Modelling and Calibration of Swap Market Models · 2016. 2. 22. · The forward swap rates...

Modelling and Calibration of Swap Market Models

Zhijiang Huang,FAME & HEC, University of Lausanne

Olivier Scaillet,FAME & HEC, University of Geneva

May 2003

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Abstract

This paper derives a general framework for swap market models,which includes the co-terminal swap market model, the co-initial swapmarket model and the sliding swap market model. The standard Li-bor market model appears as a special case of the sliding swap marketmodel. European options on the chosen family of swap rates can bepriced directly using Black formula, which is phase with market prac-tice. Other interest rate derivatives need to be priced by approximationsor simulations. Approximation and simulation approaches are numeri-cally confronted to check the accuracy of the approximating formulas.Furthermore we compare different swap volatility specifications and cal-ibration methodologies from an goodness-of-fit point of view. Resultsfrom calibration are then used to price exotic swap rate derivatives in-cluding Bermudan swaptions.

1 Introduction

Recently interest rate market models have attracted much attention ofacademics and practitioners. They distinguish themselves from instan-taneous short rate and instantaneous forward rate modelling on severalaspects. First, they directly use simple interest rates like forward Liborrates and forward Swap rates, as modelling basis. This is an obviousadvantage since these interest rates act as underlying asset for many in-terest rate derivatives and their quotes are readily available on the mar-ket. Pricing of interest rate claims and calibration of the model becomesignificantly more simple. Second, they rely on a family of lognormallydistributed interest rates so that the associated European options canbe priced using Black formula for option on forward contracts. This pro-vides a pricing procedure consistent with market practice, and explainswhy they are called market models.

Up to now, the primary focus of interest has been the Libor marketmodel (LMM). Indeed LMM is often perceived as more easy to handle,and this may explain its popularity. We do not share this view and claimthat swap market models (SMM) are in fact better choices both in theirtheoretical and practical aspects. From a theoretical or mathematicalpoint of view, SMM and LMM are remarkably close in their construc-tion. Both models start with a family of lognormally distributed simpleinterest rates. The chosen set of simple interest rates are forward Liborrates in LMM and forward swap rates in SMM. Furthermore, as it willbe clear from later developments, SMM are much more general and ac-commodate LMM as a special case. We believe however that it is moreimportant to judge them from a practical point of view. To such extentSMM have far more attractive features. First of all, banks typically useinterest rates with long maturities to hedge their positions. Libor ratematurities are basically shorter than 1 year, and thus LMM should notbe of first importance for risk management purposes. Second, a swaprate can be viewed as a basket of several Libor rates while a Libor ratecan be expressed as a combination of only two successive swap rates.This means that any error in the calibration of short dated Libor instru-ments will impact calibration of long dated instruments. This problemwill not happen in the calibration of SMM. Third, LMM is more deli-cate to calibrate because of the inherent nature of the volatility structureback out from observed caplet prices.

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2 Framework

2.1 Tenor structure, forward swap and Libor rates

We assume that we are given a pre-specified collection of reset/settlementdates T0 = 0 < T1 < ... < TM , referred to as the tenor structure (see Fig-ure ??). Let us denote δj = Tj −Tj−1 for j = 1, · · · ,M . Then obviously

Tj =j∑

i=1

δj for every j = 0, · · · ,M .

T0=0 T1 T2 Tk Tk+1 TM-1 TM

Tenor structure.

We write B(t, Tj), j = 1, ...,M, to denote the price at time t of aTj-maturity zero-coupon bond. The forward Swap rate S (t, Tj, Tk) , forany integers j and k satisfying 1 ≤ j < k ≤ M, is defined as usuallythrough

S (t, Tj, Tk) =B (t, Tj)−B (t, Tk)

G (t, Tj, Tk), ∀t ∈ [0, Tj] . (1)

Let us further consider the fixedmaturity coupon price processG (t, Tj, Tk),corresponding to the the level numeraire, and defined by:

G (t, Tj, Tk) =k∑

l=j+1

δlB (t, Tl) , ∀t ∈ [0, Tj] . (2)

A probability measure PTj ,Tk on(Ω, FTj

), equivalent to the historical

probability measure P , is said to be the forward Swap probability mea-sure, associated with the date Tj and Tk, if for every i = 1, 2, ...,M, the

relative bond price B(t,Ti)G(t,Tj,Tk)

, ∀t ∈ [0, Ti ∧ Tj+1], follows a local martingale

under PTj,Tk . Obviously, G (t, Tj, Tk) is the price of the numeraire linkedto the probability measure PTj ,Tk and the forward Swap rate S (t, Tj, Tk)is a martingale under PTj ,Tk . We denote the corresponding Brownianmotion under PTj,Tk by W Tj,Tk .

If we are ready to assume that forward Swap rates follow diffusionprocesses, S (t, Tj, Tk) should be martingale and its drift term shouldvanish under its corresponding probability measure PTj,Tk ,:

dS(t, Tj, Tk)

S(t, Tj, Tk)= λ(t, Tj, Tk)

′

dW Tj ,Tk , ∀t ∈ [0, Tj] ,

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where the volatility function λ(t, Tj, Tk) is left unspecified for themoment. Note that if λ(t, Tj, Tk) is deterministic, S(t, Tj, Tk) will belognormally distributed.

Let us observe that the forward Libor rate L (t, Tj) , j = 1, ...,M −1,defined as

L(t, Tj) =B(t, Tj)−B(t, Tj+1)

δj+1B(t, Tj+1), ∀t ∈ [0, Tj] , (3)

obtains as a special case of the forward Swap rate S(t, Tj, Tk) whentaking k = j + 1.

We denote PTj and W Tj the corresponding forward Libor probabilitymeasure of L(t, Tj) and the Brownian motion under PTj . Then, for every

i = 1, 2, ...,M, the relative bond price B(t,Ti)δj+1B(t,Tj+1)

, ∀t ∈ [0, Ti ∧ Tj+1],

follows a local martingale under PTj .To ease reading, we introduce the following compact notations for

discount bonds, level numeraires, forward Swap rates, forward Liborrates and volatility functions:

Bj (t) = B (t, Tj) , Gjk (t) = G (t, Tj, Tk) ,

Sjk (t) = S(t, Tj, Tk), λjk (t) = λ(t, Tj, Tk),

Lj (t) = L(t, Tj), λj (t) = λ(t, Tj).

Furthermore we will often omit time indexing.

2.2 The three types of SMM

As alreadymentioned there are basically three types of SMM: co-terminal,co-initial and sliding.

The forward swap rates underlying the co-terminal Swap marketmodel are shown in Figure 1. All forward swap rates share the sameterminal date TM . Co-terminal Swap markets models (CTSMM) arethus build on a family SjM , j = 1, · · · ,M − 1, of forward Swap rates,a collection of mutually equivalent probability measures PTj ,TM , j =1, · · · ,M − 1, and a family W Tj,TM , j = 1, · · · ,M − 1, of processes insuch a way that: (i) for any j = 1, · · · ,M − 1, the process follows a d-dimensional standard Brownian motion under the probability measurePTj,TM , (ii) for any j = 1, · · · ,M −1, the forward Swap rate satisfies theSDE:

dSjM = SjMλjM′

dWTj,TMt , ∀t ∈ [0, Tj] ,

with the initial condition

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Figure 1: Co-terminal Swap rates.


Figure 2: Co-initial swap rates.

SjM (0) =Bj (0)−BM (0)∑M

l=j+1 δlBl (0).

In the co-initial Swap market model all forward swap rate have thesame initial date T1 (see Figure 2). Co-initial Swap markets models(CISMM) are build on a family S1j, j = 2, · · · ,M, of forward Swaprates, a collection of mutually equivalent probability measures PT1,Tj ,j = 2, · · · ,M , and a family W T1,Tj , j = 2, · · · ,M, of processes in such away that: (i) for any j = 2, · · · ,M , the process follows a d-dimensionalstandard Brownian motion under the probability measure PT1,Tj , (ii) forany j = 2, · · · ,M , the forward Swap rate satisfies the SDE:

dS1j = S1jλ1j′

dWT1,Tjt , ∀t ∈ [0, T1] ,


S1j (0) =B1 (0)−Bj (0)∑j

l=2 δlBl (0).

Finally the forward swap rates associated with the sliding Swap mar-ket model are plotted in Figure ??. Here forward swap rates share the

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Figure 3: Sliding swap rates.


Figure 4: Spcial case of sliding swap rates –– Libor rates.

same time interval between tenor dates. Sliding Swap markets models(SLSMM) are build on a family Sj,j+n, j = 1, · · · ,M−n, of forward Swaprates, a collection of mutually equivalent probability measures PTj ,Tj+n ,j = 1, · · · ,M −n, and a family W Tj ,Tj+n , j = 1, · · · ,M −n, of processesin such a way that: (i) for any j = 1, · · · ,M − n, the process follows ad-dimensional standard Brownian motion under the probability measurePTj,Tj+n , (ii) for any j = 1, · · · ,M − n, the forward Swap rate satisfiesthe SDE:

dSj,j+n = Sj,j+nλj,j+n′

dWTj ,Tj+nt , ∀t ∈ [0, T1] ,


Sj,j+n (0) =Bj (0)−Bj+n (0)∑j+n

l=j+1 δlBl (0).

Figure 4 shows the LMM which can be viewed as a special case ofSLSMM by simply taking n equal to 1.

2.3 Co-Terminal Swap Market Model

The co-terminal Swap market model takes the dynamics of a set of co-terminal Swap rates SjM as given and express other swap rates andderivatives as functions of the given set. This model has been analyzedinitially by Jamshidian (1997), in which the following quantities are in-troduced:

νij ≡ νij,M ≡M−1∑k=j

δk+1

k∏l=i+1

(1 + δlSlM) ,

νi ≡ νii,

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Note that these expressions only use co-terminal swap rates, and thatthe following relations hold:

GjM = BMνj, (4)

Bj

BM

= 1 + νjSjM . (5)

2.3.1 Swap rate dynamics

First observe that the forward swap measure PTM−1,TM associated to levelnumeraire GM−1,M is the same as the terminal forward forward Libormeasure PTM associated to the discount bond BM since GM−1 = δMBM .

This also implies that WTM−1,TMt = W TM

t . Jamshidian (1997) has shownthat the swap rate dynamics under the terminal forward measure PTM

is then given by:

dSjMSjM

= −λ′jM

M−1∑i=j+1

νji1 + δiSiM

δiSiMλiMνj

dt + λ′jMdW TM

t .

The above expression only involve swap rates and swap rate volatil-ities.

More generally it can be shown that under PTl,TM :

dSjMSjM

= λ′jM

[M−1∑i=l+1

νli1 + δiSiM

δiSiMλiMνl

−M−1∑i=j+1

νji1 + δiSiM

δiSiMλiMνj

]dt+λ′

jMdW Tl,TMt .

2.3.2 Libor rate dynamics

Using the fact that Libor rates are martingale under their own forwardLibor measure, it can be easily deduced from the relationship

dWTjt − dW TM

t = −SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1

δlSlMλlMνjl1 + δlSlM

)dt.

induced by the change of measure between P Tj and P TM that

dLj

Lj

= −λ′jSj+1,M

1 + νj+1Sj+1,M

(λj+1,Mνj+1 +

M−1∑l=j+2

δlSlMλlMνj+1,l1 + δlSlM

)dt

+ λ′jdW

TMt .

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Similarly by using the change of measure between P Tj and P Tk,TM

yielding

dWTjt − dW Tk,TM

t =

(M−1∑i=k+1

νki1 + δiSi

δiSiλiνk

−SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1


))dt,

we get

dLj

Lj

= λ′j

(−

Sj+1,M1 + νj+1Sj+1,M

(λj+1,Mνj+1 +

M−1∑l=j+2


)+

M−1∑i=k+1

νki1 + δiSi

δiSiλiνk

)dt

+ λ′jdW

Tk,TMt .

2.4 Co-Initial Swap Market Model

The co-initial Swap market model relies on the dynamics of a set of co-initial Swap rates S1j. Here we need to define quantities only dependingon co-initial swap rates:

υij ≡

j∑l=2

δl

i∏k=l

(1 + δkS1k)−1 ,

υj ≡ υjj.

These are the counterparts of the former νij and νj of the co-terminalmodel. It is then easy to check that the following equations are satisfied:

Bj = 1− υjS1j, (6)

G1j = υj. (7)

2.4.1 Swap rate dynamics

The swap rate dynamics in the initial forward measure PT1 are:

dS1jS1j

= λ′1j

j∑l=2

δlS1lλ1lυj

υjl1 + δlS1l

dt+ λ′1jdW

T1t .

Besides we also have under PT1,Tk :

dS1jS1j

= λ′1j

[j∑

l=2

δlS1lλ1lυj

υjl1 + δlS1l

−k∑l=2

δlS1lλ1lυk

υkl1 + δlS1l

]dt+λ′

1jdWT1,Tkt .

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2.4.2 Libor rate dynamics

The Libor rate dynamics in the initial forward measure P T1 and in theforward swap measure P T1,Tk are given by:

dLj

Lj

=λ′jS1j+1

1− υj+1S1j+1

(λ1,j+1υj+1 +

j+1∑l=2

δlλ1lS1lυj+1,l1 + δlS1l

)dt+ λ′

jdWT1t

and

dLj

Lj

= λ′j

(S1j+1

1− υj+1S1j+1

(λ1,j+1υj+1 +

j+1∑l=2


)−

k∑l=2

δlS1lλ1lυk

υkl1 + δlS1l

)dt

+ λ′jdW

Tk,TMt

2.5 Sliding Swap Market Model

The sliding Swap market model (SLSMM) starts with the dynamics ofa set of swap rates Sj,j+n, j = 1, ...,M − n. When n = 1, SLSMM coin-cides with LMM and has been developed in Brace, Gatarek and Musiela(1997), Jamshidian (1997) and Miltersen, Sandmann and Sondermann(1997). When n>1, we have more general sliding swap market models.

To be developed.

2.6 Pricing and Approximations

In the following section, our discussion is only for co-terminal Swap mar-ket models. Results suitable for the co-initial and sliding swap marketmodels can be easily derived with minor and straightforward modifica-tions.

2.6.1 Swaption pricing

Under a deterministic volatility structure co-terminal swap rates arelognormally distributed, so the corresponding Swaption can be pricedvia Black-Scholes formula. This is in phase with market convention:

Swaption (t, Tj, K) = δj+1GjM (t)EPTj,TMt

[(SjM (Tj)−K)+

]= δj+1GjM (t) [SjM (t)N (d1)−KN (d2)] ,

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where

d1 =ln(SjM (t) /K) + 1

2σ2BS,j (Tj − t)

σBS,j√Tj − t

,

d2 = d1 − σBS,j√Tj − t,

σ2BS,j =1

Tj − t

∫ Tj

t

λ′jMλjMds.

2.6.2 Caplet pricing

Let us consider the caplet price:

C (t, Tj+1, K) = δj+1B (t, Tj+1)EPTj+1t

[(L (Tj, Tj)−K)+

],

where the expectation is taken w.r.t. the forward measure PTj+1

under which the following forward LIBOR rate is a martingale:

dLj (t) = Lj (t)λj (t) dWTj+1t .

By using the standard change of numeraire technique we get:

C (t, Tj+1,K) = δj+1GjM (t)EPTj,TMt

[1

GjM (Tj+1)(L (Tj, Tj)−K)+

],

or

C (t, Tj+1, K) = δj+1BM (t)EPTMt

[1

BM (Tj+1)(L (Tj, Tj)−K)+

].

Libor rates are not lognormally distributed, so we can not pricecaplets using Black-Scholes formula directly. Two approximation ap-proaches are commonly used to price swaptions in the Libor marketmodel: the Hull and White approach and the Rebonato approach. Theyhave been proposed in Hull and White (1999) and Rebonato (1988)respectively. In the following we modify both methodologies to pricecaplets in the co-terminal swap market model.

2.6.3 Hull and White approach

Let us start from L (Tj, Tj) =1

δj+1

(1+νjSj

1+νj+1Sj+1− 1). We know that the

forward LIBOR rate is a martingale under the forward measure P Tj+1.Hence we have

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dL (t, Tj) =M−1∑l=j

∂L (t, Tj)

∂S (t, Tl, TM)S (t, Tl, TM)λ (t, Tl)

′ dWTj+1t .

Direct computation leads to:

∂L (t, Tj)

∂S (t, Tl, TM)=

1

δj+1

(∂νj∂Sl

Sj + νj∂Sj∂Sl

)(1 + νj+1Sj+1)−

(∂νj+1∂Sl

Sj+1 + νj+1∂Sj+1∂Sl

)(1 + νjSj)

(1 + νj+1Sj+1)2

,

where

∂νj∂Sl

=

νjl

δl1+νlSl

, l > j,

0, l ≤ j.

So,

∂L (t, Tj)

∂S (t, Tl, TM)

=

νjδj+1(1+νj+1Sj+1)

, l = j,νj+1(δj+1Sj(1+νj+1Sj+1)−(1+νjSj))

δj+1(1+νj+1Sj+1)2 , l = j + 1,

δl(νjlSj(1+νj+1Sj+1)−νj+1,lSj+1(1+νjSj))δj+1(1+νj+1Sj+1)

2(1+νlSl), j + 2 ≤ l ≤ M − 1,

0, otherwise.

Let us consider

dL (u, Tj)

L (u, Tj)=

M−1∑l=j

∂L (u, Tj)

∂S (u, Tl, TM)

S (u, Tl, TM)

L (u, Tj)λ (u, Tl)

′ dWTj+1t .

We will freeze the swap rates and consider:

dL (u, Tj)

L (u, Tj)=

M−1∑l=j

wl (t)λ (u, Tl)′ dW

Tj+1

t .

with wl (t) =∂L(t,Tj)

∂S(t,Tl,TM )S(t,Tl,TM )L(t,Tj)

.

The volatility parameter to plug into the Black-Scholes caplet priceis then simply given by the square root of:

σ2BS,j =1

Tj − t

M−1∑l=j

M−1∑k=j

wl (t)wk (t)

∫ Tj

t

λ (u, Tl)′ λ (u, Tk) du.

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The above approximation is only valid if the weightswl (t) =∂L(t,Tj)

∂S(t,Tl,TM )S(t,Tl,TM )L(t,Tj)

do not vary too much.We have

d < wl, wk >=SkLj

SlLj

<∂Lj

∂Sk,∂Lj

∂Sl> +

SkLj

∂Lj

∂Sl<

∂Lj

∂Sk,SlLj

> +

∂Lj

∂Sk

SlLj

<SkLj

,∂Lj

∂Sl> +

∂Lj

∂Sk

∂Lj

∂Sl<

SkLj

,SlLj

>

From the expression of ∂L∂S

it is quite easy to observe that the numer-ator and denominator are of the same order for the two first subscriptsj and j + 1, namely δ (1 +O (S)) which means that the ratio is close toone. The numerator is of order δO (S) for higher j. The other ratio L

Sis

also close to one since swap rate and LIBOR rate share the same orderof magnitude. Similarly computing

d <SkLj

,SlLj

>=1

Lj

1

Lj

d < Sk, Sl > +SkL2j

SlL2j

d < Lj, Lj >

−1

Lj

SlL2j

d < Sk, Lj > −SkL2j

1

Lj

d < Lj, Sl >

shows that each term is made of ratios with denominators and nu-merators of the same order of magnitude. Computations of the otherterms in the sum lead to the same conclusions.

Numerical studies show that the first two weights are much largerthan others, with the difference about 1000 times. The price or impliedvolatility of caplet will only change marginally if other weights apartfrom the first two are neglected.

2.6.4 Rebonato approach

This approach is analogous to the approach advocated by Rebonato(1998). The basket approach is based on an approximation recognizingthat the basket process dynamics, here a basket of only two swap rates,is close to lognormal.

Rewrite Libor rate as a basket (actually a spread) of two co-terminalswap rates:

Lj = wjSjM − wj+1Sj+1,M ,

where

wj =νj

δj+1 (1 + νj+1Sj+1,M),

wj+1 =νj+1

δj+1 (1 + νj+1Sj+1,M),

wj −wj+1 = 1.

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Empirical evidences show that the weights wj and wj+1 are much lessvolatile than co-terminal swap rates, so we can neglect its contributionto the total uncertainty.

Under its forward Libor measure P Tj+1, we have

dLj (u)

Lj (u)=(wj (u)λ

′jM (u)− wj+1 (u)λ

′j+1,M (u)

)dW Tj+1

u ,

with

wj =νj

δj+1 (1 + νj+1Sj+1,M)

SjMLj

,

wj+1 =νj+1

δj+1 (1 + νj+1Sj+1,M)

Sj+1,MLj

,

wj − wj+1 = 1.

Then we may freeze the weights to obtain a deterministic approxi-mation of the lognormal volatility:

dLj (u)

Lj (u)=(wj (t)λ

′jM (u)− wj+1 (t)λ

′j+1,M (u)

)dW Tj+1

u .

The volatility parameter to plug into the Black-Scholes caplet priceis then simply given by the square root of:

σ2BS,j =1

Tj − t

∫ Tj

t

[w2j (t)λ

′jM (u)λjM (u) + w2

j+1 (t)λ′j+1,M (u)λj+1,M (u)

−2wj (t) wj+1 (t)λ′jM (u)λj+1,M (u)

]du.

Note that∫ Tjt

λ′jM (u)λjM (u) du can be obtained from the price of

the swaption on SjM . The integral∫ Tjt

λ′j+1,M (u)λj+1,M (u) du is part of∫ Tj+1

tλ′j+1,M (u)λj+1,M (u) du obtained from the price of the swaption on

Sj+1M . The integral∫ Tjt

λ′jM (u)λj+1,M (u) du is unknown and needs to

be calibrated while respecting the constraints on the previous integrals.

2.6.5 Spread option approach

Instead of using the fact that the spread dynamics is close to lognormal,we may use the close form solution for an option on a spread afterfreezing of the weights: Since the caplet is an expectation under theforward measure , we will then need the dynamics of SjM and Sj+1,Munder P Tj+1. These dynamics will involve some drifts, which will needto be frozen.

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2.7 Simulation

To check the accuracy of the approximation pricing approach, we needto do simulation and compute the prices of caplet and other derivatives.Swap rates and Libor rates are martingale underlying their appropriateforward measure, however, this property will be easily lost in discretetime simulation. Therefore, as pointed out by Glasserman and Zhao(2000), it is very important to choose adequately the quantities to besimulated.

dνjνj

=1

νj

M−1∑l=j+1

(λl (νl−1 − δl − νl)

l−1∏s=j+1

νs−1 − δsνs

)dW TM

t .

To be typed in.

2.8 Calibration

When we deal with calibration, it is more convenient to use the followingscalar specification of co-terminal swap rates for j = 1, ...,M − 1, undertheir appropriate forward swap measures:

dSjMSjM

= λjMdWTj,TMt .

where λjM = |λjM | is the norm of the corresponding instantaneous

vector λjM and WTj ,TM

t =λ′jMdW

Tj,TMt

|λjM |are one dimension Brownian mo-

tion under the forward measure P Tj,TM with numeraire Gj. Notice thereexist simple transformation between these two dynamics specifications:

λ′iMλjM = ρij(t)λiMλjM

where ρij(t) is the instantaneous correlation between Brownian mo-tions, as well as between swap rates, which is normally taken as inputand treated as constant ρij by practitioners in calibration.

ρij(t)dt = d < WTi,TM ,W

Tj,TM >

So far we have not specified the instantaneous volatility function ofthe co-terminal swap rates. The instantaneous volatility of swap ratesare not observable in the market, but two observations help us to for-mulate them. First, we observe today the ”hump” shape of the Blackimplied volatilities (market volatilities) of the swaptions with the sameunderlying ( The same underlying here means the forward swap rates ofthe same length of life disregarding when they become spot rate.). In

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other words, the Black implied volatilities increase at the early beginningand then decrease with maturity. Second, we observe that the Black im-plied volatility of a swaption increases during the time evolution whenit is approaching to maturity, and decreases at the end. Moreover, weobserve that the ”hump” shape of implied volatility is preserved duringtime evolution. This suggests the local volatility is time-homogeneousto some extend. We take the following form of instantaneous volatility:

λjM(t) = φj(t)ψj(Tj − t)

The second part, ψj(Tj−t), is to fit the hump of the implied volatilityof the swaption with the same underlying as SjM , due to the briefs frompractitioners that the swap rate family of SjM share something commonthrough ψj(Tj−t). The first part can be a constant or a slowly decreasingfunction of time t, in order to press down the hump a bit but not todestroy the shape.

Further more, by careful consideration, only an increasing instan-taneous volatility specification of time t is consistent with the secondobservation. We further take the parametric form of the second part:

ψj(Tj − t) = (aj(Tj − t) + bj)e−cj(Tj−t) + dj

This specification of ψj(Tj−t) is a decreasing function of Tj−t, but anincreasing function of t ∈ [0, Tj] for a given Tj. From now on, we take it asfunction of one single variable Tj−t. Term e−cj(Tj−t) is used to model thedown shape and term aj(Tj− t)+bj is to model the small increase at thebeginning and together (aj(Tj − t) + bj)e

−cj(Tj−t) will produce an humpshape. Term dj sets the level. As we mentioned above, by modellingψj(Tj−t) a function of time to maturity, the instantaneous volatility willbe time homogeneous if the first component is quite flat and close to oneconstant. This feature is desirable and allows us to fit the shape withthe hump of implied volatility of the corresponding swaption family. Theset of parameters θj = (aj, bj, cj, dj) will be difference resulting from thedifference among hump shapes of swaptions.

Before we discuss calibration methodology, it is worth to emphasizethat the importance of choosing appropriate market data to calibrate themodel according to the financial product priced or risk managed, due tothe fact that financial market is not always consistent or even completeas financial theories suggest. Therefore, it is important to identify therisk source and use the appropriate market data to catch the right in-formation which is crucial to the product. In the case of Swap marketmodel, when the product is only linking to co-terminal swaption, we willonly calibrate the model with co-terminal swaption market data. One

16

example is Bermudan swaption, which has an natural co-terminal swapmarket structure. When the product concerning both swap and liborrisk, we will calibrate the model on both swaption and caplet marketdata to catch both information for pricing and risk management pur-pose. We will try to calibrate on both swaption and caplet market datain this paper.

As mentioned above, we do calibration in two steps. First, we fitthe hump shape of market volatility of swaptions with ψj(Tj − t) andthen adjust the level to exactly recover the corresponding co-terminalswaption. Second, by choosing φj(t) and correlation ρij we try to fitboth caplet and swaption market volatilities. φj(t) is a constant or afunction of time close to 1. If it is an function, it is used to slightlychange the shape of the instantaneous volatility in order to fit bothcaplet and swaption market volatility better. If the correlation is takenas input, the only freedom left is the first component of the instantaneousvolatility of swap rates.

We will apply two kinds of calibration methodologies in the secondstep: bootstrap and global minimization.

17

3 Numerical results

3.1 Pricing approximations

In this section we analyze the performance of the approximations sug-gested in the previous section, namely approximations of caplet pricesrelying on the so-called Hull and White and Rebonato approaches aswell as the spread approach. The performance is measured with re-spect to the ”true” price obtained by the Glasserman and Zhao (2000)Monte Carlo method with 100 000 simulations and 16 steps per pe-riod. We consider caplets on 1-year LIBOR and time-to-maturitiesup to 9 years for TM = 10. We use one Brownian motion for eachswap rate and instantaneous correlation between SjM and SkM is set to1− 0.01|j − k|. Instantaneous volatilities of swap rates are taken equalto λjM(t) = (a(Tj − t) + b) exp(−c(Tj − t)) + d. This form is flexibleenough to cover most shapes possibly observed in the market. We use arange of volatility values observed in practice, i.e. values between 10%and 30%, and four different shapes: increasing, decreasing, bump andhump. These shapes are plotted on Figure 5 and Table 1 gathers theassociated values of a, b, c, d.

0

10

20

30

40

0 1 2 3 4 5 6 7 8 9

Time to Maturity (year)

Inst

anta

neo

us

vola

tilit

y (%

)

dec.

bump

inc.

hump

Figure 5: Four types of instantaneous volatilities of swap rates, as func-tions of time to maturity.

In Table 2 we report mean absolute relative errors (MARE) in termsof implied volatilities and caplet prices. The average is computed onthe nine maturities for the four different shapes. All approaches deliverrather good approximations with a maximum MARE of 2.2%. The Hulland White approach seems to be the one to be preferred in practice.

18

a b c ddec. .05 .20 .45 .10bump .20 .15 .60 .10inc. -.05 -.20 .45 .30hump -.20 -.15 .60 .30

Table 1: Parameter values of volatility shapes

Indeed it beats the two other approaches in all the situation. Let usfurther remark that only the two adjacent weights wj(t) and wj+1(t)really matter in the HW approximation of the dynamics of L(t, Tj). Theweights wj+2(t), ..., wM−1(t) are in fact 1000 times smaller than thetwo first weights wj(t) and wj+1(t). It can be checked that neglectingthese additional terms has a marginal impact on the performance of theHW approximation only (Figure ??). We will see later that using twoweights instead of the full set has some advantage in the calibrationphase. Moreover, we can see that the sum of all the weights is slightlylarger than 1. Recall that in Rebonato approach the sum of weightsis exactly equal to 1. This may explain partly that Rebonato is lessaccurate than Hull and White approach in Swap Market Model.

Imp. Vol. HW Rebo. Spreaddec. 0.76% 1.4% 2.2%bump 0.92% 1.7% 2.2%inc. 0.95% 1.4% 1.5%hump 1.4% 1.5% 2.0%

Prices HW Rebo. Spreaddec. 0.72% 1.3% 2.1%bump 0.88% 1.6% 2.1%inc. 0.94% 1.4% 1.5%hump 1.4% 1.5% 2.0%

Table 2: Mean absolute relative error

19

1 2 3 4 5 6 7 8 9 Sum1 12.1812 -11.0065 -0.0164 -0.0148 -0.0130 -0.0109 -0.0085 -0.0059 -0.0030 1.10232 9.3412 -8.2440 -0.0103 -0.0090 -0.0076 -0.0059 -0.0041 -0.0021 1.05823 7.4225 -6.3704 -0.0064 -0.0053 -0.0042 -0.0029 -0.0015 1.03194 6.0591 -5.0305 -0.0041 -0.0032 -0.0022 -0.0012 1.01795 4.9358 -3.9204 -0.0026 -0.0018 -0.0009 1.00996 3.9141 -2.9066 -0.0015 -0.0008 1.00517 2.9127 -1.9107 -0.0005 1.00158 1.9670 -0.9666 1.00049 1 1

Weights of the ith Libor rate ( ith row ) on the jth co-terminal swaprate ( jth column ) computed from market data on March 17, 2003.

3.2 Calibration

To be done.

20

APPENDIX A: Co-terminal swap rate market model

We start with the dynamics of the discount bond since it is thebuilding block linking all swap rates. The discount bond dynamics underthe risk neutral measure are taken as usually equal to:

dBj

Bj

= rtdt + σ′jdWt,

for some volatility function σj.

A.1. Swap rate dynamics

By Ito’s lemma and from (1) we get:

dSjMSjM

= d

(Bj −BM

GjM

)/

(Bj −BM

GjM

)=

d (Bj −BM)

Bj −BM

−dGjM

GjM

−d < Bj −BM , GjM >

(Bj −BM)GjM

+d < GjM , GjM >

G2jM

=

Bjσ′j −BMσ′

M

Bj −BM

−

M∑l=j+1

δlBlσ′l

M∑l=j+1

δlBl

dWt −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

dt

.

Now using the fact that the swap rates are martingale under theirown forward swap measure, we deduce:

λjM =Bjσj −BMσM

Bj −BM

−

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

,

dWTj ,TMt = dWt −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

dt.

Since the forward swap measure PTM−1,TM coincides with PTM , we

have WTM−1,TMt = W TM

t and we obtain:

21

dWTj ,TMt − dW TM

t = −

M∑

l=j+1

δlBlσl

M∑l=j+1

δlBl

− σM

dt.

This expression can be expressed only in terms of swap rates usingthe definitions of νij, νi, and the relations (4), (refbond):

M∑l=j+1

δlBl (σl − σM)

M∑l=j+1

δlBl

=

M∑l=j+1

δl (1 + νlSl) (σl − σM)

νj.

Now since

λjM =Bjσj −BMσM

Bj −BM

−

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

= (σj − σM)(1 + νjSjM)

νjSjM−

M∑l=j+1

δl (1 + νlSlM) (σl − σM)

νj,

and defining uj = (σj − σM) (1 + νjSjM), we deduce that

uj − Sj

M∑l=j+1

δlul = λjνjSj.

Rewriting this formula in matrix form and inverting the relation weget

uj = λjMνjSjM + SjM

M−1∑l=j+1

δlλlMνlSlM

l−1∏k=j+1

(1 + δkSkM) .

This gives

M∑l=j+1

δl (1 + νlSl) (σl − σM)

νj=

M∑l=j+1

δlul

νj,

and finally

22

dWTj ,TMt = dW TM

t −

M∑l=j+1

δlul

νjdt.

The above expression may also be rewritten in order to show ex-plicitly the dependence in the swap rate volatilities, i.e. the lambdas(Jamshidian (1997)).

First note that for i<j

νj ≡ νjj ≡M−1∑k=j

δk+1

k∏l=j+1

(1 + δlSl) =M−1∑k=j

δk+1

∏k

l=i+1 (1 + δlSl)∏j

l=i+1 (1 + δlSl)=

νij∏j

l=i+1 (1 + δlSl).

Then we may reorganize

1

νj

M−1∑i=j+1

δiui =1

νj

M−1∑i=j+1

δiSi

(λiνi +

M−1∑l=i+1

δlλlνlSl

l−1∏k=i+1

(1 + δkSk)

)by the order of λi, i.e. according to λj+1, λj+2, λj+3, ...λj+1 term:

1

νj(δj+1Sj+1λj+1νj+1) =

νj,j+11 + δj+1Sj+1

δj+1Sj+1λj+1νj

.

λj+2 term:

1

νj(δj+2Sj+2λj+2νj+2 + δj+1Sj+1δj+2Sj+2λj+2νj+2)

=νj+2 (1 + δj+1Sj+1) (1 + δj+2Sj+2)

1 + δj+2Sj+2

δj+2Sj+2λj+2νj

=νj,j+2

1 + δj+2Sj+2

δj+2Sj+2λj+2νj

.

...λn term:

1

νj

δnSnλnνn + δn−1Sn−1δnSnλnνn + δn−2Sn−2δnSnλnνn (1 + δn−1Sn−1)

+... + δn−pSn−pδnSnλnνn∏n−1

k=n−p+1 (1 + δkSk)

+...+ δj+1Sj+1δnSnλnνn∏n−1

k=j+2 (1 + δkSk)

=

δnSnλnνnνj

n−1∏k=j+1

(1 + δkSk)

=νjn

1 + δnSn

δnSnλnνj

.

23

Therefore we rewrite

1

νj

M−1∑i=j+1

δiSi

(λiνi +

M−1∑l=i+1

δlλlνlSl

l−1∏k=i+1

(1 + δkSk)

)=

M−1∑i=j+1

νji1 + δiSi

δiSiλiνj

,

and we get the final expression:

dWTj,TMt = dW

TMt −

M−1∑i=j+1

νji1 + δiSi

δiSiλiνj

dt.

We also conclude that

dWTj,TMt = dW Tl,TM

t +

[M−1∑i=l+1

νli1 + δiSi

δiSiλiνl

−M−1∑i=j+1

νji1 + δiSi

δiSiλiνj

]dt.

Replacing in the expression defining the swap rate dynamics givesthe drift restriction in the terminal forward measure PTM and in the for-ward swap measure PTl,TM (see expressions (2.3.1) and (2.3.1)).

A.2. Libor rate dynamics


dLj

Lj

= d

(Bj −Bj+1

δj+1Bj+1

)/

(Bj −Bj+1

δj+1Bj+1

)=

d (Bj −Bj+1)

Bj −Bj+1−

dBj+1

Bj+1−

d < Bj −Bj+1, Bj+1 >

(Bj −Bj+1)Bj+1+

d < Bj+1, Bj+1 >

B2j+1

=Bj(σ

′j − σ′

j+1)

Bj −Bj+1(dWt − σj+1dt) .

Now using the fact that the Libor rates are martingale under theirown forward Libor measure, we deduce

λj =Bj(σj − σj+1)

Bj −Bj+1,

dWTj+1t = dWt − σj+1dt.

We know that

24

uj = (σj − σM) (1 + νjSjM)

= λjMνjSjM + SjM

M−1∑l=j+1

δlλlMνlSlM

l−1∏k=j+1

(1 + δkSkM)

= SjM

(λjMνj +

M−1∑l=j+1


).

This gives

σj − σM =SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1


), (8)

and finally

λj =Bj ((σj − σM)− (σj+1 − σM))

Bj −Bj+1

=

(1 + νjSjM)SjM

1+νjSjM

(λjMνj +

M−1∑l=j+1

δlSlMλlMνjl1+δlSlM

)(1 + νjSjM)− (1 + νj+1Sj+1,M)

−

(1 + νjSjM)Sj+1,M

1+νj+1Sj+1,M

(λj+1,Mνj+1 +

M−1∑l=j+2

δlSlMλlMνj+1,l1+δlSlM

)(1 + νjSjM)− (1 + νj+1Sj+1,M)

=

SjM

(λjMνj +

M−1∑l=j+1


)νjSjM − νj+1Sj+1,M

−

Sj+1,M (1 + νjSjM)

(λj+1,Mνj+1 +

M−1∑l=j+2

δlSlMλlMνj+1,l1+δlSlM

)(νjSjM − νj+1Sj+1,M) (1 + νj+1Sj+1,M)

.

From the change of measure between P Tj and P TM , we have

dWTjt − dW TM

t = − (σj − σM) dt

= −SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1


)dt,

25

and from the change of measure between P Tj and P Tk,TM , we have:

dWTjt − dW Tk,TM

t =

M∑

l=k+1

δlBlσl

M∑l=k+1

δlBl

− σj

dt

=

M∑

l=k+1

δlBl (σl − σM)

M∑l=k+1

δlBl

− (σj − σM)

dt

=

(M−1∑i=k+1

νki1 + δiSi

δiSiλiνk

−SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1


))dt.

Replacing in the expression defining the swap rate dynamics givesthe drift restriction in the terminal forward measure P TM and the swapforward measure P Tk,TM :

dLj

Lj

= −λ′jSj+1,M

1 + νj+1Sj+1,M

(λj+1,Mνj+1 +

M−1∑l=j+2


)dt

+ λ′jdW

TMt ,

and

dLj

Lj

= λ′j

(−

Sj+1,M1 + νj+1Sj+1,M

(λj+1,Mνj+1 +

M−1∑l=j+2


)+

M−1∑i=k+1

νki1 + δiSi

δiSiλiνk

)dt

+ λ′jdW

Tk,TMt .

A.3. Summary of changes of measure.

The link between the Brownian motions in the two probability mea-sures P1 and P2 will be indexed by ψt:

dWP1t = dWP2

t + ψtdt.

The measures P1 and P2 can be the risk neutral measure Q, theforward terminal measure P TM , the forward Swap measure P Tj,TM orthe forward Libor measure P Tj . According to what we have derived in

26

the previous lines, we list ψt in the following table with P1 row measureand P2 column measure.

Q P TM P Tk,TM P Tk

Q 0 σM

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

σk

P TM −σM 0

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

− σM σk − σM

P Tj ,TM −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

σM −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

−

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

σk −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

P Tj −σj σM − σj

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

− σj σk − σj

Measure changes among terminal measure, forward Swap measureand forward Libor measure only involve the difference of discount bonds’volatility, σj − σM , which can be computed from formula (8). However,measure changes between risk-neutral measure and other measures in-volve discount bond’s volatility σj. The system represented by (8) hasM unknown variables but only n − 1 equations for j = 1, ...,M − 1, sowe can not solve σj.

To solve this problem, we notice that (8) is also valid for j = 0 if weadd the present Swap rate S0M into the co-terminal Swap rate set:

σ0 − σM =S0M

1 + ν0S0M

(λ0Mν0 +

M−1∑l=1

δlSlMλlMν0l1 + δlSlM

)where σ0 = λ0M = 0 since both B0 and S0M are known. So

σM = −S0M

1 + ν0S0M

(M−1∑l=1


)and

σj =SjM

1 + νjSjM

(λjMνj +

M−1∑l=j+1


)−

S0M1 + ν0S0M

(M−1∑l=1


)(9)

Substituting (9), (4) and (5) we have the final results.

27

QQ 0

P TM S0M1+ν0S0M

(M−1∑l=1

δlSlMλlMν0l1+δlSlM

)

P Tj ,TM

M∑i=j+1

δl(1+νlSlM )

(S0M

1+ν0S0M

(M−1∑l=1


)−

SiM1+νiSiM

(λiMνi+

M−1∑l=i+1

δlSlMλlMνil1+δlSlM

))

νj

P Tj S0M1+ν0S0M

(M−1∑l=1


)−

SjM1+νjSjM

(λjMνj +

M−1∑l=j+1


)P TM

Q − S0M1+ν0S0M

(M−1∑l=1


)P TM 0

P Tj ,TM −

M∑i=j+1

SiMδl(1+νlSlM )1+νiSiM

(λiMνi+

M−1∑l=i+1


)

νj

P Tj −SjM

1+νjSjM

(λjMνj +

M−1∑l=j+1


)P Tk,TM

Q

M∑i=j+1

δl(1+νlSlM )

(−

S0M1+ν0S0M

(M−1∑l=1


)+

SiM1+νiSiM

(λiMνi+

M−1∑l=i+1


))

νj

P TM

M∑i=k+1


(λiMνi+

M−1∑l=i+1


)

νk

P Tj ,TM

M∑i=k+1


(λiMνi+

M−1∑l=i+1


)

νk−

M∑i=j+1


(λiMνi+

M−1∑l=i+1


)

νj

P Tj

M∑i=k+1


(λiMνi+

M−1∑l=i+1


)

νk−

SjM

(λjMνj+

M−1∑l=j+1

δlSlMλlMνjl

1+δlSlM

)

1+νjSjM

P Tk

Q SkM1+νkSkM

(λkMνk +

M−1∑l=k+1

δlSlMλlMνkl1+δlSlM

)− S0M

1+ν0S0M

(M−1∑l=1


)P TM SkM

1+νkSkM

(λkMνk +

M−1∑l=k+1


)

P Tj ,TM

SkM

(λkMνk+

M−1∑l=k+1


)

1+νkSkM−

M∑i=j+1


(λiMνi+

M−1∑l=i+1


)

νj

P Tj

SkM

(λkMνk+

M−1∑l=k+1


)

1+νkSkM−

SjM

(λjMνj+

M−1∑l=j+1

δlSlMλlMνjl

1+δlSlM

)

1+νjSjM

28

APPENDIX B: Co-initial swap rate market model

Denote Bj =Bj

B1and G1j =

G1jB1

. We want to express it only usingco-initial swaps rates. From

S1j =B1 −Bj

G1j

,

We have

S1j

j−1∑l=2

δlBl + (1 + δjS1j) Bj = 1 (10)

for j = 2, ...,M. Write it down in matrix form:

1 + δ2S12 0 0 · · · 0δ2S13 1 + δ3S13 0 · · · 0δ2S14 δ3S13 1 + δ4S14 · · · 0...

......

. . ....

δ2S1M δ3S1M δ4S1M · · · 1 + δMS1M

B2

B3

B4...

BM

=

111...1

Notice that the above system is a linear system with M−1 unknown

and M − 1 equation, so it has unique solution.

Bj =1

1 + δjS1j− S1j

j−1∑l=2

δl

j∏i=l

(1 + δiS1i)−1

for j = 2, ...,M.

B.1. Swap rate dynamics


dS1jS1j

= d

(B1 −Bj

G1j

)/

(B1 −Bj

G1j

)=

d (B1 −Bj)

B1 −Bj

−dG1j

G1j−

d < B1 −Bj, G1j >

(B1 −Bj)G1j+

d < G1j, G1j >

G21j

=

B1σ′

1 −Bjσ′

j

B1 −Bj

−

j∑l=2

δlBlσ′

l

j∑l=2

δlBl

dWt −

j∑l=2

δlBlσl

j∑l=2

δlBl

dt

29

Now using the fact that the swap rates are martingale under theirown forward swap measure, we deduce

λ1j =B1σ1 −BjσjB1 −Bj

−

j∑l=2

δlBlσl

j∑l=2

δlBl

(11)

dWT1,Tjt = dWt −

j∑l=2

δlBlσl

j∑l=2

δlBl

dt. (12)

From the definitions of υij, υj and the relations (6) and (7) we mayrewrite (11) as

λ1j =B1σ1 −BjσjB1 −Bj

− σ1 +

j∑l=2

δlBl (σ1 − σl)

j∑l=2

δlBl

=Bj

B1 − Bj

(σ1 − σj) +

j∑l=2

δlBl (σ1 − σl)

G1j

=1− υjS1jυjS1j

(σ1 − σj) +

j∑l=2

δl (1− υlS1l)

υj(σ1 − σl)

=

j−1∑l=2

δl (1− υlS1l)

υj(σ1 − σl) +

(1 + δjS1j) (1− υjS1j)

υjS1j(σ1 − σj) .

Define uj = (1− υjS1j) (σ1 − σj), then

S1j

j−1∑l=2

δlul + (1 + δjS1j) uj = υjS1jλ1j,

for j = 2, ...,M. Write it down in matrix form:

1 + δ2S12 0 0 · · · 0δ2S13 1 + δ3S13 0 · · · 0δ2S14 δ3S13 1 + δ4S14 · · · 0...

......

. . ....

δ2S1M δ3S1M δ4S1M · · · 1 + δMS1M

u2u3u4...

uM

=

υ2S12λ12υ3S13λ13υ4S14λ14

...υMS1Mλ1M

30

Solve the matrix equation we get

uj = λ1jυjS1j − S1j

j∑l=2

δlλ1lυlS1l

j∏k=l

(1 + δkS1k)−1

Denote PT1 the initial measure, which is associated to the discountbond B1. Easily we have

dW T1t = dWt − σ1dt

So

dWT1,Tjt − dW T1

t =

j∑l=2

δlBl (σ1 − σl)

j∑l=2

δlBl

dt

=

j∑l=2

δl (1− υlS1l) (σ1 − σl)

υj

=

j∑l=2

δlulυj

Finally

dWT1,Tjt −dW T1

t =1

υj

j∑l=2

δlS1l

(λ1lυl −

l∑i=2

δiS1iλ1iυi

l∏k=i

(1 + δkS1k)−1

)The above expression may also be rewritten differently in order to

show explicitly the dependence in the swap rate volatilities.λ2 term:

δ2S12λ12υ2

(1−

δ2S121 + δ2S12

− · · · −δjS1j

(1 + δ2S12) · · · (1 + δjS1j)

)=

δ2S12λ12υ2(1 + δ2S12) · · · (1 + δjS1j)

λ3 term:

δ3S13λ13υ3

(1−

δ3S131 + δ3S13

− · · · −δjS1j

(1 + δ3S13) · · · (1 + δjS1j)

)=

δ3S13λ13υ3(1 + δ3S13) · · · (1 + δjS1j)

31

...λj term:

δjS1jλ1jυj

(1−

δjS1j1 + δjS1j

)=

δjS1jλ1jυj1 + δjS1j

Therefore we rewrite

dWT1,Tjt − dW T1

t =1

υj

j∑l=2

δlS1lλ1lυl

j∏k=l

(1 + δkS1k)−1

=1

υj

j∑l=2

δlS1lλ1lυjl

1 + δlS1l

By substituting

υjl =l∑

i=2

δi

j∏k=i

(1 + δkS1k)−1 = υl

j∏k=l+1

(1 + δkS1k)−1

for l<j.We also conclude that

dWT1,Tjt = dW T1,Tk

t +

[1

υj

j∑l=2

δlS1lλ1lυjl

1 + δlS1l−

1

υk

k∑l=2

δlS1lλ1lυkl

1 + δlS1l

]dt

Replacing in the expression defining the swap rate dynamics givesthe drift restriction in the terminal forward measure PTM and in theforward swap measure PT1,Tk .

B.2. Libor rate dynamics.


dLj

Lj

= d

(Bj −Bj+1

δj+1Bj+1

)/

(Bj −Bj+1

δj+1Bj+1

)=

d (Bj −Bj+1)

Bj −Bj+1−

dBj+1

Bj+1−

d < Bj −Bj+1, Bj+1 >

(Bj −Bj+1)Bj+1+

d < Bj+1, Bj+1 >

B2j+1

=Bj(σ

′

j − σ′

j+1)

Bj −Bj+1(dWt − σj+1dt) .

Now using the fact that the Libor rates are martingale under theirown forward Libor measure, we deduce

32

λj =Bj(σj − σj+1)

Bj −Bj+1

dWTj+1t = dWt − σj+1dt

We know that

uj = (1− υjS1j) (σ1 − σj)

= λ1jυjS1j − S1j

j∑l=2

δlλ1lυlS1l

j∏k=l

(1 + δkS1k)−1

= S1j

(λ1jυj +

j∑l=2

δlλ1lS1lυjl1 + δlS1l

)

This gives

σ1 − σj =S1j

1− υjS1j

(λ1jυj +

j∑l=2


)(13)

and finally

λj =Bj ((σ1 − σj)− (σ1 − σj+1))

Bj −Bj+1

=

(1− υjS1j)S1j

1−υjS1j

(λ1jυj +

j∑l=2

δlλ1lS1lυjl1+δlS1l

)(1− υjS1j)− (1− υj+1S1,j+1)

−

(1− υjS1j)S1,j+1

1−υj+1S1,j+1

(λ1,j+1υj+1 +

j+1∑l=2

δlλ1lS1lυj+1,l1+δlS1l

)(1− υjS1j)− (1− υj+1S1,j+1)

=

S1j

(λ1jυj +

j∑l=2

δlλ1lS1lυjl1+δlS1l

)υj+1S1,j+1 − υjS1j

−

S1,j+1 (1− υjS1j)

(λ1,j+1υj+1 +

j+1∑l=2

δlλ1lS1lυj+1,l1+δlS1l

)(υj+1S1,j+1 − υjS1j) (1− υj+1S1,j+1)

In the change of measure between P Tj and P T1, we have:

33

dWTjt − dW T1

t = (σ1 − σj) dt

=S1j

1− υjS1j

(λ1jυj +

j∑l=2


)dt,

while in the change of measure between P Tj and P T1,Tk , we have:

dWTjt − dW T1,Tk

t =

k∑l=2

δlBlσl

k∑l=2

δlBl

− σj

dt

=

(σ1 − σj)−

k∑l=2

δlBl (σ1 − σl)

k∑l=2

δlBl

dt

=

(S1j

1− υjS1j

(λ1jυj +

j∑l=2


)−

k∑l=2

δlS1lλ1lυk

υkl1 + δlS1l

)dt.

Replacing in the expression defining the swap rate dynamics gives thedrift restriction in the initial forward measure P T1 and in the forwardswap measure P T1,Tk :

dLj

Lj

=λ′jS1j+1

1− υj+1S1j+1

(λ1,j+1υj+1 +

j+1∑l=2


)dt+ λ′jdW

T1t

and

dLj

Lj

= λ′j

(S1j+1

1− υj+1S1j+1

(λ1,j+1υj+1 +

j+1∑l=2


)−

k∑l=2

δlS1lλ1lυk

υkl1 + δlS1l

)dt

+ λ′jdWTk,TMt

B.3. Summary of changes of measure.

Changes of measure between P1 and P2 will be characterized by ψt,which links Brownian motions in two probability measures.

dWP1t = dWP2

t + ψtdt.

34

The measures P1 and P2 can be the risk neutral measure Q, theforward initial measure P T1, the forward Swap measure P Tj ,TM or theforward Libor measure P Tj . According to what we derived above, we listψt in the following table with P1 row measure and P2 column measure.

Q P T1 P T1,Tk P Tk

Q 0 σ1

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

σk

P T1 −σ1 0

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

− σ1 σk − σ1

P T1,Tj −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

σ1 −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

−

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

σk −

M∑l=j+1

δlBlσl

M∑l=j+1

δlBl

P Tj −σj σ1 − σj

M∑l=k+1

δlBlσl

M∑l=k+1

δlBl

− σj σk − σj

APPENDIX C: The sliding Swap Market Model.

4 References

Bates, D., 1996, ”Jump and Stochastic Volatility: Exchange Rate ProcessesImplicit in Deutsche Mark Options”, The Review of Financial Studies,9, 69-107.

Bjork, T., Y. Kabanov, and W. Runggaldier, 1997, Bond MarketStructure in the Presence of Market Point Processes, Mathmatical Fi-nance, 7, 211-239.

Brace, A., D. Gatarek, andM. Musiela, The Market Model of InterestRate Dynamics, Mathematical Finance, 7, 127-155 (1997).

Bremaud, P., 1981, Point Processes and Queues, Martingale Dynam-ics. New York: Springer-Verlag.

De Jong, F., J. Driessen, and A. Pelsser, 2000, Libor and Swap Mar-ket Models for the Pricing of Interest Rate Derivatives: An EmpiricalAnalysis, European Financial Review,

Duffie, D., J. Pan, and K. Singleton, Transform Analysis and AssetPricing for Affine Jump-Diffusions, Econometrica, 68, 1343-1376 (2000).

Glasserman, P., and X. Zhao, 2000, Arbitrage-free discretization oflognormal forward Libor and swap rate models, Finance and Stochastics,4, 35-68.

35

Heath, D., R. Jarrow, and A. Morton, Bond Pricing and the TermStructure of Interest Rates: a New Methodology for Contingent ClaimsValuation, Econometrica, 60, 77-105 (1992).

Heston, S. L., A Closed-Form Solution for Options with StochasticVolatility with Applications to Bond and Currency Options, The Reviewof Financial Studies, 6, 327-343 (1993).

Hull J. and White A. (1999), Forward Rate Volatilities, Swap RateVolatilities and the Implementation of the LIBOR Market Model.

Jamshidian, F., LIBOR and Swap Market Models and Measures,Finance and Stochastics, 1, 293-330 (1997).

Jamshidian, F., LIBOR Market Model with Semimartingales, Work-ing Paper, NetAnalytic Ltd., London (1999).

Protter, P., 1990, Stochastic Integration and Differential Equations.New York: Springer-Verla.

Rebonato R. (1998) ”Interest Rate Option Models”, 2nd edition,Wiley, Chichester.

36

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