek and CIR models have T -bond pricesthat have the aine form

P(t, T ) = exp(A(t, T )− B(t, T )r(t))

• An aine transformation in mathematics is a linear transformationfollowed by a translation (i.e., x 7→ Bx + a).• The form of A and B depend on the model• Q: Are there any other models that have an aine form for P(t, T )?



Aine Models

• Consider the abstract SDE for r(t)

dr(t) = m(t, r(t))d t + s(t, r(t))dfW (t)

wherefW is a BM under EMMQ

• Suppose thatP(t, T ) = exp(A(t, T )− B(t, T )r(t))

• Apply Itô’s formula with f (t, x) = exp(A(t, T )− B(t, T )x) to get

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

∂ A∂ t−∂ B∂ t

r(t)− Bm+12

Bs2

d t − BsdfW (t)



Aine Models• But underQ, we know that

dP(t, T ) = P(t, T )[r(t)d t + S(t, T )dfW (t)]

where S(t, T ) is the volatility of P(t, T )• This follows as all tradable assets must have expected returns given by therisk-free rate r(t) underQ

• SetG(t, r) :=

∂ A∂ t−∂ B∂ t

r(t)− Bm(t, r) +12

Bs(t, r)2

• We need G(t, r) = r(t) for all t and r. Equivalently,

g(t, r) := G(t, r)− r(t) = 0, ∀t, r



Aine Models

• Now, to ensure this holds we need

∂ 2g∂ r2

= −B(t, T )∂ 2m(t, r)∂ r2

+12

B(t, T )2∂ 2(s(t, r)2)∂ r2

= 0

• Clearly we need B(t, T ) 6= 0 for all t, T which implies

−∂ 2m(t, r)∂ r2

+12

B(t, T )∂ 2(s(t, r)2)∂ r2

= 0

• Hence we conclude

∂ 2(s(t, r)2)∂ r2

= 0 and∂ 2m(t, r)∂ r2

= 0

• Linear functions f (x) = a+ bx satisfy f ′′(x) = 0 for all x



Aine Models

TheoremIf the dri and volatility of r(t) take the form

m(t, r(t)) = a(t) + b(t)r(t) and s(t, r(t)) =Æ

γ(t)r(t) +δ(r)

this is suicient for P(t, T ) to have the aine form

P(t, T ) = exp(A(t, T )− B(t, T )r(t))

where a(t), b(t), γ(t) and δ(t) are deterministic functions



Aine Models

Example (Vasicek Model)γ= 0, δ = σ2, b = −α and a = αµwhich gives

dr(t) = α(µ− r(t))d t +σdfW (t).

We already know it has the aine form.



Aine Models

Example (CIR Model)δ = 0, γ= σ2, b = −α, and a = αµwhich gives

dr(t) = α(µ− r(t))d t +σÆ

r(t)dfW (t).

Again, we already know it has the aine form.



Aine Models

Example (Merton Model)γ= 0, δ = σ2, b = 0which gives

dr(t) = ad t +σdfW (t)

and we can determine

B(t, T ) = T − t

A(t, T ) =16σ2(T − t)3 −

12

a(T − t)2



Aine Models

Example (Pearson and Sun Model)δ = σ2, γ= −β/σ2, b = −α and a = α(µ+ β)which gives

dr(t) = α(µ− r(t))d t +σÆ

r(t)− βdfW (t).

Note:• This model is similar to CIR• Now, the minimum value for r(t) is β• One can derive A(t, T ) and B(t, T ) in terms of the A and B from the CIR



Week 9

No-Arbitrage Models

The Heath-Jarrow-Morton (HJM) Framework



No-Arbitrage Models



Introduction

• We shall now consider the class of no-arbitragemodels• We saw that with the Vasicek and CIRmodels, our theoretical prices do notmatch all the bond prices we observe in the market• By design, no-arbitrage models take the observed prices in the market asan input into the model• This means that the theoretical prices P(t, T )match the observed prices

Pobs(t, T ) at the time of calibration t



Introduction

• Market practitioners prefer no-arbitrage models over equilibriummodels• Market makers need to both buy and sell at quoted prices. If their pricingmodel gives dierent prices than the rest of the market then this will giverise to arbitrage opportunities against the market maker• No-arbitrage models allowmarket makers to check market scenarios (e.g.,what happens to other prices if the short-term rate increases?)• Market makers are typically considered with the pricing and hedging ofshort-term claims. No-arbitrage models work well in this situation• Banks typically want to price OTC derivatives in a way which is consistentwith the closest-matching traded derivatives



Markov Models

We shall consider models for which:• the current bond prices in the market are an input• the probability law of P(s, T ) givenFt where t ≤ s < T is the same as

P(s, T ) given the current value of the short-rate r(t)



Aine Term Structures

• Recall that a model is called aine if the T -bond prices are given by

P(t, T ) = exp (A(t, T )− B(t, T )r(t))

for functions A and B• Some no-arbitrage models are also aine



Aine Term Structures

TheoremGiven an aine short-rate model

dr(t) = (b(t) + β(t)r(t))d t +Æ

a(t) +α(t)r(t)dfW (t)

then the functions Aand B satisfy the system of ODE:

∂tA(t, T ) = −12

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

∂t B(t, T ) =12α(t)B2(t, T )− β(t)B(t, T )− 1, B(T, T ) = 0.

andP(t, T ) = exp (A(t, T )− B(t, T )r(t))



Ho and Lee Model

• In 1986, Ho and Lee considered the model

dr(t) = θ (t)d t +σdfW (t)

wherefW (t) is a BM under the EMMQ

• This is a more general version of the Merton model where the dri θ (t) isnow a function of time• We also have the current term structure as an input:

P(0, T ) for all T > 0



Ho and Lee Model

• Ho and Lee model is


• So in the aine term structure theorem:

b(t) = θ (t), β(t) = 0, a(t) = σ2, α(t) = 0

• To find A and B, we need to solve the ODE:

∂tA(t, T ) = −12σ2B2(t, T ) + θ (t)B(t, T ), A(T, T ) = 0,

∂t B(t, T ) = −1, B(T, T ) = 0.



Ho and Lee Model

• Set g(s) := B(s, T ) and integrate g ′(s) = ∂sB(s, T ) from t to T :∫ T

t

g ′(s) ds =

∫ T

t

−1 ds

• Use the boundary condition B(T, T ) = 0 to get

g(T )=0

−g(t) = −(T − t) ⇒ B(t, T ) = T − t



Ho and Lee Model

• By integrating ∂sA(s, T ) from t to T :∫ T

t

∂sA(s, T ) ds =

∫ T

t

−12σ2B2(s, T ) ds+

∫ T

t

θ (s)B(s, T ) ds

• Substituting B(t, T ) = T − t and using the condition B(T, T ) = 0we get

A(t, T ) =

∫ T

t

12σ2(T − s)2 ds−

∫ T

t

θ (s)(T − s) ds

=σ2

6(T − t)3 +

∫ T

t

θ (s)(T − s) ds



Ho and Lee Model

• Hence, T -bond prices in the Ho and Lee model are given by

P(t, T ) = exp (A(t, T )− B(t, T )r(t))

withB(t, T ) = T − t

A(t, T ) =σ2

6(T − t)3 −

∫ T

t

θ (s)(T − s) ds



Ho and Lee Model

• In an aine model, T -bond prices are given by

P(t, T ) = exp (A(t, T )− B(t, T )r(t))

• The forward curve is given by

f (t, T ) = −∂

∂ Tlog P(t, T )

= −∂

∂ T[A(t, T )− B(t, T )r(t)]



Ho and Lee Model

• For the Ho and Lee model, we have

∂T B(t, T )r(t) = r(t)

∂T A(t, T ) =σ2

2(T − t)2 +

∫ T

t

θ (s) ds

• Hence,

f (t, T ) = −σ2

2(T − t)2 +

∫ T

t

θ (s) ds+ r(t)



Ho and Lee Model

• Suppose we have all the market prices Pobs(0, T ) for T > 0

• Then we have the market forward curve

fobs(0, T ) = −∂

∂ Tlog Pobs(0, T )

• And our theoretical forward curve

f (0, T ) = −σ2

2T 2 +

∫ T

0

θ (s) ds+ r(0)

• Q: How do we calibrate our model so that f (0, T ) = fobs(0, T )?



Ho and Lee Model

• We now chooseθ (s) :=

∂

∂ sfobs(0, s) +σ2s

• Then∫ T

0

θ (s) ds =

∫ T

0

∂

∂ sfobs(0, s) ds+

∫ T

0

σ2s ds

= fobs(0, T )− fobs(0,0) +σ2T 2

= fobs(0, T )− r(0) +σ2T 2

• We now have thatf (0, T ) = fobs(0, T )

• Our theoretical forward curve matches the market forward curve exactly!



Ho and Lee Model

• Given a forward curve f (t, T ) at time t , our T -bond prices are given by

P(t, T ) = exp

−∫ T

t

f (t, u) du

• In the Ho and Lee model, by integrating we get

P(t, T ) = exp

−∫ T

t

fobs(0, u) du+ fobs(0, u)(T − t)

−σ2

2t(T − t)2 − (T − t)r(t)



Ho and Lee Model

• The solution of the short-rate SDE


is easily found to be

r(t) = r(0) +

∫ t

0

θ (s) ds+σfW (t)

• With our choice of θ (s) := ∂∂ s fobs(0, s) +σ2s we get

r(t) = fobs(0, t) +σ2 t2

2+σfW (t)



Ho and Lee Model

• This means that r(t) fluctuates along a modified initial forward curve



Hull and White Model

• In 1990, Hull and White proposed a generalisation of the Vasicek modelgiven by

dr(t) = α(µ(t)− r(t)) d t +σ dfW (t),

wherefW (t) is a BM underQ and µ(t) is a deterministic function of time• In the Vasicek model, µ(t) = µ is constant




• This is an aine model with

b(t) = αµ(t), β(t) = −α, a(t) = σ2, α(t) = 0

• We get the same B(t, T ) as the Vasicek model:

B(t, T ) =1α

1− e−α(T−t)

• Similar to the Ho and Lee model, by integrating:

A(t, T ) =σ2

2

∫ T

t

B2(s, T ) ds+

∫ T

t

αµ(s)B(s, T ) ds




• The theoretical initial forward curve is

f (0, T ) = −∂T A(0, T ) + ∂T B(0, T )r(0)

• Notice that

∂T B(t, T ) = ∂T1α

1− e−α(T−t)

= e−α(T−t)

∂t B(t, T ) = −e−α(T−t)

• Hence,∂T B(s, T ) = −∂sB(s, T )




• We now get

f (0, T ) = −∂T A(0, T ) + ∂T B(0, T )r(0)

=σ2

2

∫ T

0

∂sB2(s, T )ds+

∫ T

0

αµ(s)∂T B(s, T )ds+ ∂T B(0, T )r(0)

= −σ2

2B2(0, T ) +α

∫ T

0

e−α(T−s)µ(s)ds+ e−αT r(0)

= −σ2

2α2(1− e−αT )2 +α

∫ T

0

e−α(T−s)µ(s)ds+ e−αT r(0)




• We nowwant to fit this model to the forward curve observed in themarket: fobs(0, T )• If we choose

µ(s) =1α∂s fobs(0, s) + fobs(0, t) +

σ2

2α2(1− e−αt)2

then we getf (0, T ) = fobs(0, T )



Hull and White Model• From the properties of the Ornstein-Uhlenbeck process, the short-rate r(t)evolves like

r(t) = e−αt r(0) +α

∫ t

0

e−α(t−s)µ(s) ds+σ

∫ t

0

e−α(t−s) dfW (s)

• As

α

∫ t

0

e−α(t−s)µ(s) ds = fobs(0, t)− e−αt r(0) +σ2

2α2(1− e−αt)2

we get

r(t) = fobs(0, t) +σ2

2α2(1− e−αt)2 +σ

∫ t

0

e−α(t−s)dfW (s)



Hull and White Model• As the distributional properties of the Hull and White model are similar tothe Vasicek model, pricing European options is straight-forward• The price at time t of a European Call option on an S-bond with maturitydate T and strike price K is

C(t) = P(t, S)Φ(d1)− KP(t, T )Φ(d2)

where Φ is the CDF of a standard Normal and

d1 =1σp

logP(t, S)

KP(t, T )+σp

2, d2 = d1 −σp

σp :=σ

α(1− e−α(S−T ))

√

√1− e−2α(T−t)

2α



Dothan Model

• In 1978, Dothan proposed the model

dr(t) = σr(t) dW (t)

whereW (t) is a P-BM• Themarket price of risk is assumed constant, which gives

dr(t) = β r(t) d t +σr(t) dfW (t)

underQwherefW (t) is aQ-BM• Notice that this is a lognormal model like the Black-Scholes model for ashare price



Dothan Model

• The solution to the SDE is

r(t) = r(s)exp

β −σ2

2

(t − s) +σ(fW (t)−fW (s))

, s ≤ t

• So theFs-conditional distribution of r(t) is lognormal

EQ[r(t)|Fs] = r(s)eβ(t−s)

VarQ[r(t)|Fs] = r2(s)e2β(t−s)

eσ2(t−s) − 1

• In lognormal models, r(t) cannot go negative which is desirable



Dothan Model• No closed-form solution for T -bond prices exists• Recall that the cash-bond is given by

B(t) = exp

∫ t

0

r(u) du

• Take ε > 0 small, then

EQ[B(ε)] = EQ

exp

∫ ε

0

r(u) ds

≈ EQ

exp

r(0) + r(ε)2

ε

≈ EQ

eeX

, X ∼ Normal

=∞



Dothan Model

• This means that the Dothanmodel cannot be used to price Eurodollarfutures• This is whyMarket Models have been introduced



Black-Karasinski Model

• In 1991, Black and Karasinski introduced the following model.• Start with Y (t) = log(r(t)), whereby

dY (t) = α(t)(logµ(t)− Y (t)) d t +σ(t) dfW (t)

wherefW (t) is aQ-BM• Applying Itô’s formula gives

dr(t) = α(t)r(t)

logµ(t) +σ(t)2

2α(t)− log r(t)

d t +σ(t)r(t) dfW (t)



Black-Karasinski Model

• Under this model, r(t) has a local mean-reversion level of

µ(t)exp

σ2(t)2α(t)

• Like the Dothan model, there is no closed-form solution for T -bond prices• It is a lognormal model• Similar to the Dothanmodel, we get

EQ[B(ε)]→∞ as ε→ 0



The Heath-Jarrow-Morton (HJM) Framework



The HJM Framework

• The HJM approach gives a very general framework that we can work in todevelop a no-arbitrage model• It describes how a general model for the term-structure should evolve in away that is arbitrage-free and which the initial forward-rate curve is part ofthe input



The HJM Framework

• We start with infinitely many processes f (t, T ) for 0≤ t < T : one processfor each T ∈ R+• Wewill consider the one-factor version: all f (t, T ) are driven by the sameBrownian motionW (t)• We have that f (t, T ) satisfies

d f (t, T ) = α(t, T )d t +σ(t, T )dW (t)

for any fixed maturity T



The HJM Framework

• Or equivalently,

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

∫ t

0

α(s, T ) ds+

∫ t

0

σ(s, T ) dW (s)

• The coeicients α andσmay depend upon f (t, T ) or the whole forwardcurve at time t• Or evenmore generally, uponFt = σ(W (s) : s ≤ t)• As all f (t, T ) depend on the same BM, the changes over the wholeforward rate curve are perfectly related to each other (in a nonlinear way)



The HJM FrameworkTechnical Conditions

• For all T , α(t, T ) andσ(t, T ) are previsible and depend only upon thehistory ofW (s) up to time t

•∫ T

0

∫ T

0

|α(s, t)| dsd t <∞ for all T

• sups,t≤T‖σ(s, t)‖<∞ for all T (for eachω-path)



The HJM Framework

• We have the forward curve dynamics under P given by

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

∫ t

0

α(s, T ) ds+

∫ t

0

σ(s, T ) dW (s)

• The key insight of HJM (1980) is: if we assume that the dynamics of f (t, T )lead to an arbitrage-free model then there must be a “dri condition”• Our aim is now to determine what this dri condition is



The HJM Framework

• Define the risk-free asset• Define the tradable assets• Change measure from P to EMMQ

• Determine the replicating strategy for a claim X• Workout the “dri restriction” for this all to work



The Risk-Free Asset

• We now need to define the risk-free asset in the HJM framework• The risk-free asset is simply the cash account B(t)which satisfies

dB(t) = r(t)B(t) d t

• Recall that it is called “risk-free” as there is no BM term: there is no pricerisk, in the worst case you get back your initial deposit B(0)• Solving the SDE, we get

B(t) = B(0)exp

∫ t

0

r(u) du



The Risk-Free Asset

• Since

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

∫ t

0

α(s, T )ds+

∫ t

0

σ(s, T )dW (s)

we get

r(T ) = limtT

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

∫ T

0

σ(s, T ) dW (s) +

∫ T

0

α(s, T ) ds

• r(T )may or may not be Markov depending on the form ofσ(s, T )



The Risk-Free Asset

• We now plug the HJM formula for r(t) into B(t) to get

B(t) = B(0)exp

∫ t

0

f (0, u) du+

∫ t

0

∫ t

s

α(s, u) duds

+

∫ t

0

∫ t

s

σ(s, u)du

dW (s)

• Noticed that we changed the order of integration∫ t

0

∫ u

0

σ(s, u)dW (s)

du=

∫ t

0

∫ t

s

σ(s, u)du

dW (s)



Tradable Assets• Our tradable assets will be all the zero-coupon bonds with prices P(t, T )for T > t• Our T -bond prices are given by

P(t, T ) = exp

−∫ T

t

f (t, u) du

• Plugging in our forward curve f (t, T )we get

P(t, T ) = exp

−∫ t

0

∫ T

t

σ(s, u) du

dW (s)−∫ T

t

f (0, u) du

−∫ t

0

∫ T

s

α(s, u) duds



Tradable Assets

• Define the discounted asset price

Z(t, T ) :=P(t, T )

B(t)

• Combining integrals and taking B(0) = 1, we get

Z(t) = exp

∫ t

0

S(s, T )dW (s)−∫ T

0

f (0, u) du−∫ t

0

∫ T

s

α(s, u) duds

with

S(s, T ) := −∫ T

s

σ(s, u) du



Tradable Assets

• S(s, T ) is interpreted as the volatility of P(t, T )• Note that S(s, T ) is negative which makes sense as bond prices go downwhen interest rates go up• What are the dynamics of Z(t, T )?



Tradable Assets

• Using Itô’s formula,

dZ(t, T ) = Z(t, T )

12

S2(t, T )−∫ T

t

α(t, u)du

d t + S(t, T )dW (t)



Change of Measure

• We nowwant to make the discounted asset price into a martingale• We achieve this by changing measure• The required change of measure dri (market price of risk) for the T -bondis

γ(t) =12

S(t, T )−1

S(t, T )

∫ T

t

α(t, u) du

• We assume that γ(t) satisfies the Novikov condition



Change of Measure

• By Girsanov’s theorem, there exists a measureQ equivalent to P such that

fW (t) =W (t) +

∫ t

0

γ(s) ds

is aQ-BM• UnderQwe have

dZ(t, T ) = Z(t, T )S(t, T )dfW (t)

• Therefore, Z(t, T ) is a martingale underQ (assuming S(t, T ) satisfies thetechnical condition)



Change of Measure

• It follows that the dynamics of the T -bond prices underQ are given by

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

r(t) d t + S(t, T )dfW (t)

• So all T -bonds have expected return equal to the risk-free rate r(t) underQ

• We now need to determine the relationship between α(t, T ),σ(t, T ), andγ(t) so that the model is arbitrage-free



Replicating Strategies

• Suppose that we have a claim X at time S with S < T on a T -bond• We now determine a hedging strategy for X in terms of the cash holdingwith value B(t) and the T -bond with value P(t, T )




• Find the equivalent measureQ under which Z(t, T ) is a martingale• Define theQ-martingale D(t) = EQ[B(S)−1X |Ft]• Find the previsible processφ(t) such that

D(t) = D(0) +

∫ t

0

φ(s) dZ(s, T )

• Defineψ(t) = D(t)−φ(t)Z(t, T )• The trading strategy (ψ(t),φ(t)) represents the number of units of B(t)and P(t, T ) respectively is self-financing, replicating strategy for the claimX at time S




• We have already found the EMMQ under which Z(t, T ) is a martingale• We define D(t) = EQ[B(S)−1X |Ft]• D(t) is automatically aQ-martingale by the tower property for conditionalexpectations




• By the Martingale Representation Theorem, there exists a previsibleprocessφ(t) such that

D(t) = D(0) +

∫ t

0

φ(s) dZ(s, T )

• Alternatively written dD(t) = φ(t)dZ(t, T )




• Suppose we have a trading strategy which holdsφ(t) units of the T -bondwith price P(t, T )• Setψ(t) = D(t)−φ(t)Z(t, T ) to be the amount of cash B(t) that we holdat time t• The value of the portfolio at time t is given by

V (t) = B(t)D(t) = B(t)EQ[B(S)−1X |Ft]




• The instantaneous change in the portfolio is

dV (t) = d(B(t)D(t))= D(t)dB(t) + B(t)dD(t)= r(t)B(t)D(t)d t + B(t)φ(t)dZ(t, T )

as dB(t) = r(t)B(t)d t and dD(t) = φ(t)dZ(t, T )




• We have

dP(t, T ) = d(B(t)Z(t, T )) = r(t)B(t)Z(t, T )d t + B(t)dZ(t, T )

• The instantaneous investment gain is

ψ(t)dB(t) +φ(t)dP(t, T )= [D(t)−φ(t)Z(t, T )]B(t)r(t)d t

+φ(t)[r(t)B(t)Z(t, T )d t + B(t)dZ(t, T )]= r(t)B(t)D(t)d t +φ(t)B(t)dZ(t, T )= dV (t)




• The investment is self-financing as

ψ(t)dB(t) +φ(t)dP(t, T ) = dV (t)

• Therefore, V (t) is the value at time t of the claim X with maturity S



The Arbitrage-Free Market

• Suppose that X = 1 at maturity S then X is an S-bond payo• We have determined that the arbitrage-free price is then

P(t, S) = B(t)EQ

B(S)−1|Ft

= EQ

exp

−∫ S

t

r(u) du

Ft




• The discounted S-bond is

Z(t, S) =P(t, S)B(t)

= EQ

B(S)−1|Ft

so that Z(t, S) is aQ-martingale• This must be true for any maturity S that we choose• This implies that all bonds must be turned intoQ-martingales by the samechange of measure: they all have the samemarket price of risk γ(t)




• Thus for all maturities T ,

12

S(t, T )−1

S(t, T )

∫ T

t

α(t, u) du= γ(t)

• Or equivalently,∫ T

t

α(t, u) du=12

S(t, T )2 − γ(t)S(t, T )




• Recall that

S(t, T ) = −∫ T

s

σ(s, u) du

so that∂

∂ TS(t, T ) = −σ(t, T )

• And dierentiating∫ T

t α(t, u) duwith respect to T we get

α(t, T ) = σ(t, T ) [γ(t)− S(t, T )]




• Now recall that our original model for the forward dynamics under Pwas

d f (t, T ) = α(t, T )d t +σ(t, T )dW (t)

• Substituting in our “arbitrage-free dri” α(t, T ) and changing toQ gives

d f (t, T ) = α(t, T )d t +σ(t, T )

fW (t)− γ(t)d t

= σ(t, T )[γ(t)− S(t, T )]d t +σ(t, T )dfW (t)−σ(t, T )γ(t)d t

= −σ(t, T )S(t, T )d t +σ(t, T )dfW (t)




• Therefore, underQwe have

d f (t, T ) = −σ(t, T )S(t, T )d t +σ(t, T )dfW (t)

• Notice that the dynamics are solely determined from the volatilityσ(t, T )• Consequently,

r(t) = f (0, t)−∫ t

0

σ(s, T )S(s, t) ds+

∫ t

0

σ(s, t)dfW (s)



Week 10

Multifactor Models



Multifactor Models



Introduction

• We shall now look at models which include more than one source ofrandomness• When you look at historical interest rate data to see that changes ininterest rates with dierent maturities are not perfectly correlated aspredicted by one-factor model• We can look at some data fromhttp://www.rba.gov.au/statistics/tables/


http://www.rba.gov.au/statistics/tables/


Cash Yield Vs. Bond Yield

2000 2005 2010

34

56

78

Year

Yie

ld (%

)



Cash Yield Vs. Bond Yield

3 4 5 6 7 8

34

56

78

Short Yield (%)

Long

Yie

ld (%

)



Introduction

• Data analysis shows that multi-factor models may be more appropriate• Clearly, a one-factor time-homogeneous model is inappropriate



Introduction

• One of the main reasons is to deal with more-complex interest rateoptions which refer to two or more stochastic underlying quantities• For example, consider the option with the following payo at time T :

X =max [R(T, T + 5)− R(T, T + 10), 0]

• A one-factor model would probably overprice this contract due to theperfect (non-linear) correlation between the rates in such a model



Multi-factor Aine Models

• We have already seen one-factor aine models: Vasicek (1977),Cox-Ingersoll-Ross (1985), Ho-Lee (1986), and Hull-White (1990)• We now look at multi-factor models which have the aine form




• Consider an n-dimensional diusion model

X (t) =

X1(t)X2(t)...

Xn(t)

• Themodel is called aine if the T -bond prices are of the form

P(t, T ) = exp

A(t, T ) +n∑

j=1

B j(t, T )X j(t)




• Or written in ‘vector notation’ as

P(t, T ) = exp

A(t, T ) + B(t, T )T X (t)

where

B(t, T ) =

B1(t, T )B2(t, T )

...Bn(t, T )

• B(t, T )T is the transpose of B(t, T )




• Themodel is time homogeneous if X is time homogeneous and A(t, T )and B(t, T ) are functions of T − t only• We shall now restrict ourselves to the time homogeneous models forsimplicity and use A(T − t) and B(T − t)




• Recall that for one-dimensional aine models of the short-rate r(t)wemust have

dr(t) = (a+ br(t))d t +Æ

γr(t) +δdfW (t)

• Q: What formmust the SDE of X (t) have in the n-dimensional case if X (t)is aine?



Multi-factor Aine ModelsTheorem (Duie and Kan 1996)Suppose P(t, t +τ) = exp[A(τ) + B(τ)T X (t)]. Then X (t) has the SDE

dX (t) = (α+ BX (t))d t + SD(X (t))dfW (t),

where α= (α1, . . . ,αn)T is a constant vector, B = (bi j) and S = (σi j) are aconstant matrices and D(X (t)) is the diagonal matrix

D(X (t)) =

q

γT1 X (t) +δ1 0 · · · · · · 0

0Æ

γT2 X (t) +δ2 0 · · · 0

... . . . ...

... . . . ...0 · · · · · · 0

Æ

γTn X (t) +δn

where each δi ∈ R and each γi = (γi1, · · · ,γin)T is a constant vector.Dale Roberts (c) 2011-2014 Last Modified: 14/10/2014 08:19 465 / 561



• Within this framework, we need to ensure that each of the volatilityprocesses

γTi X (t) +δi,

for i = 1, . . . , n remain positive (preferably strictly positive)• Duie and Kan (1996) provide conditions on the parameters for thevolatility processes to remain positive




• Consider the spot rates in this framework

R(t, t +τ j) = −1τ j[A(τ j) + B(τ)T X (t)]

for j = 1, . . . , n for terms to maturity τ1 < τ2 < · · ·< τn

• In vector notation, R(t) = (R(t, t +τ1), . . . , R(t, t +τn))T , giving

R(t) = aR + BRX (t)

for a constant vector aR ∈ Rn and constant matrix BR ∈ Rn×n




• If the matrix BR is invertible (i.e., exists B−1R such that B−1

R BR = I ) then

X (t) = B−1R (R(t)− aR)

• Hence,P(t, t +τ) = exp

A(τ) + B(τ)T B−1R (R(t)− aR)

= exp

A(τ) + B(τ)R(t)

for suitable functions A(τ) and B(τ)• Therefore, it the model is aine in X (t) then it can be reformulated in away which is aine in R(t)




• These transformations make the n-factor model simple to calibrate if weare given at time t the market spot rates

R(t, t +τ1), . . . , R(t, t +τn)

• Choosing a specific model gives A(τ) and B(τ) then the bond prices are

P(t, t +τ) = exp

A(τ) + B(τ)R(t)




• Models that are time-homogeneous underQ do not need to be timehomogeneous under the real-world measure P

• Assume that the vector of market prices of risk

λ(t) = (λ1(t), . . . ,λn(t))T

is previsible and satisfies the Novikov condition• LetfW (t) be a n-dimensionalQ-BM. Then

W (t) :=fW (t) +

∫ t

0

λ(s) ds

is a (n-dimensional) P-BM




• The SDE for X (t) under P then becomes

dX (t) = [α+ BX (t)]d t + SD(X (t))[dW (t) +λ(t)d t]= [α+ BX (t) + SD(X (t))λ(t)]d t + SD(X (t))dW (t)




We shall now consider three special cases in the aine framework:• Gaussian multi-factor models• Generalized CIR models• The Longsta and Schwartz model (1992)



Gaussian Multi-factor Models

• Gaussian models allow interest rates to become negative• The first multifactor extension of the Vasicek (1977) model was developedby Langetieg (1980)• Later works include Beaglehole-Tenney (1991) for the general theory andBabbs-Nowman (1999) for model calibration• We shall now consider the theory of time-homogeneous Gaussian models




• Let X (t) be an n-dimensional diusion process with SDE

dX (t) = BX (t)d t + KfW (t)

where B and K are real-valued constant n× nmatrices andfW (t) is an-dimensionalQ-BM• The risk-free rate of interest is

r(t) = µ+ θ T X (t)

where θ = (θ1, . . . ,θn)T ∈ Rn

• If all θi 6= 0we can scale X (t) and assume θi = 1




• Thematrix B has a spectral decomposition B = BRΛBL

• Λ= diag(λ1, . . . ,λn) is the diagonal matrix of eigenvalues of B (somethese eigenvalues may be complex)• BL and BR are the matrices of le and right eigenvectors of B. Column i of

BR is the right eigenvector of B corresponding to λi

• The columns of BR are scaled in a way which ensures BRBL = I




• This decomposition is not unique• For X (t) to be stationary we require that the real parts of all eigenvaluesλ1, . . . ,λn are negative• We have Bk = BRΛ

kBL which is useful for defining the exponential of amatrix




• Recall that ex = 1+ x + x2

2! +x3

3! + . . . so we have

exp(B) = BR exp(Λ)BL

where, aer definingΛ0 = I ,

exp(Λ) = I +Λ+Λ2

2!+Λ3

3!+ . . .

=∞∑

k=0

Λk

k!




• DefineY (t) := e−tΛBLX (t)

• Applying the n-dimensional Itô’s formula,

dY (t) = e−tΛBLKdfW (t)

• Hence, we have

Y (t) = Y (0) +

∫ t

0

e−uΛBLKdfW (u)




• But as Y (t) = e−tΛX (t) and BRBL = I we have

X (t) = BRetΛY (t)

• Using the fact that BRetΛBL = etB, it follows that

X (t) = etBX (0) +

∫ t

0

e(t−u)BKdfW (u)

• This is an n-dimensional Orstein-Uhlenbeck process (i.e., solution ton-dim Vasicek model that mean-reverts to zero)




• The requirement that the real parts of the eigenvalues λ1, . . . ,λn arenegative ensures that

exp(tΛ)→ 0 and exp(tB)→ 0

as t →∞• Think of the 1-dimensional case where a ∈ R, then

e−ta→ 0 as t →∞




• Now let

R(T ) =

∫ T

0

r(t) d t

= µT +

∫ T

0

θ T X (t) d t

• This is Normally distributed with

EQ[R(T )] = µT + θ T BRΛ−1(eTΛ − I)BLX (0)

VarQ[R(T )] =

∫ T

0

θ T BRΛ−1(eTΛ − I)BLKK T BT

L (eTΛ − I)Λ−1BT

R θd t




ExampleSimplest 2-dimensional case is

B =

−1 00 −1

B = BRΛBL BR = BL = I , Λ= −I

thenEQ[R(T )] = µT + θ TΛ−1(eTΛ − I)X (0)

= −µT − θ TΛ−1(I − etΛ)X (0)

and compare to the solution to the 1-dimensional Vasicek model




• We then have that T -bond prices at t = 0 are given by

P(0, T ) = EQ

∫ t

0

r(t) d t

F0

= EQ[ex p(−R(T ))|X (0)]

= exp

−EQ[R(T )|X (0)] +12

VarQ[R(T )|X (0)]

• Recall ‘identifying Normals’: If Z ∼ N(µ,σ2) then

E

eθ Z

= exp

θµ+12θ 2σ2




Example(Beaglehole and Tenney 1991) We take

r(t) = (X1(t) +µ1)︸︷︷︸

inst. price inflation

+(X2(t) +µ2)︸︷︷︸

inst. real interest

wheredX1(t) = −α1X2(t)d t +σ11dfW1(t),

dX2(t) = −α2X1(t)d t +σ21dfW1(t) +σ22dfW2(t).

Q: What are the matrices B and K?




ExampleWe take

r(t) = µ+ X1(t),

wheredX1(t) = −α1(X2(t)− X1(t))d t +σ11dfW1(t),

dX2(t) = −α2X2(t)d t +σ21dfW1(t) +σ22dfW2(t).

In this model, µ+ X2(t) behaves like a stochastic (local) mean reversion levelfor r(t).



Generalized CIR Models

• A number of multi-factor CIR models have been proposed• In terms of the general framework

dX (t) = (α+ BX (t))d t + SD(X (t))dfW (t),

where D(X (t)) = diag(q

γT1 X (t), . . . ,

Æ

γTn X (t)) and

P(t, t +τ) = exp[A(τ) + B(τ)T X (t)]

• The risk-free rate of interest is then defined as

r(t) =n∑

i=1

X i(t)



Generalized CIR Models• As D(X (t)) is diagonal, the X i(t) are independent• We get that

P(t, T ) = EQ

exp

−∫ T

t

r(u) du

Ft

= EQ

exp

−∫ T

t

n∑

i=1

X i(u) du

Ft

=n∏

i=1

EQ

exp

−∫ T

t

X i(u) du

Ft

= exp

n∑

i=1

Ai(T − t)−n∑

i=1

Bi(T − t)X i(t)



Generalized CIR Models

• P(t, T ) is the product of n one-dimensional CIR• The coeicients are

Ai(τ) =2αiµi

σ2i

log

2γie(γi+αi)τ/2

(γi +αi)(eγiτ − 1) + 2γi

Bi(τ) =2(eγiτ − 1)

(γi +αi)(eγiτ − 1) + 2γi, γi

q

α2i + 2σ2

i

• If we have 2αiµi > σ2i then the probability that X i(t) hits zero is zero



The Longsta and Schwartz Model

• A variation of the two-factor CIR model was proposed by Longsta andSchwartz in 1992• Take for i = 1,2 the SDEs

dYi(t) = αi(µi − Yi(t)) d t +Æ

Yi(t)dfWi(t)

wherefW1 andfW2 are independentQ-Brownian motions• For constants c1 and c2, define

r(t) := c1Y1(t) + c2Y2(t), V (t) := c21Y1(t) + c2

2Y2(t)




dr(t) = c1dY1(t) + c2Y2(t)

= c1

α1(µ1 − Y1(t))d t +Æ

Y1(t)dfW1(t)

+ c2

α2(µ2 − Y2(t))d t +Æ

Y2(t)dfW2(t)

=

c1α1µ1 + c2α2µ2 −(c2α1 − c1α2)r(t) + (α2 −α1)V (t)

c2 − c1

d t

+

√

√

√ c1(c2r(t)− V (t))c2 − c1

dfW1(t) +

√

√

√ c2(V (t)− c1r(t))c2 − c1

dfW2(t)




• It can be shown that instantaneous variance of r(t) is V (t)d t which is thereason why the model was formulated this way• Also, by using r(t) and V (t)we focus on the variables that are believed toinfluence the pricing of derivatives, instead of using Y1(t) and Y2(t)• r(t) is the level and V (t) is the volatility• If c1 < c2 then the form of r(t) and V (t)means that V (t) is limited to theregion (c1r(t), c2(t))• Themodel has richer dynamics when c1 and c2 are far apart



Multifactor Heath-Jarrow-Morton Models

• We can easily extend the one-factor HJM framework to the multi-factorsetting• We assume the existence of a given model under the risk-neutral measure

Q

• Wemodel the T -bond dynamics

dP(t, T ) = P(t, T )[r(t) d t + S(t, T )T dfW (t)]

where the volatility function S(t, T ) = (S1(t, T ), . . . , Sn(t, T ))T isprevisible andfW (t) is a standard n-dim BM underQ




• The forward dynamics are given by

d f (t, T ) = −σ(t, T )T S(t, T )d t +σ(t, T )T dfW (t)

whereσ(t, T ) = (σ1(t, T ), . . . ,σn(t, T ))T

σi(t, T ) = −∂

∂ TSi(t, T )




Example(Rebonato 1996)• Suppose that primary sources of randomness in forward-rate curve are:slope of curve and twist in curve• In a Gaussian setting, this can be modelled by

σ1(t, T ) = σ11e−α1(T−t)

σ2(t, T ) = σ21e−α1(T−t) +σ22e−α2(T−t)

• Example: 0< α2 < α1 then can we achieve a twist by adding theconstraintsσ22 > 0 andσ21 < −σ22



Week 11

The Forward-Measure Approach




• In pricing problems that we have seen so far we have expressed prices interms of expectations under the risk-neutral equivalent martingalemeasuresQ

• Sometimes the calculation can be made easier by making a secondchange of measure fromQ to the forwardmeasure• In recent years, this idea has been critical in the development of totallynew approaches to interest rate modelling



A New Numeraire

• For a T -bond, we have the dynamics

dP(t, T ) = P(t, T )[r(t) d t + S(t, T )dfW (t)]

wherefW (t) is aQ-BM• Previously, we considered the discounted price process

Z(t, U) =P(t, U)

B(t)

where B(t) = exp

∫ t

0 r(s) ds

is the cash account at time t

• The discounted price was a martingale under the samemeasureQ for allmaturity dates U



A New Numeraire

• Let’s change the numeraire from B(t) to P(t, T ). Thus we consider

Y (t, U) =P(t, U)P(t, T )

• Then we get

dY (t, U) =1

P(t, T )dP(t, U)+P(t, U)d

1P(t, T )

+dP(t, U)d

1P(t, T )



A New Numeraire• Then

dY (t, U) = Y (t, U) (S(t, U)− S(t, T )) dfW (t)

+ Y (t, U)S2(t, T ) d t − Y (t, U)S(t, U)S(t, T ) d t

• Now suppose we define a new processcW (t)with

cW (0) = 0 and dcW (t) = dfW (t)− S(t, T ) d t

• Then

dY (t, U) = Y (t, U) (S(t, U)− S(t, T )) [dcW (t) + S(t, T )d t]− Y (t, U)S(t, T )[S(t, U)− S(t, T )]d t

= Y (t, U)[S(t, U)− S(t, T )]dcW (t)



A New Numeraire

• So we have

dY (t, U) = Y (t, U)[S(t, U)− S(t, T )]dcW (t)

• It appears that we can find a suitable measure under which Y (t, U) for allU > T are martingales



Change of Measure

• Define γ(t) = −S(t, T )• By Girsanov’s theorem, there exists a measure PT which is equivalent toQsuch that

cW (t) =fW (t) +

∫ t

0

γ(s) ds

is a Brownian motion under PT

• Under the assumption that Novikov’s condition holds:

EQ

exp(

∫ U

0

12γ(t)2d t)

<∞

• PT is called a forwardmeasure



Change of Measure

• The forward measure depends on the forward maturity date T but notupon the maturity date U of the bond under consideration• The Y (t, U) for U > T are all martingales under the measure PT



Derivative Payments

• Let X be a claim contingent uponFT payable at time T• Define

D(t) = EPT

XP(T, T )

Ft

= EPT[X |Ft]

• D(t) is martingale under PT

• Y (t, U) is also a martingale under PT

• By MRT, there exists a previsible processφ(t) such that

dD(t) = φ(t)dY (t, U)



A Replicating Strategy

• Letψ(t) = D(t)−φ(t)Y (t, U)• Consider the investment strategy which holds at time t ,ψ(t) units of thenumeraire P(t, T ) andφ(t) units of U-bond• The value of this portfolio at time t is

V (t) = P(t, T )D(t)

• Now

dP(t, T ) = P(t, T )[r(t)d t + S(t, T )dfW (t)]

= P(t, T )[r(t)d t + S(t, T )(dcW (t) + S(t, T )d t)]

= P(t, T )([r(t) + S2(t, T )]d t + S(t, T )dcW (t))




• And we have

dD(t) = φ(t)dY (t, U)

= φ(t)Y (t, U)(S(t, U)− S(t, T ))dcW (t)




• Thus

dV (t) = D(t)dP(t, T ) + P(t, T )dD(t) + dP(t, T )dD(t)

= P(t, T )D(t)((r(t) + S2(t, T ))d t + S(t, T )dcW (t))

+φ(t)P(t, T )Y (t, U)(S(t, U)− S(t, T ))dcW (t)

+φ(t)P(t, T )Y (t, U)[(r(t) + S2(t, T ))d t + S(t, T )dcW (t)]

× [(S(t, U)− S(t, T ))dcW (t)]

= V (t)[(r(t) + S2(t, T ))d t + S(t, T )dcW (t)]

+φ(t)P(t, U)(S(t, U)− S(t, T ))dcW (t)+φ(t)P(t, U)S(t, T )(S(t, U)− S(t, T )) d t




• The instantaneous investment gain is thus

ψ(t)dP(t, T ) +φ(t)dP(t, U)

= (D(t)−φ(t)Y (t, U))P(t, T )((r(t) + S2(t, T ))d t + S(t, T )dcW (t))

+φ(t)P(t, U)(r(t)d t + S(t, U)(dcW (t) + S(t, T )d t))= dV (t)

• Hence (ψ(t),φ(t)) is a self-financing strategy• V (T ) = X so it is also a replicating strategy




TheoremThe price at time t for the claim X payable at time T is

V (t) = P(t, T )EPT[X |Ft]

• This price is independent of the choice of the hedging asset P(t, U)• But the hedging strategy depends upon the choice of U



Evaluation of a Derivative Price

• Suppose that the claim X at T is a function of P(T, U)• Under the martingale measureQwe have

V (t) = EQ

exp

−∫ T

t

r(s) ds

X

Ft

• Evaluation of this expectation requires the knowledge of the jointdistribution of P(T, U) and exp(−

∫ T

t r(s) ds) underQ




• On the other hand, under the forward measure approach

V (t) = P(t, T )EPT[X |Ft]

• We only need to establish the distribution of X under PT

• This is a lot easier to calculate




LemmaThe forward price at time t for delivery of P(T, U) at time T is given by

EPT[P(T, U)|Ft].

Proof: Since Y (t, U) is a martingale under PT we have

EPT[Y (T, U)|Ft] = Y (t, U).

Therefore,

EPT

P(T, U)P(T, T )

Ft

= EPT[P(T, U)|Ft] =

P(t, U)P(t, T )

,

which is the forward price.




LemmaSuppose that the T -bond volatility S(v, V ) is deterministic. Then under PT ,P(t, U) is lognormal with

VarPT[log P(T, U)|Ft] = b2 =

∫ T

t

(S(v, U)− S(v, T ))2 dv

andEPT[log P(T, U)|Ft] = a = log

P(t, U)P(t, T )

−12

b2



Vasicek Model

ExampleUnder the Vasicek model,

S(v, U) =σ

α(1− e−α(U−v))

Which gives

b2 =σ2

α2

∫ T

t

(e−α(T−v) − e−α(U−v))2 dv

=σ2

2α3(1− e−α(U−T ))2(1− e−2α(T−t))




LemmaSuppose that a random variable Y has a lognormal distribution withparameters a and b2; that is, log Y ∼ N(a, b2). Then for any constant K > 0:

E[(Y − K)+] = E[Y ]Φ(h)− KΦ(h− b),

where

h=a+ b2 − log K

b=

log(E[Y ]/K) + 12 b2

bandΦ is the CDF of a standard Normal distribution.



Evaluation of a Derivative Price• We can apply this lemma to price a call option with maturity T and strikeprice K on a U-bond with price P(t, U)• We get

EPT[(P(T, U)− K)+|Ft] = EPT

[P(T, U)|Ft]Φ(h)− KΦ(h− b)

whereh=

1b

logP(t, U)P(t, T )

− log K +12

b2

=1b

logP(t, U)

KP(t, T )+

12

b

• Recall

b2 =

∫ T

t

(S(v, U)− S(v, T ))2 dv




• Hence, the price at time t of the call option is

V (t) = P(t, U)EPT[(P(T, U)− K)+|Ft]

= P(t, U)Φ(h)− KP(t, T )Φ(h− b)

• Notice the similarity to the Black-Scholes formula• This formula only holds when bond prices are lognormally distributed



Equity Options with Stochastic Interest

• The forward measure approach can also be used to price equity optionswhen the cash rate r(t) is stochastic• Suppose we have the usual model

dP(t, T ) = P(t, T )[r(t)d t + S(t, T )dfW1(t)]

wherefW (t) is a BM under the martingale measureQ

• Consider an equity index R(t)which includes full reinvestment ofdividends• UnderQ suppose that R(t) has the SDE

dR(t) = R(t)[r(t)d t +σ1dfW1(t) +σ2dfW2(t)]

wherefW2(t) is second BM underQ independent offW1(t)




• Let D(t) = EPT[ f (R(T ))|Ft]. This is a martingale under PT

• The processes

bP(t, U) =P(t, U)P(t, T )

, bR(t) =R(t)

P(t, T )

are both martingales under PT

• By the multidimensional MRT, there exist previsible processesφP(t) andφR(t) such that

dD(t) = φP(t)dbP(t, U) +φR(t)dbR(t)




• Defineψ(t) = D(t)−φP(t)bP(t, U)−φR(t)bR(t)

• Consider the portfolio process which holds at time t

φP(t) of P(t, U)φR(t) of R(t)ψ(t) of P(t, T )

• The value at time t is

V (t) = P(t, T )D(t) = P(t, T )EPT[ f (R(T ))|Ft]

• This portfolio is self-financing and replicating (t = T ) and V (t) is theunique no-arbitrage price at time t




TheoremConsider the European call option f (R(T )) = (R(T )− K)+. The price for thisoption at time t is

V (t) = P(t, T )[F(t)Φ(h)− KΦ(h− b)]

where

h=log(F(t)/K) + 1

2 b2

b

b2 =

∫ T

t

(σ1 − S(v, T ))2 dv +σ22(T − t)

F(t) =R(t)

P(t, T )[forward price for R(T )]



Week 12

Market Models



Market Models



Introduction

• Instantaneous forward rates are an unobservable quantity• This makes the forward curve T 7→ f (t, T ) diicult to estimate• This lead to the suggestion that maybe one should develop a model ofdirectly observable (i.e., market quoted) quantities such as LIBOR



Introduction

• A key modelling assumption was that relevant market rates are lognormal• This was an assumption already used by traders to price some interestrate derivatives• Themarket practice was/is to use Black’s formula



Black’s Formula

DefinitionA capletwith reset date T and settlement date T +δ pays the holder thedierence between a simple market rate F(T, T +δ) and the strike rate κ

• Its cash flow at time T +δ is

δ(F(T, T +δ)−κ)+

• Wewrite Cpl(t; T, T +δ) for the price at time t of the caplet with resetdate T and settlement date T +δ



Black’s Formula

DefinitionA Cap is a strip of caplets. It consists of:• A number of future dates T0 < T1 < · · ·< Tn with Ti − Ti−1 = δ• A cap rate κ

• Wewrite Cp(t) for its price at time t• We have

Cp(t) =n∑

i=1

Cpl(t; Ti−1, Ti)



Black’s Formula

• It is market practice to price a cap/floor according to Black’s formula• Let t ≤ T0. Black’s formula for the value of the ith caplet is

Cpl(t; Ti−1, Ti) = δP(t, Ti)[F(t; Ti−1, Ti)Φ(d1(i; t))− κΦ(d2(i; t))]

where

d1,2(i; t) =log

F(t;Ti−1,Ti)κ

± 12σ(t)

2(Ti−1 − t)

σ(t)p

Ti−1 − t

• Hereσ(t) is the cap implied volatility (it is the same for all capletsbelonging to a cap)



Black’s Formula

• Typically, t = 0, T0 = δ and δ = 3months (US) or δ = 6months (Europe)• Caps are quoted in the market in terms of their implied volatilities• So to price a Cap, one takes the implied volatilityσ(t) quoted at time tand plugs it into the Black model



Black’s FormulaATM Cap Volatilities

0 5 10 15 20 25 30

1314

1516

1718

19

Maturity (in years)

ATM

vol

s (in

%)



Black’s Formula

• This approach is completely heuristic!• It is based on the assumption that the simple market rates F(t; Ti−1, Ti)are lognormal• It is a challenge for any market realistic interest rate model to match thegiven volatility curve observed in the market



Market Models

• We are now going to look at the development ofMarket Modelswhichattempt to construct a model of observable interest rates such as LIBOR• Sometimes denoted LMM (in job ads): LIBOR Market Model• The first developments all happened very quickly in 1997• Miltersen, Sandmann & Sondermann (1997): formulae for zero-couponbond options, caps and floors• Further improvements were obtained by Brace, Gatarek, and Musiela(1997), Jamshidian (1997), and Musiela and Rutkowski (1997)



LIBOR

• There are two types of market interest rates: LIBOR and forward LIBOR• LIBOR is an annualized, simple rate of interest that will be delivered at theend of a defined period• The δ-LIBOR denoted by F(T, T +δ)means that an investment of 1 attime T will grow to 1+δF(T, T +δ) at time T +δ• LIBOR is always quoted as an annual rate of interest even though ittypically applies over shorter periods



LIBOR

• For a fixed δ > 0, the forward δ-period LIBOR for the future date Tprevailing at time t is the simple forward rate

L(t, T ) := F(t; T, T +δ) =1δ

P(t, T )P(t, T +δ)

− 1

• With this notation, L(T, T ) = F(T ; T, T +δ) is the LIBOR rate



LMM: Heuristic Derivation

• Let PT+δ be the (T +δ) forward measure• Recall that

P(t, T )P(t, T +δ)

is a martingale under PT+δ




• We also know that

d

P(t, T )P(t, T +δ)

=P(t, T )

P(t, T +δ)σT,T+δ(t)dW T+δ(t)

whereσT,T+δ(t) =∫ T+δ

T σ(t, u) du• Using this with the definition of L(t, T ) gives

d L(t, T ) =1δ

d

P(t, T )P(t, T +δ)

=1δ

P(t, T )P(t, T +δ)

σT,T+δ(t)dW T+δ(t)

=1δ(δL(t, T ) + 1)σT,T+δ(t)dW T+δ(t)




• We have

d L(t, T ) =1δ(δL(t, T ) + 1)σT,T+δ(t)dW T+δ(t)

• Assumption: there exists aRn-valued deterministic function λ(t, T ) suchthat

σT,T+δ(t) =δL(t, T )δL(t, T ) + 1

λ(t, T )

• Plugging this in to the above formula, we get

d L(t, T ) = L(t, T )λ(t, T )dW T+δ(t)




• The solution of that SDE is given by

L(t, T ) = L(s, T )exp

∫ t

s

λ(u, T )dW T+δ(u)−12

∫ t

s

‖λ(u, T )‖2 du

• So the PT+δ distribution of log L(t, T ) conditional onFt is Gaussian withmean

log L(t, T )−12

∫ T

t

‖λ(s, T )‖2 ds

and variance∫ T

t

‖λ(s, T )‖2 ds




• The time t price of a caplet with reset date T , settlement date T +δ andstrike rate κ is thus

EQ

h

e−∫ T+δ

t r(s)dsδ(L(T, T )− κ)+

Ft

i

= P(t, T +δ)EPT+δ[δ(L(T, T )−κ)+|Ft]

= δP(t, T +δ)(L(t, T )Φ(d1(t, T ))−κΦ(d2(t, T )))

where

d1,2(t, T ) :=log

L(t,T )κ

± 12

∫ T

t ‖λ(s, T )‖2 ds

∫ T

t ‖λ(s, T )‖2 ds1/2




• This is just Black’s formula for the caplet price with

σ(t) =

√

√

√ 1T − t

∫ T

t

‖λ(s, T )‖2 ds

• This means that any HJMmodel satisfying

σT,T+δ(t) =δL(t, T )δL(t, T ) + 1

λ(t, T )

for some deterministic λ(t, T ) yields Black’s formula for caplet prices




• The question is: Does such a HJMmodel exist?• The answer is yes, but the construction and proof are not easy• This is carried out in the paper by Brace, Gatarek and Musiela (1997)



LMM: Direct Approach

• We can also construct the LMM using a direct approach• We don’t use the HJM framework (although it can be implicitly implied)• Wework with the forward measure concept




• Fix a time horizon TM = Mδ with M ∈ N• Fix our probability space (Ω,F , (Ft)t∈TM

,PTM)

• Under PTMwe have a d-dimensional Brownian motionW TM

• PTMis the TM -forward measure

• Define the dates Tm := mδ form= 0, 1, . . . , M




• We now construct a model of M forward LIBOR rates with maturitiesT0, T1, . . . , TM−1

• For everym≤ M − 1, we define λ(t, Tm) as the volatility of L(t, Tm)• We assume there is an initial positive and nonincreasing term structure of

M bond pricesP(0, T0), . . . , P(0, TM−1)



LMM: Direct ApproachDefinitionFor notational convenience, if S in an Itô process and H is a predictableprocess we write

H · S =∫

H(u)dS(u)

Examples: Recall the stochastic exponential of an Itô process X is given byEt(X ) := exp(X − 1

2⟨X ⟩t) then

(λ(·, T ) ·W T )t =

∫ t

0

λ(u, T )dW T (u)

Et(λ(·, T ) ·W T ) = exp

∫ t

0

λ(u, T )dW T (u)−12

∫ t

0

|λ(u, T )|2 du




• First, let’s postulate the dynamics of L(t, TM−1) = F(t; TM−1, TM) as

d L(t, TM−1) = L(t, TM−1)λ(t, TM−1)dW TM (t), t ∈ [0, TM−1]

with initial condition

L(0, TM−1) =1δ

P(0, TM−1)P(0, TM)

− 1

• The solution is

L(t, TM−1) = L(0, TM−1)Et(λ(·, TM−1) ·W TM )




• Or plugging in the definition of L(0, TM−1)we get

L(t, TM−1) =1δ

P(0, TM−1)P(0, TM)

− 1

Et(λ(·, TM−1) ·W TM )

• Nowwe assume that

σTM−1,TM(t) =

δL(t, TM−1)δL(t, TM−1) + 1

λ(t, TM−1), t ∈ [0, TM−1]




• Recall that we changed fromQ to the forward measure PT and obtained

dW T (t) = dfW (t)− ν(t, T ) d t, ν(t, T ) := −∫ T

t

σ(t, u) du

in the HJM setup

d f (t, T ) = α(t, T )d t +σ(t, T )dW (t), under P

d f (t, T ) = σ(t, T )

∫ T

t

σ(t, u)du

d t +σ(t, T )dfW (t) underQ




• Change fromQ to PTMgives new PTM

Brownian motion

dW TM (t) = dfW (t)− ν(t, TM)d t

• Change fromQ to PTM−1gives new PTM−1

Brownian motion

dW TM−1(t) = dfW (t)− ν(t, TM−1)d t

• Equate both expression to get transformation from PTMto PTM−1

:

dW TM−1(t) = dW TM (t)− (ν(t, TM−1)− ν(t, TM))d t

= dW TM (t)−σTM−1,TM(t)d t




• Again, we postulate

d L(t, TM−2) = L(t, TM−2)λ(t, TM−2)dW TM−1(t)

L(0, TM−2) =1δ

P(0, TM−2)P(0, TM−1)

− 1

• DefineσTM−2,TM−1

(t) =δL(t, TM−2)δL(t, TM−2) + 1

λ(t, TM−2)

• And so forth... to obtain a family of M processes

L(t, T0), . . . , L(t, TM−1), t ∈ [0, TM]



LIBOR Dynamics Under Dierent MeasuresLemmaLet 0≤ m, n≤ M − 1. Then the dynamics of L(t, Tm) under PTn+1

is form< ngiven by

d L(t, Tm) = L(t, Tm)

−λ(t, Tm)n∑

r=m+1

σTr ,Tr+1(t)T d t +λ(t, Tm)dW Tn+1

Form= n,d L(t, Tm) = L(t, Tm)λ(t, Tm)dW Tn+1(t)

and form> nwe have

d L(t, Tm) = L(t, Tm)

λ(t, Tm)m∑

r=n+1

σTr ,Tr+1(t)T d t +λ(t, Tm)dW Tn+1



Bond Price Dynamics

• We have constructed a market model for LIBOR rates• However, to price bonds and options we need to determine the bond pricedynamics• So now we’ll determine the bond price dynamics and then derive a capletpricing formula



Bond Price Dynamics

• Form= 1, . . . , M we can define the forward price process

P(t, Tm−1)P(t, Tm)

= δL(t, Tm−1) + 1, t ∈ [0, Tm−1]

• Sinced


= δd L(t, Tm−1)

= δL(t, Tm−1)λ(t, Tm−1)dW Tm(t)

=P(t, Tm−1)P(t, Tm)

σTm−1,Tm(t)dW Tm(t)



Bond Price Dynamics

• Therefore, we get that


=P(0, Tm−1)P(0, Tm)

Et

σTm−1,Tm·W Tm

, t ∈ [0, Tm−1]

which is a PTmmartingale

• This is sometimes call the Tm-forward price process of the Tm−1-bond



Bond Price Dynamics

• We can extend this concept and define the Tm-forward price processes forall Tk-bonds• When k < mwe define

P(t, Tk)P(t, Tm)

=m−1∏

r=k

P(t, Tr)P(t, Tr+1)

=P(t, Tk)

P(t, Tk+1)· · ·


• For k > m, we define

P(t, Tk)P(t, Tm)

=

P(t, Tm)P(t, Tm+1)

−1

· · ·

P(t, Tk−1)P(t, Tk)

−1



Bond Price Dynamics

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 theRd -valued bounded progressive process

σTk,Tm=

¨∑m−1

r=k σTr ,Tr+1, k < m

−∑k−1

r=mσTr ,Tr+1, k > m

Hence, P(t,Tk)P(t,Tm)

for t ∈ [0, Tk ∧ Tm] is a positive PTm-martingale.



Bond Price Dynamics

• We can now derive the nominal Tn-bond prices

P(Tm, Tn) =n∏

k=m+1

P(Tm, Tk)P(Tm, Tk−1)

=n∏

k=m+1

1δL(Tm, Tk−1) + 1

at dates t = Tm

• We have derived an arbitrage-free market model for the bonds withmaturities T0, . . . , TM since we have shown that PTm

is a martingalemeasure for the Tm-bond as numeraire



LMM: Pricing Claims

• Any Tm-contingent claim X with EPTm[|X |]<∞ can now be priced at time

t ≤ Tm in terms of the Tm-bond numeraire via

V (t) = P(t, Tm)EPTm[X |Ft]

• We can also express this price relative to any future Tn-bond



LMM: Pricing Claims

LemmaThe Tm-bond discounted Tm-claim price satisfies

V (t)P(t, Tm)

=P(t, Tn)P(t, Tm)

EPTn

XP(Tm, Tn)

Ft

for allm< n≤ M .



LMM: Pricing Claims

CorollaryLetm+ 1< n≤ M . The time Tm price of the nth caplet with reset 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

d1,2(n; Tm) =log

L(Tm,Tn−1)κ

± 12

∫ Tn−1

Tm‖λ(s, Tn−1)‖2 ds

r

∫ Tn−1

Tm‖λ(s, Tn−1)‖2 ds



LMM: Pricing Claims

• This is Black’s formula for the caplet price with

σ(Tm)2 =

1Tn−1 − Tm

∫ Tn−1

Tm

‖λ(s, Tn−1)‖2 ds



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