fixed income: swaption, futures, irs, rate model

Giulio Laudani #27 Cod. 20251

FIXED INCOMEWhat is about?.........................................................................................................................................................................2

Definition and notation:...........................................................................................................................................................2

Rate used in Fixed Income...................................................................................................................................................2

Day Convention...................................................................................................................................................................3

Compounding convention:..................................................................................................................................................4

Yield Curve...........................................................................................................................................................................4

Bootstrapping techniques:..................................................................................................................................................5

How we construct the term spot curve?.............................................................................................................................6

Interest rate risk: what and how to measure......................................................................................................................7

Financial Product in the Fixed Income world:..........................................................................................................................8

Euro-deposit:.......................................................................................................................................................................8

Repo rate agreement:..........................................................................................................................................................8

FRA contract and Swap:.......................................................................................................................................................8

Futures contract:...............................................................................................................................................................12

Floating rate note:.............................................................................................................................................................13

Caps and Swaption:...........................................................................................................................................................13

Exotic corporate bond:......................................................................................................................................................15

No-Arbitrage Pricing:.............................................................................................................................................................15

Change of Numeraire:.......................................................................................................................................................15

Black Formula:...................................................................................................................................................................17

Why do we consider a risk-neutral probability measure?.................................................................................................18

Short term interest Model.....................................................................................................................................................18

Merton model...................................................................................................................................................................19

Vasicek Model...................................................................................................................................................................19

Hull and White...................................................................................................................................................................20

Dothan Model....................................................................................................................................................................21

Cox Ingersoll and Ross (CIR) Model...................................................................................................................................21

The HJM Model (1992) or Entire Curve Model:......................................................................................................................21

The Market Model.................................................................................................................................................................22

LMM..................................................................................................................................................................................23

SMM..................................................................................................................................................................................23

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What is about?This course is focus on delivering us the tools and the ability to use, to develop and to understand interest model. This subject is important1 since we need to understand market dynamics, implement hedging strategy and price all kind of interest rate instruments. The next paragraphs are an introduction to a brief description on what we are going to study, we are going to give a description, usage and features of all the main financial goods.

On the Corporate Bond side we could have: Floating note, fixed rate bond and Callable bond. Those goods have different source of risk and stochasticity: when buying a floating bond the investor is bearing the risk of change in the market yield, while by buying callable bond the investor is bearing a maturity risk, reinvestment risk is the one for fixed rate. On the Derivatives side we have (starting from the plainest vanilla one to the most exotic good): FRA contract, IRS, Swaption

Definition and notation:In Fixed income it is crucial to properly specify basically everything since the matter of this subject is to be precise, no matter what. Hence I’m going to spend some word on this issue.

Rate used in Fixed Income

The most important rate used is the LIBOR, which is the rate at which a panel of select bank based in London are available to offer money, it is an average rate computed at 11:00 am each day. Another rate is the LIBID which is the rate at which the same banks are willing to accept deposit. This rate is always higher than the Government bond market (AAA), since there is a higher risk of default, this spread has changed over time following the economy condition, however is/was used as a reference risk free rate to price option and bonds. Some other comments are need related its nature:

The panel bank must be a AA-AAA agency, so if one bank belong to the panel is going to deteriorating its market expectation it will come outside the Libor sooner or later. Libor is a trimmed average, in the sense that the highest and lowest quartile of the contributions are taken away before averaging; when risk of default is stable and homogenous, this fact is negligible, but when there is more variability this may have a relevance. This two statements determine that the expected future spread of Libor is constantly lower than the spread of the original panel

A bank can exit the Libor market, even if it does not exit the Libor panel. This was the case for many banks during the subprime crisis. Basically banks that cannot take unsecured loan will post rate similar to other institution

The Libor quotes with a spread over the T-Bills rate, the difference can be used as proxy of risk perception/price in the market

Another reference rate (EU market) is the Euribor, which is the rate earned by deposing in euro-dollar (foreign money) deposit. The maturity available are the overnight, week, one month interval till 6, than jumps to 9 and one year, then 1 year step till 5 years with the exception of the 18 month tenor rate. The day convention is the Act/360 and it has an add-on as opposed to discount factor security (treasury bond), note that the usual transaction day lag is two days (however there exit overnight transaction). The discount rate relation with price is P (t , t+1 )=1−d∗α

Other rate used are the forward rate of the previously measure. We can add those definitions to complete our overview the yield to maturity is the yield earned by investing in a security and by reinvesting all the intermediate cash flow at that rate; the par yield is the coupon rate which allow the bond to price at par.

Just for academic purpose (they are not a traded quantity) has been introduced the instantaneous spot and forward rate for infinitesimal tenor, note that those rates do not depend any longer to their maturity: for the first one the point

1 Note that the Bond market has a greater extension than the stock market

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considered in the curve, for the second the time when the “contract” starts (the time for resetting is always the same and the tenor is infinitesimal)These rates can be compute by extrapolating them from the discount factor, the reverse is not possible (we need to assume a deterministic process). The relationship between spot rate and instantaneous forward rate is possible if and only if we knew all the future evolution of the spot curve. Following the formal mathematical definition is provided:

r ( t )=limT →t

R ( t ,T )=−dP ( t ,T )dT

f (t )= lim∆t→0

F (t ,T 1, T 2 )=− lim

∆t→0( ln (P ( t , T ,T+∆ t ) )−ln (P (t , T ) ) )

∆ t=

−dln (P ( t , T ) )dT

R (t , T )= e−∫

t

T

f (s )ds

T−t the spot rate R is an average of the instantaneous forward rate

f ( t ,T )=R ( t ,T )+ (T−t )∗dR (t , T )dt

level and slope of term structure

P ( t , T )=e−∫

t

T

r (s )ds=¿> possible only∈adeterministic world

f (u ,u )=r (u ) ,hence P (t ,T )=E [e−∫t

T

f (s , s )ds ]The short and forward instantaneous rate are related (if we know all the future path evolution), by the relationship that “r” is a particular case of “f” (the starting point coincides with the resetting day).

The last relation in real life should be made with the expectation, hence we should associate a stochastic process to the infinitesimal rate. this quantity is used in the short term and whole curve model as reference entity.

Day Convention

There exist three day conventions in the market, that was the result of the global tendency to homogenize the market convention thanks to the work of the ISMA, which has developed a new yield methodology:

The Actual/365, often used in the money market or bond and Swap accrued interest computation: UK and Cad The Actual/360, as the first one: EU and USA ; Libor and Euro-Deposit quotation The 30/3602: This accrual method assumes 30 days per month and 360 days per year. Hence, the accrual factor is

simply the number of accrued days divided by 360. However, if the first date of an accrual period is not the 30th or 31st, and the last date of the period is the 31st of the month then that month is considered to have 31 days. In addition, if the last date of the period is the last day of February, the month of February shall not be extended to a 30-day month.

Act/Act is used for Government bond in EU, UK and USABeside the convention used there is another topic common to all of them: the Data Rolling procedure, meaning how to consider not open market/holiday days3, it is important to adjust it to properly compute the interest accrued. The most commonly used practices are:

Following business day: The payment date is rolled to the next business day.

2 The formula is 〖(max〗 (30−d1 )+min (d2,0 )+360∗( y2− y1 )+30∗(m2−m1−1 ) )

3603 A possible source for EU Holidays is TARGET holidays (as published by the European Central Bank)

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Modified following business day: The payment date is rolled to the next business day, unless doing so would cause the payment to be in the next calendar month, in which case the payment date is rolled to the previous business day. Many institutions have month-end accounting procedures that necessitate this.

Previous business day: The payment date is rolled to the previous business day. Modified previous business day: The payment date is rolled to the previous business day, unless doing so would

cause the payment to be in the previous calendar month, in which case the payment date is rolled to the next business day. Many institutions have month-end accounting procedures that necessitate this

To compute accrued interest (Coupon Bond) the used convention is A

∫ ¿=

cm

∗d 1

d2

¿, where c is the coupon rate, m the

coupon frequency, d1 is the number of day from the last coupon payment, and d2 is the number of days between the last and following coupon day. The market price is the sum of clean price plus accrued interest.

Compounding convention:

The compounding conventions used are: The simply one for the FRA and Caps market P(t ;T )[1+L( t ;T )∗r∗(t ;T )] The option pricing formula refers to the continuously compounded one. P ( t , T )eR (t , T )(T−t )

Another compounding convention is the annually one used to price Treasury bond P (t , T ) (1+Y (t , T ) )T−t

Interest rate curvesNow we are going to describe how to build an interest rate curve and the discount curve, going through a description on which rates are in the market, how to treat them, moving forward bootstrapping techniques and finally how to end up with our final curve to be used to fit our models.

Yield Curve

First of all what is a yield curve: “It is the graph of the function mapping maturities into rates at time t, it associates at each maturity the value of the spot rate from time t to the desired maturity”. Now how many of them there exist: the nature of interest rates determines the nature of the term structure: depending on which rate4 we are interested in, we can get different types of term structures: spot curve, swap curve, yield curve, par yield, forward curve and so on. Furthermore we can define a curve for each different issuer, that determines the term structure of yield spreads between different issuer and finally we distinguish from market and implied curve: the first ones (yield and swap curve) are read directly from the market, the others (spot, par and forward) are constructed using market data.

The yield spread displays the yields of coupon-bearing bonds as function of time to maturity. The absolute yield spread between any two bond issues, bond X and bond Y, is computed as the difference of each yield. This traditional yield spread is also known as the nominal spread. The drawback of this approach is that there is no reason to expect the credit spread to be the same regardless the timing of the cash flow received there is a term structure of credit spreads. Dealer firms typically estimate a term structure for credit spreads for each credit rating and market sector. Generally, the credit spread increases with maturity: typical shape for the term structure of credit spreads. In addition, the shape of the term structure is not the same for all credit ratings. Typically, the lower the credit rating, the steeper the term structure of credit spread

The zero-volatility spread or Z-spread is a measure of the spread that the investor would realize over the entire Government spot rate curve if the bond is held to maturity. It is not a spread of one point on the Treasury yield curve, as is the nominal spread. The Z-spread, also called the static spread, is calculated as the spread that will make the present value of the cash flows from the non-Treasury bond, when discounted at the Treasury spot rate plus the spread, equal to

4 Those fundamental securities are: ZCB, Annuity, Floating note and Coupon Bond

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the non-Treasury bonds price. A trial-and-error procedure is required to determine the Z-spread. It represents a spread to compensate for the non-Treasury securities credit risk, liquidity risk, and any option risk (i.e., the risks associated with any embedded options).

Before going deep in the topic some empirical features, really important later on for modeling purpose: The distribution of interest rate is not normal for short maturity, while it get closer to it for longer maturity The volatility of interest rate is not constant/equal for all maturity, but it is higher in the short term The correlation between rates is not perfect, it is high for closed rate and it gets smaller as the tenor difference

increase. This not trivial correlation structure force us to implement multifactor5 model to properly address this feature when we need to price option on more than one rate.

By performing a PCA on interest rate matrix we note(i) the first component explain the movement of the term structure (ii) the second one will represent slope movement (iii) the third will capture the relative movment of the two extremes.

During the course we are mainly referring to: the yield curve, which is the representation of the ZCB’s simply compounded rate and the equivalent discount curve6. Those two curves use as reference rate the spot yield. The relationship between

the discount rate and the investment/spot rate is

d

1−d∗act

360

∗365

360

where d is the discount rate and the second term is

the day convention adjustment needed to move form a 360 year to a 365 one.

However there is another rate mostly used in the market, i.e. the forward rate. Those rates are traded in the market via FRA contract, with which today investors can lock for a given time in the future a return set today. This rate is related with

the spot price by the P (t , T 1 , T 2)=P ( t ,T 2 )P ( t ,T 1 )

. This formula is obtained with the combination of two zero coupons Bond

with NA arguments, we need to create a strategy with 0 cash flow at time t, a negative cash flow at time “T1” and receive the investment procedure at time “T2”:

We go long on a ZCB with tenor till “T2” we pay P ( t , T 2 ) We sell k-times ZCB with tenor “T1” we pay K∗P (t ,T 1 ) At time 0 there won’t be any cash flow, at time “T1” we need to pay K and at “T2” we receive 1. K is the forward

discount factor which is solved by imposing the first cash flow to be zero: −P ( t , T 2 )+K∗P (t , T 1 )=0 From that quantity we can obtain the forward rate by applying a compounding rule: If it’s the simple one:

F (t ,T 1, T 2 )=

1T 2−T1

∗(P (t ,T 1 )−P (t , T 2 ) )

P (t , T 2 ) which is used in forward Libor where the underlying

contract is named Forward Rate Agreement (FRA).

If it’s the continuous one: F (t ,T 1, T 2 )=−lnP (t ,T 2 )−ln P ( t , T 1 )T 2−T 1

The floating amount can be replicated in two different ways: Difference of discount price []. It relies on the NA argument, no counterpart rik

5 But how many factors are necessary in order to generate the observed structure? An help in answering to this question comes from the so called Principal Component Analysis (Scheinkman, 1991) (Weiss, 1991), those papers show that three is good number of factors.6 Note that the discount rate is not a rate of return, since it is a percentage of the final value, not the initial one

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Forward rate as proxy of future Libor rate []. This method is preferred if there is a sizable counterparty risk (see below)

Bootstrapping techniques:

The spot term structure is constructed from shorter maturities to longer maturities using market instruments like Libor, futures, FRA and swaps. We won’t use Treasury bond to stick with the market convention because the swap curve are more liquid than government bond, there are less regulation hence those rate are more comparable across country, moreover there is no country risk premium and more maturity available. It can be used for pricing basic derivatives instruments, like coupon bonds. The procedure that allows the construction of the term structure is called bootstrapping. We need to develop some methodology to infer discount factors at certain "grid points" and at the intermediate dates, the most common used is the linear; however there exist more exotic interpolation between values at the grid points.

How we construct the term spot curve?

Since the securities used are different depending on the maturity we need to implement three different methods to convert that rate/price into something homogenous:

We will use Euribor or Libor (depending on the currency used) quotation for short maturity, till 6 month Then we will use futures7 quotation for maturity lower than 2 year, There after the at par swap rate8.

At first we need to extrapolate the discount factor form the available rate: Directly from the Libor (money market), using the simple compounding rule and the ACT/360 day convention Speaking about Futures we need to decide if we are going to use the convexity adjustment or not, when this

decision is made we need to perform a preliminary task, i.e. interpolate for stub date9.

o The futures rate is computed with 100−FRA

100 which a forward rate, from whom we obtain a forward

discount factor. So now we need to convert those “futures discount factor” into the spot one by using P t , t+t 1∗P t , t+ t 1 ,t+t 1+t 2=P t , t+t 1+ t 2 or vice versa

P t , t+t 1+ t 2=P t ,t+ t∗11+F t , t+1 ,t+1+2∗∆ t

o Since the first forward discount factor start from a different date from the Libor nock we need to interpolate the stud date.

The Swap rate curve need as well as the FRA some preliminary step:o We need to extrapolate the implied swap rate for the shorter maturity, to use them to compute the

annuity, which is needed to extrapolate from the Swap rate the discount factor. This task is performed using the discount factor computed previously with geometrical interpolation if we need to exactly

match a swap maturity P (t , Ti )=1−S ( t , Ti )∗Annuity1+S (t , Ti )aTi−1 ,Ti

o We need to interpolate the missing maturity: usually we use linear interpolation of rates

There are some considerations to be made: the money market beard more credit risk than a Swap, hence there will be jump/step when changing tool to bootstrap the yield curve; futures quote are not equal to FRA rate; the linear interpolation is not suggested because may create problem in computing the equivalent forward rate curve, basically if I’ll use it on Discount factor, the forward curve will be discontinuous; moreover the DF should like more an exponential decaying function not like a straight line. That why has been introduced other techniques:

The Constant forward rate, where basically we solve to find a forward rate which is constant in the interpolation period (the simplest solution).

7 They are used as if they are FRA contract, margin payment are ignored, it is a strong simplification8 We won’t use anymore the Treasury curve since the Swap market provide more maturity and sometimes is more liquid9 It is the adjustment needed to reconcile the first futures starting on Wednesday with the last Libor instrument so that we obtain the proper maturity date. It is usually used the geometrical interpolation that is equivalent to the liner when using annually compounded rate

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Other interpolation techniques propose to use line or other equation formulation to fit the forward rate The convexity adjustment: Regarding the usage of Futures contract is more than advisable to correct for the bias

toward short position, which makes those rate higher than the equivalent FRA rate. This property is called convexity, and its adjustment requires a term structure model, i.e. some assumption about the dynamics of interest rates since it depends on its future evolution. The adjustment following the Howel model is:

σ2

2(t 2−t )∗(t 1−t), which it must be subtracted from futures rates

Another problem/situation to deal with is the interpolation techniques to be used to extend to all the other grid point (for pricing reason, we cannot use the liner or the previous one); there exit two methodologies: the Nelson-Siegel (NelsonC. and A. F. Siegel, 1987) and the Cubic equation or spline method. The first is a parametric model, very parsimonious indeed, while the second consists on dividing the curve in knock point, where the function is continuous.

In details the first approach assume that the instantaneous forward rate f (t , t+ t)=b0+(b1+b 2t)exp(−t k ), where the parameters stand for

b0 specifies the long rate to which the fwd rate horizontally asymptotically, b1 is the weight attached to the short term component (spread short/long-term), b2 is the weight attached to the medium term component, k measures the point of the beginning of decay When t=0 the intercept is given by b0+b1, and t=∞ the function will be equal to b0, those two condition are

required to bound the equation to the market oneBy integrating f(x) we came out with the spot yield which will be used to compute the discount factors, and the resulting will have parameter chosen to minimize the distance with the market data. This model is really easy to implement, but is not really flexible and the fit is never perfect, however is a good methodology (actually it is the suggested one when dealing with bond pricing).

The Cubic method (McCulloch J. Huston, 1977) is based on dividing the curve into piecewise and fit the function, however this method may show some erratic behavior in the spot curve and most importantly the forward rate may show jumps or counter intuitive behavior.

Interest rate risk: what and how to measure

Measuring the exposure to interest rate risk can be interesting at least for two reasons: Hedging: matching the exposure to interest rate risk of the assets with the interest rate risk of the liabilities; Exploiting views: given a view about future changes in interest rates to determine which securities (or

combination of securities) will perform best if their view does, in fact, obtain.

Before starting describing how to assess the risk we need to specify how we are going to model the interest rate shift. We can assume a simple parallel shift of the interest rate curve and that all the instruments have same sensitivity regardless the maturity of their components. Or we can try something more realistic, like assuming different sensitivity for each spot rate maturity, a possible parallel shift of forward rate (instead of spot) and considering as a risk-factor the yield to maturity.

The first possible way to assess this risk is to re-price the instrument for given interest rate movement; hence we are going to re-apply the pricing formula each time. This method is the most precise; however it is time consuming and for large portfolio with exotic payoff might be unfeasible.

On the other hand we can use proxy; the most used one is the Duration ∑ CF∗T i

(1+ y )iT

Price

, more precisely the modified

duration. The measure can be compute in absolute or in relative terms. Of course it is a proxy and it works only for small

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change and under parallel shift hypothesis. The Babcock's formula gives a simple (approximate) expression for the holding period return in terms of “w” the holding period, “y” the YTM, “D0” the Duration at the beginning of the investment period

and ∆y the change in the YTM. yw= y−D 0−w

w∗∆ y . Where it is easy to see that the risk is higher the higher is the

excess Duration.

Besides the system used to compute the change in value there is a common market standard to express change in value, i.e. the DV0110, it is the first derivatives of the security with respect to the reference rate. Note it can be computed with

numerical solution of the partial derivatives: −B ( y¿+∆ y )−B ( y¿−∆ y )

2Dy∗0.01

100. This quantity is linear, in fact to

compute the DV01 of portfolio we can simply sum the weighted single asset value.We can use DV01 to hedge the portfolio by jointly imposing the portfolio DV01 equal 0 and the portfolio value to a given level/price.

Here a list of DV01 for the main contracts. It is defined so that if the rate increase the price decrease:

DV 01FRA=−αT ,T+s∗P(T ;T +s )∗N∗1

10000 where “T” is time “0”, spot rate

DV 01hedge=DV 01unhedge−n∗DV 01hedgingInstruments

DV 01caplet=−¿∗P (T ;T+s )∗N (d1 )∗110000

∗Q∗αT ,T+swhereQis the survivor prob∧αthe tenor .

DV 01CorN=¿∗P (T ;T+s )

¿ αT ,T +s∗1

10000∗Q∗N (d 2 )

σ √α T ,T + s

where ¿ is thenumber of contracts

DV 01bond=

−∑ cs∗T−t

(1+is (T−t ) )2∗1

10000

DV 01FRN=α0 , t

(time¿next payment )∗1+c∗α0 , t+r(whole coupon time)

(1+L (0 , t )∗α 0 ,t )2 ∗1

10000where t is t he first reset date

DV 01 futures=−tick(25dollari )∗1

10000

Financial Product in the Fixed Income world:Here we will analyze and summarize the main features and valuation techniques on the principal fixed income securities. Note that those securities do not need an interest rate, since they can be replicate by non-arbitrage argument using other securities

Euro-deposit:

10 It is an acronym for dollar value of an .01%, i.e. the change in the value of a fixed income security for a one-basis point decline in rates

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Those deposits are posted in a foreign currency (London is the main market). They are add-on opposed to discount instrument (such as Treasury note). The capitalization used is the simple one and the most used reference rate is the Libor the market quotation convention is to post annualized rate for each tenor with the day counting convention of Act/360. The face value of the deposit is the present value since it is considered risk free we use its own rate to discount the cash flow. For deposit with maturity longer than one year at each anniversary the interest is paid

Repo rate agreement:

The repo desk acts as the intermediary between the investors who want to borrow cash and lend securities and the investors who want to lend cash and borrow securities. More precisely, the borrower of cash will pay the bid repo rate times the amount of cash borrowed, while the lender of cash will get the ask repo rate times the amount of cash lent. The repo desk gains the bid{ask spread on all the transactions that it makes.

Overnight repo if repo maturity is one day, term repo if repo maturity exceeds one day; Special repo rates apply for particular Treasury securities in high demand, opposed to generalized collateral rates (GC) that are the repo rates on most Treasury securities. Rates on repo are different from LIBOR rates, since repos are considered a secured loan whereas the LIBOR is used for unsecured interbank lending.

Short-term borrowers against collateral. Short-term lenders in a very liquid short-term market, and, more generally, over tailor-made horizons, by rolling over either several overnight transactions or different repo transactions with various maturity horizons. Commonly used by traders and portfolio managers to finance either long or short positions (usually in government securities).

FRA contract and Swap:

The FRA and Swap contracts are the most plain vanilla available in the market, and both of them do no actually need any interest model to be price, in fact the first one can be replicated by trading with bonds as well as the second. However speaking about Swap and FRA we have noticed that the spread between the swap traded rate and the price of the replicating strategy is getting larger and larger in those days not only due to Fundamental and Technique difference between the two strategies.

This is a direct consequence of the increase of the credit risk of the counterparty (this problem is made more severe due to the nature of the Market OTC); in fact the two strategies have different risk profile: the Libor market incorporate the default risk of a panel of selected bank. Nowadays there is lots of paper to try to incorporate the existence of counterparty risk into the pricing formula [CVA]; basically what is happening is that the return of one leg of the strategy is not equal to the other: two rates for different risk profile [when we lend our money the counterpart might have a credit rating different from the Bank panel contributing to the Libor, which is the rate used to invest our money ]. This problem is quite severe in a practical sense since we cannot use any more the usual pricing formula (without referring to any interest model, but simply basing our strategy on trading bonds) we need to introduce a model.

PwithRisk (0 ,t )=P (0 , t )∗e−cst

q=e−cswhich is the survivor probability

This two formulas are used to add to the discount factor the credit spread, note that the reverse is possible, we might use a numerical solution by imposing the market price and re-writing the pricing formula as function of the risk free DF time the survivor probability “cs”. Remember to use the risk free rate to compute the fwd rate, while the risky one to compute the present value.

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The trading convention is: the trade date is when the two counterparts made the agreement; the value/fixing day is when the FRA rate is fixed; the settlement date is when the cash flow is netted and the position closed. The compounding rule is the simple one.

FRA is used to hedge point in time transaction, while Swap are used to hedge periodic transaction. The risk hedged is the interest rate risk; we can use the definition of FRA or the concept of DV01 to ensure to pay a fix rate in the future equal to the FRA rate itself. The FRA securities need to be valued at inception to ensure arbitrage free and during its own life:

At inception the FRA rate is chosen so that to be (assuming no counterpart risk)FRA= 1α∗(

P ( s ,T )P (s ,T +τ )

−1),

where alpha is the day fraction convention and Tao the rate period. This relationship is set by imposing the equality between the Present value of the fixed leg and the floating one. Note there could be off-market rate (different form this relationship) where the initial position has a value, and there could be a floor or a cap set up.

The value during the life period is α∗[F (s , t+τ )−F ( t , t+τ ) ]∗P ( s , t+τ )∗Nwhere “s” is the inception and

“t” the valuation date [long position]

Referring to the FRA contract we have two possible payment systems and so two pricing rules depending on if we are considering FRA in arrears or in Advance. In the first case the reset day and the payment day are at different date, while the second, which is the standard one, is when the payment is made at the some date of the resetting. The first is always greater than the standard one

[L (T ,T+r )−K ]∗r∗P (t , T+r )advance payoffat time 0 The arrears payoff is

PV (FR Aarrears)=P (t , T+r )∗( (F ( t , T ,T+r )−x )∗r+(F (t , T ,T+r )eσ2 (T−t )−x )∗r2∗F ( t , T ,T+r ) ). It

is obtained by capitalizing the advance payoff [L (T ,T+r )−k ]∗r∗(1+F ( t ,T .T +r ) r ). From this formula we

see that this payoff is the advance one plus the time adjustment, moreover the new equilibrium FRA rate is greater than the other one.

o [F (T ,T ,T +r )−k ]∗r+[F (T ,T ,T +r )−k ]∗r∗F (T ,T ,T +r )∗r where K is the fixed rate

o [F (T ,T ,T +r )−k ]∗r+F (T ,T ,T +r )2∗r2−k∗F (T ,T ,T +r )∗r2

o We need to compute the expectation under the T+r zcb measure

[F ( t ,T ,T +r )−k ]∗r−k∗r2∗F (t , T ,T +r )+r2∗ET+r[F (T ,T ,T +r )2]. The term inside the

expectation is solved by noting that the dynamics of the forward rate squared (given the normal one)

F (T )=F (t )2 e−σ2dt+2σdW t+r

together with the MGF we have

o [ (F ( t , T ,T+r )−k )∗r−k∗r2∗F (t , T ,T+r )+r2∗F ( t ,T ,T+r )2 eσ2dt ]∗P (t , T+r )

o k=F ( t )∗r∗P ( t )+F ( t )2eσ

2dt∗r 2∗P (t )(r+r2F (t ) )∗P(t)

=F ( t )+F ( t )2 eσ

2dt∗r1+rF (t)

The FRA market is an OTC market and it has a value right after the fixing day.

If the FRA rate is different from the NA arbitrage we can exploit the risk free profit by: Go Long/short the FRA and replicate it with NA argument:

o Buy the zcb with maturity T at time 0o Sell “k” zcb with maturity “t”o So the net pay off is 0 at time “0”, we pay K at time t and we receive 1 at time “T”

Compute (FR Amkt−FR ANA )∗α∗P (T )

10


Swap:

Referring to the Swap: The Swap rate is given by the general formula at generic time “t”S(t ,T )=(P ( t )−P (T ))/¿, so

that the position at time 0 has value equal zero. The derivation of this formula is made by imposing that the fixed and floating leg must have same value at inception (no-arbitrage argument)

The fixed leg present value for one payment is P ( t , T )∗α∗K∗N , note that when priced a forward starting

swap the first tenor of the fixed payment starts form the reset/future date, while all the other payment have the defined tenor

The floating leg present value is α∗L (t i−1 , T i )∗N the Libor rate fixed at time t i−1 is equivalent to the Forward

rate, hence we can rewrite the previous formula in terms of Forward rate and by applying the its definition

FRA=( P (t )P (T )

−1)∗1

α

the final formula of the floating leg in expressed as function of DF is (P (t )−P (T ) )∗N

By computing the summation of all the payments days and solving for K (Fixed rate = Swap) we have the inception swap rate so that the position has value “0”

To value a seasoned Swap by Non arbitrage is made by showing the equivalency of a position on a fixed coupon bond and a FRN (we are long/short on one of those legs depending if we are the receiver or the payer):

The FRN leg is N∗P (t , T )P ( t0 , T )

where T0 is the last payment day and “t” is the current date. Alternative the present

value of the floating leg can be computed as the usual formula P (t , T 1)−P(t ,T 2) where t1 is the first reset

day and t2 is the last payment day

The fixed leg is P (t , T N )∗N+∑ P (t ,T i )∗N∗α∗Swapwhere “Tn” is the last date

The value is the difference of those two legs (the notional will disappear), for the sing remember that the payer has a short position on the fixed part (negative)

It can be shown that the seasoned Swap has value equal to (S ( t )−K )∑ α∗P (t i )∗N where K is the new Swap

rate with same frequency, maturity and reset day (basically the forward swap rate with same date) If at inception we want to create a position which grants an immediate income to the payer Swap we just need to

set the NA Swap rate equal to the income instead of “0”

The duration of a Swap is (since we can considered a position on a coupon bond and a negative position on floating note) the difference between the duration of a coupon bond and a FRN. We can write Swap price by using bond price, forward rate or the definition of Swap rate, furthermore the swap rate is a coupon rate, from which we can bootstrap the interest

rate P (T )=

1−S∑i

n−1

P (t i )

1+S. Other important characteristics are: The swap rate can be viewed as an average of FRA rate,

the frequency of the payment is not neutral in deciding the spread applied (however in an efficient market the decision should be neutral), higher frequency are perceived as less riskier by market participant (there exist more than one swap discount curve) and the OIS (overnight indexed Swap) is a rate set as a geometric average of the daily index level over the payment period. This rate is considered less risky than Libor one and the difference is used to acknowledge the stress of the money market (risk adverse situation).

The quotation convention depends on the market and reference currency (lect 2, 137).

11


A new framework to factor in default/credit risk:

Nowadays the replication strategy is more costly than the market FRA, the basis average 50bp. This basis arise from the different rate at whom bank may borrow/lend money at different maturity. This basis get smaller and smaller the less the time period range between floating leg payment. The overnight rate, where only bank with negligible risk is admitted, can be used as proxy of risk free rate

As describe above in the Libor section, this rate is a trim rate based on panel bank (note that banks, better than average, have rate close to the mean, hence only deterioration case is modeled), hence future labor evolution is by definition small or equal to the any panel bank chosen as counterparty risk (since if defaultable the bank will exit the panel).

L (t , T )mkt ≤L (t , T )rstr

Here in the following the step to assess this risk (viewed as an option on Libor): We will define a scheme to represent potential refreshment of the counterparty rate, where we can stay stick on

the beginning rate (Libor) or we may shift to the new higher rate

Where exit is the max spread applicable to be within Libor panel, Sd X 0 is the spread at beginning and Ssubst is the

rate if the bank is not anymore a prime one

o Sd X 0(t , α ,2α)=1α∗(P

rf (t , α ,2α )P rf (t , α ,2α )

−PdX (t ,α ,2α )PdX (t ,α ,2α ) ) it is the initial forward future Libor spread,

the difference between OIS and forward classical rate (all divided by the time range used on)

o Sexit=SdX (0 , α ,2α )=E rf 2α [SdX (α ,2α ) ], basically we are assuming that if the bank went worse than

the original expectation it is outo Ssubst=2 Sdx (0 ,α ,2α )−SdX (α ,2α )

This leads to L (α ,2α )=Ldx (α ,2α )−2 (SdX (α ,α ,2α )−SdX (0 , α ,2α ) )+¿¿, hence its expectation is

E f 2α [L (α ,2α ) ]=FL (0 , α ,2α )−2E f 2α [Sd X (α ,α ,2α )−SdX (0 , α ,2α ) ]+¿¿

o Assuming a geometric Brownian motion for the spread d SdX (t , α ,2α )=SdX ( t , α ,2α )σα dWP ( t )

we can price this option with the Black formula, short two caplet o The forward in the formula is the classic one, hence it coincide with the market one if and only if the

credit volatility is 0

Second theme is how to account for collateral (discount factor), the following formula restates the credit risk in the expected value to compute forward agreement value:

E [P (0 , t )∗1c>t∗(L (T i−1 , T i )−K )∗α ] If there is a collateral the payoff remain risk free, Collateral pay/receive interest, hence we need to account for this situation whenever we need to assess forward

contract value. OIS (Overnight indexed swaps), seen as the best proxy of risk free rate.

The formula becomes: E [P¿ ( 0 ,t )∗(L (T i−1 ,T i )−K )∗α ]. We use as discount rate the difference between the

risky one and OIS=1αé ( 1

P (t , T )−1)

12


Third theme is another scheme to define the Libor dynamics (given the risk free discount factor in point 2): we won’t use any more the MIF to extract the forward from spot rate, but we will use directly the market quote of FRA basically we won’t model the spread but the rate itself. We will define the BGM approach o multi-Curve:

1. By modeling the equilibrium rate Eti [L (T i−1, T i ) ]=R (t ,T i−1 , T i ) where the measure is the forward measure:

features are a. Be a martingale, since it is a relative price b. Convergence at maturity -> Forward rate becomes spot when t=T

c. E [E¿¿t i [L (T i−1 ,T i ) ]]=R ( 0 ,T i−1 ,T i )¿ law of interested expectation

2. We associate a dynamics to the rate d Rk ( t )=σkR ( t ) Rk ( t )dW k

R (t ) and the volatility σ k (t) is the instantaneous

one Corr (dW k , d W j) and the correlation structure between rate is the one implied from the market

3. Since we are taking into account n-rates we need to modify model to multi-curve, basically we will define to motion and a correlation between them, so that we can still use the black formula with a modifies volatility as to

account the correlation matrix √∫ σ iR ( t )2dt

The case of Swap is more complicated. The NA strategy fails in assuming how to compute the floating leg (it does not depend any more only on first and last payment day), R(t) as modeled before, is not enough.

We need to define a new framework where we need to build a instantaneous correlation matrix as follow

[“R” is the Target, “FR” is the cross correlation and “F” is the OIS rate correlation]. The dynamics must be corrected according with the Numeraire chosen, since we must have one single

process/measure, simulated jointly.

μ2=μ

1−DC ( xk)∗ρ∗DC ( ln(N 1

N 2))

Note that is correction that is similar to the classic one curve LMM model do not use the FRA rate but the risk free rate as they appeared on the discounting/Numeraire and the correlation is the cross one.

d Rk ( t )=±R ( t )k σk (t )∗∑ρk , j σ j ( t )∗τ j∗F j ( t )

1+F j (t ) τ j

dt+σk ( t )Rk (t )dW k (t)

Possible drawback is to view risk free rate higher than market Libor. This result suggests thet the rate is sort of

specified by Rk ( t )=Fk ( t )+Sk ( t ), where S is a positive spread. If this is the case we should update the

correlation between OIS and Libor risky rate as well

Futures contract:

Those instrument quotation is made as price (differently from FRA which post rate), from whom we need to extract the

implied futures rate IR=1− FutPrice100

. These securities are traded into exchange, where the exchange itself gives a

guaranty on the transaction (the safeness of these securities is enhanced by the presence of margin)

Last we need to address that the difference between futures and FRA are related to the presence of margin, contracts format and for a futures the settlement occurs at the maturity whilst for a FRA there is a natural time lag between reset and payment date. We need to take into account the founding cost11 and the convexity component (since short position is advantaged and there is a negative correlation between margin and futures payoff). To value the last element we need to run a specific model. CIR has shown that the futures price is a martingale under Q, so the implied future rate must be

11 This cost will favor the short position since the loss present value is reduce (where rated decrease) while gain are invested at higher rate

13


equal to the expected future floating rate (FRA rate), the convexity is the difference between the expected future FRA rate and the FRA rate set at inception.

The price will move by tick. The DV01 of futures contract is the same regardless the maturity and it is equal to the tick size. Note that besides the cash margin the futures has an intrinsic difference form FRA, the first has the settlement day and the payment day at the same day.

Floating rate note:

A FRN is a bond paying a floating rate and (eventually) a spread over this rate. There are two possible pricing solutions:

The first one is made by computing the present value of the FRN with understanding each element of the security ( zcb method), the FRN is like a zcb with maturity the next rest date. Those cash flows can be divided into three sections:

1. The present value coupon already fixed N (L+δ )∗α2. The PV of following and uncertain coupon (we will use the FRA rate) + PV spread applied over the floating rate

∑ P (t i )∗αi∗(F (t i−1 , ti ))∗N+∑ P (t i )∗αi∗δ∗N . Note that if the reference rate used is the rate used for

the DF (risk free floating bond) ∑ P (t i )∗αi∗(F (t i−1 , ti ))∗N=(P (T 0 )−P (T N ))N3. At last the PV of Notional amount.

The ending formula with the simplification in point 2 is P (t , T )∗(1+c )∗N+∑ P (t ,T )∗α∗γ∗N . Thanks to this last

formula we can see that if the spread is Zero or negligible and we price the bond at coupon day the note will be priced at par, since “c” will be the same rate used in the discount factor. We can price a FRN by noting that (at different date from

the value date) P (t , T )∗(1+c )∗N=P ( t , T )P (t 0 ,T )

∗N

The second methods is to price the floating note by computing the PV of each cash flow (using the forward) (forward method). It is like discounting a cash flow

Caps and Swaption:

At first some general statements: those products do not show linear payoff as the previously one, they need a model on the Libor or the Swap rate (the underlying) for their distribution, hence the previously NA argument is not possible; they

are usually European and priced with the Black formula. [F ( t ,T ,T+r )N (d1 )−LxN (d2 ) ]∗P(t ,T +r )

Cap and Floor:Given the definition of a Cap (collection of Caplets, which are a call option written on Libor rate

N∗α∗[L (T ,T+τ )−K ]+¿∗P (T ,T + τ ) ¿[to cap the interest payment], fixing price at T-1 maturity date and payment day at

T, the time to maturity is the length to the next reset day) the reference terms are the: The reference rate (usually the 3m, 6m, 1y LIBOR rate), The strike rate that sets the cap or floor (usually the forward swap rate), The length of the agreement (from several months to 30 years), The frequency of reset (equal to the tenor of the LIBOR rate) and refers to the frequency with which the

reference rate is compared to the exercise rate. Note that the last reset day is one period before maturity The time between two payments is known as the tenor (typically monthly, quarterly, semi-annually and annually) The notional amount (which determines the size of the payments). If a cap or a floor is in-the-money on the reset date, the payment by the seller is typically made in arrears12.

12 It has two meaning, the first (widely used) is to pay at the end of the period, the other indicates the payment after the expiry date of the due payment

14


The trading practice on Caps in US and EU are: The caps quoted in the US market have maturities up to 10-30 years and are those written on the 3m Libor rate

with T0 (reset day) always equal to three months and all other Ti is equally three-month spaced (e.g. US LIBOR), the payment is 3 month from the reset (since the reference rate is the 3m Libor)

The caps quoted in the Euro market have maturities up to 10-30 years with in the first year cap T0 is equal to three months and the next Ti is (up to one year the reference rate is the 3m) equal to three months length, all other caps have T0 equal to six months and all remaining Ti is six-months spaced, the reference rate is the 6m.

Traders quote only at the money quotes (when the Libor rate is fixed equal to the forward rate) and it is usually traded in term of implied volatility then converted into price. The volatility can be Flat (a kind of average of all the caplet/floorlet) or forward volatility (one for each caplet/floorlet), those volatilities are usually plotted against maturity to obtain the term structure of volatility

The term volatility structure, using Flat one, shows a hump in the 2 year and then it starts to decrease. The forward-forward volatility is extracted from the Flat one by bootstrapping13 (since the two strategies must have same price), those volatility are useful in pricing other instruments and they are more humped than flat one. Trader has developed smoothing techniques to reduce possible abrupt change. Note that each of this volatility bootstrapping techniques will imply different price, which might have a sizable difference, moreover the one used are the ATM, hence when pricing an out of the money option the bias is increase

The Cap and Floor instruments have two different payment systems, which have different payoff:

The Asset or nothing (A-N)N∗a (T i ,T i+1 )∗L (T i,T i+1 )∗1 [L (T i,T i+1 )>Lx ]∗P(T i ,T i+1), here

you receive an asset paying you the Libor, hence the payoff on maturity is a function of the Libor given the exercise event. The payoff is discounted at present day, it is more costly than a call

The Cash or nothing (C-N) N∗a (T i ,T i+1 )∗1 [L (T i ,T i+1 )>L x ]∗P(T i,T i+1) here the payoff is

simply the value due at maturity, hence it Is constant and discontinued when in the money. The difference of the A-N minus C-N is a caplet or a generic call payoff

Caps and Floor can be view as portfolio of option written on ZCB (put for caplet, and call for floorlet), this statement is really important since the HJM model has explicit formulas for bond option values, so that caps can be priced very easily in these models and this facilitates the model calibration to market price contracts. This important result is shown here after:

Given the payoff of a caplet r∗N∗P (T ,T +r )∗[L (T ,T +r )−Lx ]+¿¿ we can rewrite the zcb price as function

of rate N∗[ L ( t+r ) r−LX r

1+L (t+r ) r ]=N∗[ L ( t+r ) r+1−LX r−1

1+L ( t+r ) r ] From that result by adding and subtracting one we came out with N∗¿ rearranging the payoff to leave only the

zcb price inside (look similar to a call on zcb) N∗(1+Lx r )∗[ 11+Lxr

−P (T+r )]Another important relationship to be point out is the so call put-call parity: caplet−floorlet=long FRA , so the difference between Cap and Floor is a Swap.

Swaption:Now we move on Swaption, those options are written on IRS and we usually refer to the leg paying fix as the payer, the reverse is called a receiver. The expiry is usually set up at the first IRS reset data and the tenor is equal to the IRS contract

13 We are going to divide the caplet in segments (piecewise) and we fix a constant volatility value for all the caplet in that range, sometimes we may add a smoothness, preserving the NA condition

15


itself and the maturity is equal to the IRS contract. This instrument seems quite unusual at first glance, however are frequently used by corporation to anticipate future exposure.

N∗[S (T ,T ,T n )−k ]+¿∗∑ αT i,T i+ 1P (T ,T i )¿

where K is the forward swap rate, the last term is the annuity, where the first term is the one after the expiry of the option

Prove, using a change of Numeraire argument, how to find: 1) the forward price on a zcb, and 2) how to obtain the most important formula.

Aggiungere dimostrazion MIF: partire da paying legs (no notional, aggiungere il notional) cpaitializzare e raccogliere e ottenereil floitng rate

The Swaption trading practice and developments: It has been introduced a standardized Swaption note by the Liffe (Chicago), however the market practice is to

operate OTC. As the Cap/Floor the Swaption can be view as a portfolio of option written on paying coupon bond (sell/put =

payer; buy/call = receiver) at par, where the coupon is the fixed amount due. As the Cap this instrument are usually quoted by trading volatility, from which we can construct the surface. Note

that the implied volatility is usually computed starting from the hp of having swap rate written as average of forward where the weights are not random (not the case), basically this will introduce a big bias if we are going to use this value for Risk Management purpose or pricing.

From Swaption quote we can bootstrap surface Volatility (function of option tenor and Swap time windows, the shorter the higher the volatility value). This volatility is different from the absolute volatility obtained by the definition of Swap rate as function of forward rate Var=∑ wi−1w j−1σ i−1σ❑j−1, there is a consistent basis between the two caused by difference in the two market segment.

Just one step before starting discussing the Black’s pricing formula; the Cap are a collection of caplet (can be decomposed additively) that to be priced need to model each single rate (for each caplet); on the other hand the Swaption cannot be decomposed since the swap rate depends on all the rate jointly, we need to model the joint density for all the forward rate. Due to these features the caps are called volatility instruments, while the second correlation product.

Add how to hedge with those instruments

Exotic corporate bond:

Add how to decompose formula into elementary components

No-Arbitrage Pricing: This section is based on the Black and Scholes statement on option pricing: “the instantaneous rate of one payoff must be the same achievable with a suitable portfolio”. This statement has been translate in mathematical term by the implementation of the martingale14 probability measure (and several accessory theorem15), which ensure to change the probability measure of a given payoff to match the risk neutral one. Note that the expected value of the distribution won’t change, that’s why it is meaningful to perform this tool.

Change of Numeraire:

When we are dealing with complicated payoff has been noted that the usual risk neutral martingale is not feasible due to the difficulty to solve the PDE, hence it has been introduced the change of Numeraire technique. Thanks to this device we

14 The expected value of a random variable relative a measure Q and a filtration F is the current value15 The Girsanov theorem and the Radon-Nykodin derivatives

16


can properly price option, basically we are going to choose any positive non-dividend paying asset16 to normalize all other asset price, so that to eliminate17 any unusual or too exotic feature.

N ( t )B ( t )

=~N (t ) d~N ( t )

~N ( t )=σ NW

Q ( t )

An important feature of this technique is that the equivalent probability measure associated with the new Numeraire is always a martingale itself, we need to change the drift to respect the risk free axiom. An important formula to know is

v t=nu mt EQ[ Payof f TnumT

]. Note that we do not to assume log normality of the underling to say that its expected value

under the suitable measure is the current value of the underling. Moreover if you need to compute the d1 you need to assume the log normality, remember that the volatility associated with the dynamics is the volatility of the underling used [for example if we are pricing a swaption with the equivalent coupon at par derivatives the volatility is the one of the coupon bond and it must be an integral (time varying to respect the condition of pricing at par at the end). Generally speaking it is better to model the rate instead of the price, the second case may lead to negative price with positive probability, we should change the dynamics, however no closed form available)

By applying this technique to solve the BS formula we end up with N (d1 )=PrSt

(ST>k ) and N (d2 )=Prq

(ST>k ), where

ST is the stock measure. c (t )=S (t ) 1S (T )>K−K 1S (T )>K=N1 (t ) EN1[ S (t )1S (T )>K

N1 (T ) ]−N 2 (t )EN2 [ K1S (T )>K

N2 (T ) ]. The

suitable N1 is the Stock, while for the N2 is the risk free bond, the formula will became

S1 ( t ) EN1[ S ( t )1S (T )>K

S (T ) ]−B ( t )EN2 [ K1S (T )>K

B (T ) ]=S ( t )Pr S (S (T )>K )− B (t )B (T )

K∗PR (S (T )>K ). Note that before it,

you need to find the dynamics under the new measure by using the risk free bond and the stock dynamics, dividing one with the other depending on which is the Numeraire chosen.

To apply the procedure the Novikov condition must apply (the variance must be finite) as well as the source of randomness must be a Brownian motion (to apply Girsanov). The steps to be followed are:

1. We need to compute the Radon-Nykodin derivative dQdP

=

N (T )M (T )

∗M (t )

N (t ) where P and M() are the old measures

2. By applying the Girsanov theorem we will compute the expected value of the new distribution (drift correction):

dWQ=dW P+k (t )dt , where k (t )=−DC [ ln(N new

Nold)]dt is obtained from the relationship

N (T )M (T )

∗M (t )

N ( t )=e

−12∫ k2( s )ds+∫k ( s)dW P. The meaning of the parameter is the risk adjustment for the excepted

excess return μn−μmσ

16 We can use ZCB, money market securities, constant maturity coupon bond17 We are not eliminating anything, we will account for the unusual feature into the probability measure by properly change the drift and diffusion coefficients

17


Alternatively we can compute with Ito the new dynamics, divide it by the Numeraire, compute the expected value and leave the randomness source in the basket. To solve the last term we can apply the moment generation function, where if

we have a random variable X (m, S2) the E (eγX )=Eγm+1

2γ 2s2

and solve it by imposing that the resulting number is

equal to the current value (the drift is zero, the parameter “μ” must be set to ensure that result).

A possible alternative explication can be done by using a Binomial model: we have two dynamics the risk free bond and the stock one. Our goal is to compute the probability f down and up movement relative to the money account and the stock. To do this we will divide each dynamics once by the money account and once by the stock, then we will compute

the expected value and solve to find the probability with respect to “u”, “d” and e−rt. Note that when dividing the

dynamics by itself the final payoff is always one. The relative probability allows us to price any payoff to the respect to the measure chosen (Numeraire) hence the expected value must multiply y the Numeraire value at time 0 (as shown by using the Nikodin and Girsanov method)

v t=nu mt EQ[ Payof f TnumT

]. The price will always be the same regardless the measure.

If you need to price a multi-period option you need to:

Find the minimum step needed to be in the money Su jd NUM− j>K where NUM is the number of available step

and K is the strike. The example is made in case of a call, remember that the j must be an integral, hence we need to ceil round it

Then we need to compute the probability that the number of jumps is greater/less than the trigger point (remember the relationship P ( j<K )=1−P ( j>K )∧P ( j<K )=P ( j<1 )+P ( j<2 )+… and exploit it to

reduce the computational burdensome). The distribution of j is a binomial one (NUMj )π j∗(1−π )NUM− j

Multiply it to the relative Numeraire at time zero.

The forward measure is related to the expectation theory, where we want to test the assumption that the expected value

of the future rates evolution can be predicted by the current forward rate f (t ,T )=E [r (t ) ]. To test this assumption we

can use empirical data since by considering the definition of zcb price:

P ( t , T )=E[e−∫t

T

r (u )du ]→taking derivatives−dP ( t ,T )

dt=E[e−∫

t

T

r (u )dur (T )]

Remembering that f (T ,T )=r (T )→−dP ( t , T )

dt=P (t , T )E [ f (T ,T ) ]

Knowing the definition of instantaneous rate we ends up with f ( t ,T )=E [ f (T ,T ) ]

Black Formula:

Referring to Cap/Floor and Swaption we need to introduce the Black Formula, originally introduced to price commodity derivatives, specifically to overcome the seasonality of those underlings. So here the Black’s intuition/step for Cap:

It has been first adapted to price option on forward rate, so we basically change Caplet/Floorlet underwriting asset from being spot to be a forward rate, which is assumed to be lognormal distributed

o The intuition to use the forward instrument instead of the goods dynamics was introduced in commodity pricing model to avoid to consider the seasonality (we do not want it), basically the forward agreement is the expected value today of the final possible spot goods price under Q

18


o The forward agreement payoff is linked to the spot price by the formula X (T )=S ( t )er (T−t ), given the

dynamic of the spot securities [must be a Generalized Brownian motion] dS (t )=μS ( t )dt+S (t )dW ( t), by applying Ito’s, we will have the forward dynamics SDE dX (t ,T )=σ F X (t ,T )dW .

o We will do the usual log transformation to find X (T )=x (t ) e−σ F

2

2(T−t ) +σ F dW (t ), hence X(T) is a martingale

with zero drift (law of iterated expectation). Note that this dynamics is too simple for the commodity itself, you need to take into account the carry cost. If interest rates are chosen to br deterministic, the forward volatility will coincide with the stock one

o The same statement can be applied on FRA rate, better option written on FRA. In this case the stochastic variable at expiry coincide with the FRA, hence by applying the Markovian property (we can priced the option using the final payoff) we can substitute the variable with the FRA and we can model it by applying the Black hp => lognormal distribution

At first the formula was directly applied, assuming that the discount factor was not correlated with underlying asset (which is not the case) hence it was assumed deterministic (not good either), however later on thanks to the change of Numeraire has been possible to rewrite the expected value to be properly defined

o The input are the usual one of the BS formula note that the volatility parameter is the instantaneous

variance of interest rate variation σ 2=Var

[ dG (t , T+τ )G (t , T+τ ) ]∗1

dto The Numeraire chosen is the DF itself

An interesting result of the Cap/Floor pricing is their parity, in fact the difference between Cap and Floor with same features is equal to the value of a IRS (payer or receiver depending if we subtract the floor or the cap).To price a Corporate Bond with embed option we have to use the previously findings by applying the followings:

o Divide the coupon into elementary component: fixed and option payoffo Price the fixed coupon with the usual techniques. The option need as input the ATM flat volatility (note

that if the strike is different from ATM quote our pricing method will be biased)o Correct the DF to incorporate credit risk (Z-score is a simple but good proxy)o Note that the reference rate of the corporate bond and DF should be the same (share same dynamics)

We need to change a little bit our pricing formulation for the Swaption, since the underlying asset is the Swap rate, which is an average of more than one interest rate:

The first formulation as in the Cap case was by assuming a determinist discount factor so that we can move it outside the expectation and that the forward swap rate is lognormal distributed [Wrong18]

So we need to change the Numeraire and to impose the expected value of the drift to be 0 under the new probability measure to respect the NA condition

However the hp of a jointly lognormal distribution of forward rate and swap rate (

S (T )=∑w i−iF (t , T i ,T i+τ )average of them) is incorrect (mathematically impossible), none the less the

market is pricing the Swaption with Black, since empirically the distribution of Swap rate is not too far from being lognormal distribution

Why do we consider a risk-neutral probability measure?

If expectation was taken under the real-world probability measure P, this way of computing prices would be completely wrong since we are missing to evaluate risk, usually measured via volatility. Practitioners preferred to keep the simplicity 18 It was assumed to have deterministic discount factor, so that the DF and the payoff can be considered to be not correlated. This assumption will allow to take about form the expectation the DF

19


of an expectation, but they understood that they had to correct somehow the real world distribution P to take into account risk; in particular they understood that, due to risk aversion, they had to reduce expected rates of returns proportionally to risk. Since the bank account B(t) has no volatility, it has no risk and it is the only asset for which there is nothing to modify. For this to help us in understanding the general features of a risk-adjusted or risk-neutral probability measure, we need to make two assumptions.

Under a risk-neutral probability measure, returns have already been diminished to neutralize risk, so all assets can be treated as if they had zero risk. But if the risk is zero for all tradable assets and the market is arbitrage-free, then the expected rate of return must be the same for all tradable assets. We know that one of them, the locally riskless bank account, has an expected rate of return equal to r(t). For the market to be arbitrage-free, other tradable assets must have the same expected rate of return, therefore they also must have it equal to r(t). Above we used another assumption: securities need to be tradable. If they are not tradable, they cannot be used to build an arbitrage opportunity, therefore they are not forced to be consistent with the locally riskless asset. This assumption can be called liquidity, while academicians often call it completeness.

Short term interest Model

Before spending any time on describing those interest models, I need to point out just one point: why we are modeling interest rate? As we have studied, we have assumed flat and deterministic interest rate structure in pricing equity or commodity option, however this assumption cannot be applied for the discount factor used in option written on interest rate since the object of the option is the interest rate itself, so this is the reason why we are studying those models.

The important features to be kept in mind when using or implementing an interest rate model are: its tractability, ensure the existence of positive rate, the distribution used should allow “t” fit with empirical evidence (fat tails), the possibility to use the model to explicitly price option, its first and second moment have properly dynamics (do not explode, mean reverting), it can be used for Monte Carlo simulation or recombining lattices, and finally how can we compute the parameters.

We have two main family: the first is the one focused on modeling the “strange” instantaneous rate from whom we will obtain the price by computing its distribution and expected value under a risk neutral measure; the second will take as given the price and looks like how should be the implied/consistent instantaneous rate.

The first one is also the first to have been developed, they have tried to price bond and option written on bond by modeling instantaneous spot rate, which is the basic variable on which bond are readily defined by NA arguments. Basically we are going to define a dynamic, solve it to allow finding the distribution and computing so that we can have the expected value. The tough part is to build a model that matches all the meaningful features (the price at maturity must be 1; the volatility is not flat across maturity, no arbitrage across different expiry).

Merton model

The simplest one is the Merton model, note the volatility is flat regardless the maturity, the rate can be negative, it doesn’t fit the market structure.

1. Given the dynamics dr ( t)=μdt+σdW (t ), by solving it we have r (s )=r ( t )+μ∗(s−t )+∫t

s

σdW (u)

2. The distribution of ∫t

T

r ( s )ds is still Gaussian (since sum of normal random variable preserve distribution

features) and it is ∫t

T

r ( s )ds=r ( t )∗(T−t )+μ∫t

T

(s−t )ds+∫t

T

∫t

s

σdW (u)ds

20


a. μ∫t

T

(s−t )ds=μ (T−t )2

2

b. ∫t

T

∫t

s

σdW (u)ds=σ∫t

T

(∫s

t

ds)dW (u) and by exploiting the property of the expected value and the

variance for a sum of independent terms, we have N (0 ,∫tT

σ 2(∫s

t

ds)2

du=σ 2

3(T−t )3)

3. The distribution P ( t , T )=E[e−∫

t

T

r (s )ds], given the distribution of

∫t

T

r ( s )ds N (r (t )∗(T−t )+ μ (T−t )2

2;σ 2

3∗(T−t )3) by exploiting the Moment generation function

P ( t , T )=e−( r (t )∗(T−t )+ μ (T−t )2

2 )+ σ 2

6∗(T−t )3

4. The spot rate is R (t , T )=−lnP ( t , T )T−t

=r (t )+ μ2∗(T−t )−σ2

6∗(T−t )2, given this last result we have that

a. The volatility is flat equal to σ , no exotic/different from one correlation b. Only parabola, not multi hump, decreasing (increasing) when μ <0 (>0)

All the model here and after presented (besides the Dothan) show an affine structure, meaning that the continuously

compounded interest rate is defined ¿a ( t ,T )+b ( t ,T ) r (t) where the two parameters “a” and “b” are deterministic

function, that’s always the case if the pricing formula of the bond is P(t ,T )=A (t , T )e−B (t ,T ) r (t ). A model in general to

have an affine structure must have its own parameters defined with an affine structure. The zcb price under the risk neural

dynamics is not a martingale dP ( t ,T )=r ( t )P ( t , T )dt+v ( t , r (t ) ) P ( t , T )dW (t)

All this class of model share the same procedure: 1. define a dynamics to the interesting rate, 2. solve it and compute its distribution,3. take the expectation (under the same measure of the distribution computed before), now we have the whole

term structure that can be summarized as function of observed rate, time and some parameters as define in point one

4. solve by using the moment generating Gaussian function. If we have a dynamics (we need to associate one) remember: it doesn’t have a drift only and if it is a relative measure such as forward rate (basically it’s like eliminating the stationary process).

Vasicek Model

This pioneeristic approach is based on the Black and Scholes intuition, Vasicek had proposed to model the interest rate as the stock motion in the Black’s world by adding a mean reverting structure:

drt=k (µ−r (t))dt+σ dWtThis approach has some attractive features: it is an affine transformation since it is linear and it allows for explicit solution; however it is endogenous in nature, meaning that the yield curve is an output. Hence the model can be used only by bond traders, but other trader category will suffer of severe mispricing. The parameter µ is the long term mean of the process.

We can solve the SDE as follow:

21


1. transform our target equation to be f ( r t , t )=eKt r t. dfdt

=k rt ekt ;

dgd rt

=ekt

2. By applying Ito’s Lemma we have df (r , t )=k rt ekt dt+ekt∗(k (μ−rt )+σdW t )=kμektdt+σ ekt dW t

3. The last result is an arithmetic Brownian motion which is solved as follow (integral of the motion)

a. df (r , t )=d ¿ ) ∫ d (eKt rt¿)=rt ekt−r 0e

k 0 ¿

b. ∫ kμektdt= kμkek 0+ kμ

kekt=μ (ekt−1 )

c. ∫ σ ek(s−t )dW t by applying theorem to find distribution of transformation of Gaussian process.

Note that we have eliminated ektfrom the other 2 part of the equation, that’s why we have (s-t) in the

integral ∫ σ2 e2k ( s−t )(∫t

T

1ds )2

dt= σ2

2k∗(1−e2kt )

4. r0 e−kt+μ (1−e−kt )+σ∫ek ( s−t )dWwhich is distributed as N (r 0e

−kt+μ (1−e−kt ) , σ2

2k∗(1−e2kt ))

5. R (t , θ )=R∞−(R∞−r t )( 1−e−kθ

θ )+ σ2

4 k3θ(1−e−kθ )2whit R∞=μ−σλ

k− σ2

2k2

a. The curve can be flat, increasing or decreasing, but do not allow for the possibility of inverted yield curves, U-shaped or humped curve (allows for little humped structure)

b. The volatility behavior well fit the empirical data in fact since V ( t ;θ )=σ (ekθ−1 )kθ

it is a decreasing

function of θ maturity.

Since the yield curve is an output it is crucial to properly define the dynamics used and to fit the parameters value to maximize the fitting ability of the model (consistency problem, the curve won’t fit). Unfortunately one factor model are very little flexible and not useful for traders to perform the basic activity of risk management and replicating the derivatives payoff for hedging purpose (the forward rate are not consistent with the market one, not consistent with market quote the call or discount factor as well)), furthermore do not allow to model not trivial (perfect correlated) assets payoff; this model can be used to find mispriced Bond (if you trust your model, hence this model is also named equilibrium model, since price should converge on its value).

Hull and White

A simple trick to use to change the nature of the model into an endogenous one is to change the parameters (the long run mean and volatility) to be time dependent, this has been done by Hull and White (1990) one factor model.

d rt=k (µ (t )−r (t ) )dt+σ (t )dW t

However this decision can lead to more problem than the one solved, in fact often the volatility traded in the market is not significant or be unrealistic given the usual market shape. This problem makes the modified version good for plain vanilla derivatives, but unfeasible for exotic and path-dependent derivatives. Even the possible extension to make the volatility (diffusion parameter) time dependent is not a good one, since the model will get instable: unrealistic and irregular future derivative prices. Very bad for exotic interest rate derivative traders, because:

1) The value of exotic interest rate derivatives, such as path-dependent and in particular early-exercise products, strongly depends on dynamic model properties and model implication about future prices and volatilities. They

22


must be realistic for reliable pricing. We will understand it better when speaking of evolution of the term structure of volatility.

2) Exotic interest rate derivatives are hedged using plain vanilla derivatives. Instability means that recalibration changes parameters a lot even if the market has changed little. This leads to inefficient hedging. We will consider this against when speaking of correlations. These models are really useful since it allows working either with real world probability ad with risk neutral one; hence we can estimate our parameter using historical analysis. This property is defined as tractability; however this model can lead to negative interest rate with positive probability.

That’s why Hull and White have introduced the two factors model (1996), where the second factor is introduced by adding a stochastic variable:

drt=k (µ ( t )+ut−r ( t )) dt+σ 1dW1

dμ=−but dt+σ2dW2

This model allows for a better fit, to properly address the correlation (we have more sophisticated structure) behavior and it can be used to price Swaption, however it cannot be used for all derivatives instruments and do not allow to control for variance and correlation from market. This error can be reduced by adding more complex relationship with more factors (like the Balduzzi model [BDFS] where the Diffusion coefficient is made as in the CIR model, and it depends on another SDE), but the computational difficulty will dramatically increase, there is no closed formula available and numerical procedure are needed even to price plain vanilla bond option.

The calibration of these models are based on the minimization of weighted the squared error of price or spot rate

min∑ wi (Pmkt (T i )−Pmodel (T i) ), where the weights can be chosen to be equally weighted or 1

T−t to give more

importance to short term rate. We can fit the instantaneous rate instead of the Discount factor price.

Dothan Model

He defined a geometric Brownian motion γrt+rσdW=dr, it is the only lognormal model in the literature, that’s why the variance explosion problem. Furthermore this model is usually solved numerically since the derivation of the closed formula is more than comberstone

Cox Ingersoll and Ross (CIR) Model

This model can be viewed as a natural extension of the Vasicek model, where the diffusion parameter is made time dependent by adding a term into a square root. This simple adjustment made the model always positive given a simple constrain.

drt=k (µ−r t)dt+√σ r tdWt

The CIR model as well as the Vasicek one allows changing the probability measure, the drift change is the same, while the diffusion will be different and we need to make some assumption to ensure a tractable formula. This model will allow for inverted shapes in the short end of the curve, however won’t allow for humps or U-shaped, and it has an asymptotic

behavior in the long run as well as the Vasicek since V ( t ;θ )=σ √rtD (θ)

θ

23


The HJM Model (1992) or Entire Curve Model:Here we are changing the target of our model, now we are focusing on modeling the forward rate, with a NA prospective. Basically we want to model the whole term structure. That’s a great innovation, instead of using few point of the market yield curve, we are going to use an infinity of them (all). Our goal is to compute the instantaneous forward rate

f ( t )= lim∆t→0

F (t ,T 1, T 2 )=

−1P (t , T+1 )

∗ lim∆t→0

(P ( t , T )−P (t , T+1 ))

∆ t=

−dln (P ( t , T ) )dT

Where P(t,T) can be written as function of instantaneous spot rate P ( t , T )=E[e¿¿−∫ r (s )ds]¿. The discount factors

are equal to the market one, but option price it is still different. We need to guarantee NA on df, it will be done by taking into accont that df is only infinitesimal bits of the same term structure, hence we need to link it to short term dynamics.

The first suggested model is the Ho and Lee (1986), however the great contribution was given by Heath, Jarrow and Merton (1989) by using forward rate with some constrains on the instantaneous forward rate to guarantee the NA condition in pricing19, this can easily done by applying Ito formula to the spot rate dynamics (properly define) from whom we obtain the forward one: this example is made using the Merton formula:

1. dr ( t )=μdt+σdW hencer N (r (0 )+μt ; σ2t )2. P ( t , T )=exp∫

−r ( s)ds by applying the MGF, solving the integral we have P (0 , T )=eσ2 t2

6−r ( 0)− μ

2t

3. Substituting into the instantaneous forward definition f ( t )=−σ 2 (T−t )2

2+r (0 )+μ(T−t)

4. By applying Ito’s formula (t not T) df (t )=σ T ( t ) (∫ σs (s )ds)dt+σT (t )dW i which is the generalized HJM result

There are two possible formulation of the HJM, starting to define an appropriate instantaneous rate dynamics: Ho and Lee has proposed to use r ( t )=ϑ (t )dt+σdW , where the volatility structure is assumed to be constant

(flat as a simple short term model). Note that the volatility of the zcb is not flat in fact

σ zcb(s ,T )=∫ σ f ( s ,T )ds and at maturity the volatility is equal zero (NA requirement), remember also that the

f(t) dynamics is not Markovian, hence we have problem in pricing. The model depends on one parameter sigma

Vasicek Extended model specify a different dynamics r (t )=(ϑ (t )−r ( t)γ )dt+σdW , in this case the volatility is

a decreasing function (exponential factor) σ (t)=σ e−γ (T−t ). The model depends on sigma and lambda.

There is also a multivariate specification the two factor Hull and White model to allow the model to fit a correlation structure different from the perfect correlated one. In general this model depends on two time number of parameters (sigma, lambda) plus number of factor minus one (the correlation)

The steps to be followed to find the closed formula are: Find the dynamics of the instantaneous rate, find also its distribution Compute the integral to find the zcb price and its expected value Use the zcb price and the definition of fwd instantaneous rate to find the deterministic value of the parameter

ϑ (t ). The first one is ϑ ( t )=d f mkt (t )dt

+σ2(T−t) the second one is ϑ ( t )=d f mkt ( t )dt

+ σ2

2 γ(1−e−2 γ (T−t ))

However since we are still dealing with non-real parameters (instantaneous variable) the calibration will be hard to be used for hedging derivatives, on the other hand this model can use both the risk neutral probability and the real one.

19 Be consistent with a unique short rate term structure

24


That’s why the practitioners have rewritten the dynamics to be expressed in term of simple capitalization by using the change of Numeraire tool. Basically we will show that under the new zcb measure the forward rate is a martingale.

The price of caplet with HJM is different form the Black one since the first is modeling the forward price (forward rates are Gaussian under both measure), the black formula is modeling the simple forward rate (it is a martingale under its measure). HJM allows to price exotic payoff via MC (markovian property), Black do not allow this facilitation, moreover in black there is no explicit dependence on tenor of the rate, while HJM makes it explicits.

The MC simulation using the HJM works as follow: Simulate the n-random variables where [lect 8]

The Market ModelThis is the highest and more used/complete model presented in the market. Brace, Gatarek and Musiela (1997) and Jamshidian (1997) model the term structure of forward LIBOR rates and of Swap rates, replicating exactly the market prices of cap and swaptions (2nd level consistency) and so they justify the use the market adoption of the Black's formula for this class of derivative contracts. Remember that the assumption on Libor rates and Swap rate are not compatible.

This model is consistent with the Black formula assumption to price caplet, however in case of payoff depending on a plurality of forward rate, we need to compute the dynamics of all the forward involved based on one

LMM

Numeraire chosen on one forward rate. So only one forward will be a martingale (drift less) all the other will be

d Fk (t )=± F ( t )k σk (t)∗∑ρk , jσ j ( t )∗τ j∗F j ( t )

1+F j ( t ) τ jdt+σk ( t )Fk ( t )dW k (t), the sign depending if the forward chosen

has a shorter maturity (positive) to respect with the chosen Numeraire. This important result is obtained by using the Girsanov theorem:

The new stochastic dynamics under the new measure is dWnew=dW

old−DC ( ln( Nold

Nnew))∗dt

In the Libor market model the Numeraire are the zcb, hence

DC ( ln( P (t ,T old )P ( t , Tnew ) ))=DC [ ln∏ ( 1

1+F ( j )α ) ]=DC [∑ ln( 11+F j ( t )α )]=−DC [∑ ln (1+F j ( t )α ) ]=−∑ α j

1+α jF j (t )DC (F ( j ) )

where the diffusion coefficient of the fwd is DC (F j ( t ) )=σ jF j ( t ) assuming perfect correlation between rate

Here some useful property on the diffusion coefficient DC(aX)=aDC(X); DC(A+X)=DC(X); DC(lnX)=1/X*DC(X) abd

The diffusion coefficient of DC (X tY t )=Y tDC ( XT )+XT DC (Y t), where the X and Y are two random

variable with some randomness source

The relationship between the black formula find before (zcb change of Numeraire) where the volatility used is the cap or

the floor volatility bootstrapped. The Libor market model need to use the instantaneous volatility ∫ σ (s )2ds=T∗σcaplet2 .

The bootstrapped technique highly influence the price, there could be a difference as high as 38%. The major volatility bootstrapping techniques used to price cap are:

General Piecewise Constant is the more general, each piece in time has its own value. There are multiple

solution possible: √ t σ n=√σn , 12 +σn ,2

2 +σ n ,32

25


Constant maturity dependent, where basically we will use the market cap volatility as the reference value for the time interval (a second year cap will provide the same value for the first and second caplet, it is a sort of average)

Time to maturity dependent, same structure that will translate, in other world the one year to maturity

volatility is always the same √nσ2−∑ φi−12

Separable piecewise constant σ k ( t )=skφk−z

Linear exponential sk ¿, which need to be calibrated with the usual squared difference of standard deviation

with the caplet volatility √T k−1σ kcaplet. √ ∫

T j

T j+k

σ j+k2 (t )dt

T j+ k−T j

this is the algorithm to compute the volatility for the

different derivatives given the maturity and tenor

SMM

In case of the Swaption, the swap rate is drift less under its dynamics dS ( t )=σswap ( t )S ( t )dW . Here we have two

different possibilities to compute the derivatives price; if we use the Swap measure the volatility is the one modeled in the dynamics and bootstrapped form market quote by reversing the Black formula – Swap Market Model), when using the Libor market model the instantaneous volatility must be compute using two approximation to have a deterministic quantity computed using the relationship between forward rate and Swap one (remember that this quantity is not traded in the market, dealer trade differently):

We know that the Swap rate can be seen as an average of forward rate, by imposing the weights to be constant

we have ∑ w (0 ) Fi(t ) [first approximation, weights are not time dependent]

The swap dynamics given the forward one isdS ( t )=∑ w (0 )F i ( t )σ1dZ (t), given that the input volatility is

∫ dS (t )dS (t )S ( t )2

we need to compute dS (t )dS ( t )→≅∑∑ wi (0 )w j (0 )F i ( t )F j ( t ) ρσ i (t )σ j ( t )dt

Then dividing the previous result by the square swap and making the second approximation, the forward rate are not time dependent

∫∑∑ wi (0 )w j (0 )F i (0 )F j (0 ) ρ σ i ( t )σ j ( t )dt

(∑ w (0 ) Fi (0 )σ 1)2 =

∑∑ wi (0 )w j (0 )F i (0 )F j (0 ) ρ

S (0 )t2 ∫ σ i(t)σ j(t)dt

The final result is ∑∑ w i (0 )w j (0 ) Fi (0 ) F j (0 )∗ρ

(∑ w (0 )F i (0 ) )2 ∗∫ σ i(t)σ j(t)dt

Note the two specification have difference since they are not compatible one to the other, note that if using LMM to run Monte Carlo simulation using the forward dynamics (discretized to run simulation) the log normality between swap and forward seems to work. Hence we can use the finding before for the volatility as function of Libor volatility and correlation to plug into the LMM black formula (we cannot use directly the swap volatility). The misprice of the LMM will get quite severe at high volatility level (stressed value, which has been reached during the recent financial crisis).

The MC procedure (with “n” enough big converge to be Gaussian hence the range is built as usual ±2,33∗std ( payoff )

√n

and the payoff must be discretized as follow

F ( t+r )=F ( t )+F ( t )σ∗∑ σ j (t )∗τ∗F j ( t )1+F (t ) τ

∆ t+σ j ( t )Fk ( t ) [Z (t+r )−Z (t ) ]26


More complex derivatives need to define a correlation matrix avoiding irregularity in fitting (we will use a parameterization) and respecting all the desired correlation properties20. Since for interest rate we do not need as high as the number of rates (waste of computational time) we need to reduce the matrix rank approximate:

Two factors exponential form (no sub-diagonals property). ρ j ,i=γ+(1−γ )e [−β|i− j|]

Rebonato three parameters [angles parametrization] form (no semi definiteness not guaranteed). This model will find the parameters to reduce the rank so that the output will be ex-ante a reduced matrix.

o ρ j ,i=γ+(1−γ )e [−|i− j|∗(β−α (max ( i , j )−1)) ]

o We use the spectral theorem, we take out the smallest elements CC '=(X Λ ) (X Λ )' (last columns)

we multiply again the rectangular new matrix BB' and we eill normalize the new correlation matrix

ρi , j=ρi , jold

√ ρi ,iold √ ρi , j

old

Morini propose a simple but really effective method to reduce correlation matrix dimensionality. At first we perform the usual Spectral decomposition. Taking the Eigen value matrix we simple delete the last n-elements, we recover the correlation matrix and we impose to the diagonal the value 1. We will reiterated this operation till the correlation matrix obtained from the Eigen value is close to a real correlation matrix. The strength of this model is computation speed and accuracy

The terminal correlation is obtained from the instantaneous volatility measure and it is different form the instantaneous correlation. It is the correlation between forward rate (not the instantaneous one)

corr (Fi ; F j)≠corr ( d F i

Fi

;d F j

F j), this measure is important since the forward rate are the actual one

used/traded. TC=∫σ i ( t )σ j ( t ) ρdt

√∫ σ i2 ( t )dt √∫ σ j

2 ( t )dt it is an approximation

Formula della volatilità per caplet dipending on the implied vol used slides 3 parte 2 morini

Derivare lo swap rate volatility usando libro hp di lognormalità che è quella usata nella black forula ipotizzando i libor market model per prezzare swwap

Nota che la matrice ridotta non è quadrata, ma rettangolaere, perché toglie le colonne, moltiplicata per se estessa da un quadrato classic

Nel modellizzare continuos rate si una il moment generating function, in pratica trovo la distribuzione del tasso, da cui ottengo quella del bond che usa quel tasso slide 31 lezione 1 morini. Che poi il tutto è usato per prezzare il caplet usando il modello short rate scelto, in pratica non black, ma la closed formula usata in relazione al modlelo che spiega la dinamica del tasso

Quando facciamo l’integrale della dinamica dobbiamo stare attenti al fatto di avere o meno dY/y o semplicmente dY infatti nel primo abbiamo il log(y)= …

Remember under which measure a statement is true

20 Besides the usual one we need deco relation and increasing sub diagonal trend

27

fixed income: swaption, futures, irs, rate model

Economy & Finance