markov functional interest rate models with stochastic ... · chapter 1 introduction this study is...

Markov Functional interest rate

models with stochastic volatility

New College

University of Oxford

A thesis submitted in partial fulfillment of the MSc in

Mathematical Finance

December 9, 2009

To Rahel

Acknowledgements

I would like to thank my supervisor Dr Jochen Theis for advising me throughout

the project and proof–reading of this dissertation. Furthermore I want to extend

my gratitude to d–fine GmbH for giving me the opportunity to attend the MSc

in Mathematical Finance programme. But above all I am indebted to my family,

especially to my wife Rahel, for their great support and patience.

Abstract

With respect to modelling of the (forward) interest rate term structure under

consideration of the market observed skew, stochastic volatility Libor Market

Models (LMMs) have become predominant in recent years. A powerful rep-

resentative of this class of models is Piterbarg’s forward rate term structure of

skew LMM (FL–TSS LMM). However, by construction market models are high–

dimensional which is an impediment to their efficient implementation.

The class of Markov functional models (MFMs) attempts to overcome this in-

convenience by combining the strong points of market and short rate models,

namely the exact replication of prices of calibration instruments and tractabil-

ity. This is achieved by modelling the numeraire and terminal discount bond

(and hence the entire term structure) as functions of a low–dimensional Markov

process whose probability density is known.

This study deals with the incorporation of stochastic volatility into a MFM

framework. For this sake an approximation of Piterbarg’s FL–TSS LMM is de-

vised and used as pre–model which serves as driver of the numeraire discount

bond process. As a result the term structure is expressed as functional of this

pre–model. The pre–model itself is modelled as function of a two–dimensional

Markov process which is chosen to be a time–changed brownian motion. This ap-

proach ensures that the correlation structure of Piterbarg’s FL–TSS is imposed

onto the MFM, especially the stochastic volatility component is inherited.

As part of this thesis an algorithm for the calibration of Piterbarg’s FL–TSS

LMM to the swaption market and the calibration of a two–dimensional Libor

MFM to the (digital) caplet market was implemented. Results of the obtained

skew and volatility term structure (Piterbarg parameters) and numeraire dis-

count bond functional forms are presented.

Contents

1 Introduction 1

2 A review of Libor Market and Markov Functional Models 3

2.1 The Libor Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Non–log–normal forward Libor dynamics . . . . . . . . . . . . . . . 5

2.1.2 Incorporation of stochastic volatility . . . . . . . . . . . . . . . . . . 6

2.2 Markov Functional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Definition and examples of Markov Functional Models . . . . . . . . 8

2.2.2 A Libor Markov Functional Model . . . . . . . . . . . . . . . . . . . 10

2.2.3 Multi–dimensional Markov Functional Models . . . . . . . . . . . . . 12

2.3 A Libor Market Model as pre–model for a Markov Functional Model . . . . 14

3 Piterbarg’s term structure of skew forward Libor model 17

3.1 The forward Libor dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Swap rate dynamics under the FL-TSS model . . . . . . . . . . . . . . . . . 19

3.2.1 Derivation of the forward swap volatility level . . . . . . . . . . . . . 20

3.2.2 Derivation of the forward swap skew . . . . . . . . . . . . . . . . . . 21

3.3 The effective skew and volatility formulation . . . . . . . . . . . . . . . . . 23

3.3.1 The effective forward swap skew . . . . . . . . . . . . . . . . . . . . 23

3.3.2 The effective forward swap volatility . . . . . . . . . . . . . . . . . . 26

3.4 Calibration of the FL-TSS model . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Forward rate volatility calibration . . . . . . . . . . . . . . . . . . . 31

3.4.2 Forward rate skew calibration . . . . . . . . . . . . . . . . . . . . . . 35

3.4.3 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 A Markov functional model with stochastic volatility 41

4.1 Piterbarg’s FL–TSS Libor Market Model as pre–model . . . . . . . . . . . . 42

4.2 The pre–model with two Brownian drivers . . . . . . . . . . . . . . . . . . . 43

4.3 A simplification of the pre–model process . . . . . . . . . . . . . . . . . . . 44

i

4.4 Construction of a two–dimensional Libor Markov functional model . . . . . 47

4.4.1 A two–dimensional Libor Markov functional model in the terminal

measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusion 53

A Mathematical details 55

A.1 The drift term in the Libor Market Model . . . . . . . . . . . . . . . . . . . 55

A.2 The derivative of the forward swap rate w.r.t the forward Libor rates . . . . 58

A.3 Derivation of the coefficient cmn . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.4 Proof of corollary 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.5 A recursion scheme for a system of time dependent Riccati equations . . . . 63

A.5.1 An analytic solution for Di

(t, Ti+1

). . . . . . . . . . . . . . . . . . . 64

A.5.1.1 The case gi 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 65

A.5.1.2 The case gi = 0 . . . . . . . . . . . . . . . . . . . . . . . . 66

A.5.2 An analytic solution for Ai

(t, Ti+1

). . . . . . . . . . . . . . . . . . . 66

A.5.2.1 The case gi 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 66

A.5.2.2 The case gi = 0 . . . . . . . . . . . . . . . . . . . . . . . . 68

A.5.3 Summary of the solution . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.6 Derivation of relation (3.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.7 2d–Markov functional integration . . . . . . . . . . . . . . . . . . . . . . . . 72

B The Heston Model 75

B.1 Specification of the model dynamics . . . . . . . . . . . . . . . . . . . . . . 75

B.2 The characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B.3 The solution of the Heston ODE . . . . . . . . . . . . . . . . . . . . . . . . 78

B.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.3.2 A system of Riccati ODEs . . . . . . . . . . . . . . . . . . . . . . . . 79

B.4 Option pricing by transformation techniques . . . . . . . . . . . . . . . . . . 80

B.5 Calibration of the Heston Model . . . . . . . . . . . . . . . . . . . . . . . . 86

C Tables and figures 87

Bibliography 93

ii

List of Figures

3.1 Volatility level λ10(t) and skew β10(t) of forward rate F10(t) for times T0 =

0 y ≤ t < T10 = 10 y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Proxy forward rate F10(T10) (4.11) as function of zt =(0, z2) at reset time

T10 = 10y. This corresponds to the zero correlation case, Γ(s) ≡ 0. . . . . . 47

4.2 The numeraire discount bond as functional of F10(T10, zT10): D(T10, T11; F10(T10, zT10)). 51

C.1 The numeraire discount bond as functional of F1(T1, zT1): D(T1, T11; F1(T1, zT1)). 89




iii

Chapter 1

Introduction

This study is dedicated to the incorporation of stochastic volatility into a Markov func-

tional framework. The class of Markov functional models (MFM) was introduced by Hunt,

Kennedy and Pelsser in [15]. A major motivation which lead to their development was the

desire to have models that can exactly replicate prices of liquid calibration instruments in

a similar fashion to market models while maintaining the efficiency of short rate models in

calculating derivative prices [13], [18].

Latter are formulated in terms of the short rate or instantaneous forward rate which

cannot be traded in the market. As a consequence the prices of derivatives in these models

are quite involved functions of the underlying process which is being modelled. This fact

makes it difficult to capture the most characteristic features of a derivative product with

models of this kind. However, their strong point is that the short rate process is easy to

follow and hence implemention is straightforward [13].

Unlike short rate models the class of market models is formulated in terms of market

rates which are directly related to tradable assets. Thus they exhibit better calibration

properties than short rate models. However, as these models capture the joint distribution

of market rates, they are high–dimensional by construction and tedious to implement. The

first formulation of a market model was provided by Brace, Gatarek and Musiela in the

context of forward Libors (LMM) [4]. A forward swap market model was developed by

Jamshidian in 1997 [16]. In these approaches the underlying rates are modelled as log–

normal martingales under their own probability measure.

However, the presence of a volatility skew in the caplet and swaption markets indicate

that a pure log–normal forward dynamics is not appropriate. In this respect modified

forward rate dynamics were introduced, e.g in the context of constant elasticity of variance

(CEV) and displaced diffusion models in which mixtures of pure normal and log–normal

dynamics are considered [5], [23]. Aiming at a proper modelling of the skew term structure,

stochastic volatility extensions of Libor and Swap Market models were introduced. This was

1

achieved by modelling the forward rate variance as CEV process. Approaches of this kind

are the ones by Andersen and Andreasen [2] and Piterbarg [19], [20]. Especially Piterbarg’s

stochastic volatility extension accounts for a Libor specific term structure of forward rate

skews and volatilities. In the study at hand this forward Libor term structure of skew

model (FL–TSS) is used in the construction of a Markov functional model with stochastic

volatility.

The MFM framework is based on formulating the numeraire and the terminal discount

bond as functionals of a low dimensional Markov process whose dynamics can be followed

easily. The functional forms in turn are obtained by calibration to prices of liquid derivatives

at particular dates which are relevant to the product being priced. As the discount bonds at

earlier times are obtained by applying the martingale property of numeraire rebased assets

the resulting model is arbitrage free by construction. Thus, MFMs combine the strong

points of market and short rate models.

The incorporation of stochastic volatility into a MFM is based on the concept of a

pre–model which depends on a low–dimensional Markov process [14],[17]. By regarding

the forward Libors and hence the discount bond processes as functions of the pre–model

process, the calibration can be formulated in terms of the latter. Thus the correlation

structure of the pre–model is incorporated into the MFM. In this study a pre–model is

selected which is an approximation of a calibrated Piterbarg FL–TSS model and depends

on a two–dimensional brownian motion. This proxy is then employed in constructing a Libor

MFM which inherits the stochastic volatility structure of Piterbarg’s FL–TSS. Tractability

is maintained since calibration involves the integration of the known probability distribution

of the two–dimensional Markov process.

The thesis is structured as follows: In chapter 2 the concepts of LMMs and MFMs are

reviewed. Working under the terminal (forward) measure drift terms for a stochastic volatil-

ity LMM are derived. Furthermore multi–dimensional extensions of MFMs are discussed.

Chapter 3 is dedicated to the detailed study of Piterbargs FL–TSS LMM. This model is

calibrated to the swaption market, and the resulting term structure of skews and volatilites

is presented. The incorporation of stochastic volatility into a MFM is the topic of chap-

ter 4. Here a two–dimensional Libor MFM is constructed which uses an approximation of

Piterbarg’s FL–TSS as pre–model. The model is calibrated to the (digital) caplet market,

and resulting numeraire discount bond funtionals are presented. The thesis concludes with

chapter 5. Besides result tables and figures, the appendices contain mathematical details

and a thorough presentation of the Heston Model.

2

Chapter 2

A review of Libor Market and

Markov Functional Models

In this chapter we review the class of Libor Market and Markov functional models which

have become prevalent in the last ten to fifteen years. In section 2.1 we discuss the LMM

under consideration of non–log–normal forward rate dynamics and stochastic volatility.

In particular, working under the terminal (forward) measure forward rate drift terms are

derived. The Markov functional framework is introduced in section 2.2. Therein multi–

dimensional extensions are discussed as well. In section 2.3 the idea of constructing MFMs

in terms of a pre–model is introduced.

2.1 The Libor Market Model

As already mentioned in the introduction the Libor Market Model focusses on modelling

the dynamics of forward Libor rates Fi

(t; Ti, Ti+1

)which reset at times Ti, i = 1, . . . , N.

In a deterministic volatility setting employing K independent Brownian drivers these are

modelled as log–normal variables with respect to their martingale measure Qi+1 which is

induced by taking the discount bond D(t, Ti+1

)as numeraire,

dFi

(t; Ti, Ti+1

)= λi(t)Fi

(t; Ti, Ti+1

) K∑

k=1

σi,k(t) dW i+1k (t) = λi(t)Fi

(t; Ti, Ti+1

)σi(t)

TdWi+1t ,

for times T0 ≤ t < Ti ≤ TN where dWi+1t is a K–dimensional vector of orthogonal Brow-

nian motions under Qi+1 and λi(t) are positive continuous, real valued functions. In

particular the relation⟨dWk, dWl

⟩= δkl dt holds. The K–dimensional vector σi(t) con-

tains the load factors of the orthogonal brownian motions onto forward rate Fi which

satisfy∑K

k=1 σi,k(t)2 = 1. Thus they define a correlation matrix through the relation

ρij(t) = σi(t)σj(t)T = ρiji=1,...,K

j=1,...,K

.1 Indeed, in this formulation the covariance of for-

1We assume that K = N, i.e., that each forward rate has its own driving brownian motion.

3

ward Libor yields is given by

⟨dFi(t)

Fi(t),dFj(t)

Fj(t)

⟩= λi(t)λj(t)

K∑

k,l=1

σi,k(t)σj,l(t)⟨dW i+1

k (t), dW i+1l (t)

⟩︸︷︷︸

δkl dt

= λi(t)λj(t)

( K∑

k=1

σi,k(t)σj,k(t)

︸︷︷︸=ρij(t)

)dt = λi(t)λj(t)ρij(t) dt,

from where it becomes apparent that the brownian correlations ρij(t) as well as the volatil-

ities λi(t) contribute to the forward rate correlation. Since the forward Libors follow a

log–normal process in their own martingale measure caplet prices are given by the Black76

formula, comp. [3]. Therefore the volatilities λi(t) can be obtained by calibration to quoted

caplet prices. For this a widespread approach is to assume a parametric shape for the

volatilities as function of time to expiry which correctly captures their dynamics observed

in the market. The parameters are then determined in the course of the calibration process,

comp. [21]. With respect to the correlation function the most reasonable approach is to

model it in parametric form as well. The reason for this is that it is not easy at all to

extract information on the instantaneous correlation ρij(t) out of quoted derivative prices,

e.g., swaption volatilities, because the latter depend on the history of λi(t)λj(t)ρij(t) on

the time interval which starts at T0 and ends on swaption expiry. The parametric form

proposed by Rebonato in [21] and [22] is given by

ρij(t) = ρ∞ + (1 − ρ∞) exp(−δ

∣∣(Ti − t)ǫ − (Tj − t)ǫ∣∣), (2.1)

with constants −1 ≤ ρ∞ ≤ 1, δ, ǫ > 0, and will be adopted in what follows.

Of course, to use the LMM in practice the dynamics of all forward Libor rates have to be

formulated in a single measure. In this respect a convenient choice is the terminal measure

QN+1 which is induced by taking the terminal discount bond D(t, TN+1

)as numeraire. As a

consequence only the forward Libor FN is a martingale, and according to Girsanov’s theorem

all other forward Libors will be modified by additional drift terms, comp. [13]. Indeed, when

changing from the terminal measure QN+1 to the martingale measure of forward rate Fi

the process dWN+1k (t) + µN+1

i (t) dt also is a brownian motion under Qi+1. Hence in the

terminal measure the forward rate processes become

dFi

(t; Ti, Ti+1

)= λi(t)Fi

(t; Ti, Ti+1

)σi(t)

T[µN+1

i (t) dt + dWN+1t

],

= Fi

(t; Ti, Ti+1

)λi(t)

[µN+1

i (t) dt + σi(t)TdWN+1

t

], (2.2)

µN+1i (t) = −

N∑

l=i+1

αl λl(t) ρil(t)Fl

1 + αlFl(t), µN+1

N (t) = 0,

, T0 ≤ t < Ti ≤ TN , i = 1, . . . , N

4

where the drifts µN+1i (t) are given by (A.10) which is derived in appendix A.1.

2.1.1 Non–log–normal forward Libor dynamics

The forward Libor process (2.2) presented above models the forward Libors Fi as a log–

normal processes with respect to their martingale measure. In the terminal measure a

log–normal behaviour is maintained for forward Libor FN . However, the caplet/floorlet

market displays a volatility surface (in terms of terms of implied Black volatilities), i.e.,

the volatility varies as option expiries and moneyness changes. Specifically, the observed

volatilities are monotone decreasing functions of the forward Libor level, a behaviour which

is denoted as volatility skew. Its presence indicates that the forward Libors do not follow a

log–normal process, for in that case the implied Black volatilities should be constant. One

proposal for an alternate forward rate dynamics provided by Rubinstein [23] is a displaced

diffusion which combines a log–normal and a normal process. Another approach is to model

the forward Libors as constant elasticity of variance (CEV) processes which was proposed

by Cox and Ross [5]. In both cases the forward Libor change can be written in terms of

a volatility function ϕ(Fi

)which imposes a rate level dependence onto the forward rate

volatilities, i.e the forward Libors are modelled as

dFi

(t; Ti, Ti+1

)= ϕ

(Fi

)λi(t)σi(t)

TdWi+1t , i = 1, . . . , N.

For example a displaced diffusion model is established with a function of the form

ϕ(Fi

(t; Ti, Ti+1

))= β Fi

(t; Ti, Ti+1

)+

(1 − β

)Fi

(T0; Ti, Ti+1

),

where the displacement parameter β is a real valued constant. Obviously the case β = 1

corresponds to a log–normal dynamics. For β = 0 a normal process is recovered. A

generalization of this with time dependent paramter β(t) will be considered in the next

chapter where Piterbarg’s forward Libor term structure of skew model (FL–TSS) will be

presented.

A CEV–model is obtained by defining

ϕ(Fi

(t; Ti, Ti+1

))= Fi

(t; Ti, Ti+1

)β, (2.3)

with 0 ≤ β ≤ 1. As with the displaced diffusion model the case β = 1 corresponds to

log–normal dynamics whereas β = 0 results in a normal model. In this model the yield or

percentage volatility, i.e., the volatility of dFi

Fiis given by ϕ(Fi)

Fiλi(t) = λi(t)Fi

(t; Ti, Ti+1

)β−1

which for 0 < β < 1 is a monotone decreasing function of forward Libor Fi in accordance

with the observed market behaviour. A model with this kind of local volatility function was

first introduced by Dupire in modelling equities, comp. [7].

5

Based on these ideas a generalised forward Libor process can be formulated in the

terminal measure QN+1,

dFi

(t; Ti, Ti+1

)= ϕ

(Fi

)λi(t)σi(t)

T[µN+1

i (t) dt + dWN+1t

],

= ϕ(Fi

)λi(t)

[µN+1


t

], (2.4)

µN+1i (t) = −

N∑

l=i+1

αl λl(t) ρil(t)ϕ(Fl

)

1 + αlFl(t), µN+1

N (t) = 0,

T0 ≤ t < Ti ≤ TN , i = 1, . . . , N

which was proposed by Andersen and Andreasen, comp. [1]. The drift terms are given by

equation (A.9) which is derived in appendix A.1.

2.1.2 Incorporation of stochastic volatility

As the market observed volatility skews cannot be solely captured by the introduction of a

(local) volatility function stochastic volatility extensions of the LMM were devised. In this

respect one approach is to extend the forward Libor process (2.4) with a stochastic variable

which accounts for the volatility level and as such modulates the local volatility function

ϕ(Fi). A convenient choice is the square root of a variance process Σt which follows a

one–dimensional CEV–process, comp. [1].

Assuming that the brownian driver of the variance process is correlated with each for-

ward rate driver, i.e.,⟨dV N+1(t), dWN+1

k (t)⟩

= Γk(t) dt (k = 1, . . . , N), the process (2.4)

can be generalised to

dFi

(t; Ti, Ti+1

)= ϕ

(Fi

)λi(t)

√Σt

[µN+1


t

]

= ϕ(Fi

)λi(t)

√Σt

[µN+1

i (t) dt + σi(t)TΩ(t)T dZN+1

t + σi(t)TΓ(t) dV N+1

t

],

(2.5a)

dΣt = Θ(Σ0 − Σt

)dt + η

√Σt dV N+1

t , (2.5b)

with⟨dV N+1(t), dWN+1

k (t)⟩

= Γk(t) dt, < dV N+1(t), dZN+1k (t)

⟩,

and T0 ≤ t < Ti ≤ TN , i = 1, . . . , N, k = 1, . . . , N,

where the K–dimensional vector of brownian drivers dWN+1t was decomposed into orthog-

onal components dZN+1t and dV N+1

t according to (A.3). The drift terms are given by

6

equation (A.8) which is derived in appendix A.1:

µN+1i (t) = −

√Σt

N∑

l=i+1

αl λl(t)ϕ(Fl

)

1 + αlFl(t)

×[(

σl(t)TΩ(t)T

)(Ω(t)σi(t)

)+

(σl(t)Γ(t)

)(σi(t)Γ(t)

)], (2.5c)

µN+1N (t) = 0,

with matrix Ω(t) =√

1 − Γk(t)2 δkj

k=1,...,Kj=1,...,K

and vector Γ(t) = Γk(t)k=1,...,K .

Of course, the introduction of an additional Brownian driver increases the dimensionality

of the model, and the additional correlation coefficients Γk(t) enlarge the parameter space.

However, for the stochastic volatility model we will work with in the following chapters,

namely Piterbarg’s FL–TSS LMM, the rate and variance processes are independent. Thus

Γk(t) = 0 and the drift reduces to

µN+1i (t) = −

√Σt

N∑

l=i+1


)

1 + αlFl(t), µN+1

N (t) = 0. (2.6)

Nevertheless the process (2.5) will be referenced in section 2.3 where the idea of a pre–model

in a Markov functional context is discussed.

2.2 Markov Functional Models

The class of Markov functional interest rate models was originally introduced by Hunt,

Kennedy and Pelsser in [15]. A major motivation which lead to their development was the

desire to have models that can fit observed prices of liquid instruments in a similar fashion

to the market models while maintaining the efficiency of short rate models in calculating

derivative prices, comp. [13], [18]. This is achieved by specifying a low dimensional process

which is Markovian in some martingale measure and formulating pure discount bond prices

as functions of this process. Since efficient algorithms to compute conditional distribution

functions are known for this set up, the valuation of derivatives in a Markov functional

framework is much more efficient when compared to pricing using market models. Although

market models are Markovian as well, they are naturally of high dimension. Moreover an

essential feature of these types of models is the freedom to choose the functional form

of the discount bond prices in such a way that market prices of calbration instruments

are replicated. This distinguishes Markov functional from short rate models in which the

functional form of discount bond prices with respect to the Markovian short rate is fixed.

Therefore Markov functional models combine the strong points of market as well as short

rate models, namely the fitting to observed prices of liquid instruments and tractability.

7

2.2.1 Definition and examples of Markov Functional Models

In this section we want to give a formal definition of Markov functional models and also

present some examples. We begin by citing the definition given by Hunt and Kennedy:

Definition 2.2.1 (Hunt and Kennedy [13]). An interest rate model is said to be Markov–

functional if there exists some numeraire pair (N, N) and some process x sucht that:

1. the process x is a (time–inhomogeneous) Markov process under the measure N;

2. the pure discount bond prices are of the form

DtS = DtS(xt), T0 ≤ t ≤ ∂S ≤ S,

for some boundary curve ∂S : [0, ∂∗] → [0, ∂∗] and some constant ∂∗;

3. the numeraire N, itself a price process, is of the form

Nt = Nt(xt) T0 ≤ t ≤ ∂∗.

Obviously the boundary curve ∂S is introduced so that the model does not need to be

defined over the entire time domain 0 ≤ t ≤ S. The most common choice for the boundary

curve is

∂S =

S, if S ≤ T

T, if S > T,

for some constant T.

Thus the main ingredients of a Markov functional model are the driving Markov process

xt which describes the state of the economy and the functional forms of

1. the discount bond D∂SS

(x∂S

)≡ D

(∂S , S; x∂S

)on the boundary curve ∂S ;

2. the numeraire Nt(xt) for times T0 ≤ t ≤ ∂∗.

The reason for this is that the functional forms of bonds at earlier times t < ∂S are deter-

mined by the functional form of the discount bond on the boundary curve by the martingale

property of numeraire rebased assets,

D(t, T ; xt

)= Nt(xt) EN

[D(∂S , T ; x∂S

)

N∂S(x∂S

)

∣∣∣∣Ft

], T0 ≤ t ≤ ∂S ≤ T, (2.7)

comp. [10].

One particular choice of measure is the (terminal) forward measure N = QN+1 which is

induced by taking the discount bond D(t, TN+1

)as numeraire. Taking TN+1 as boundary

we thus have

∂S = min(S, TN+1

), Nt(xt) = D

(t, TN+1; xt

), T0 ≤ t ≤ TN+1,

8

and the price of a discount bond maturing at S ≤ TN+1 becomes

D(t, S; xt

)= D

(t, TN+1; xt

)EQN+1

[D(S, S; xS)

D(S, TN+1; xS

)∣∣∣∣xt

]

= D(t, TN+1; xt

)EQN+1

[1

D(S, TN+1; xS

)∣∣∣∣xt

], T0 ≤ t ≤ S ≤ TN+1, (2.8)

where the expectation is conditioned on xt because of the Markov property of the underlying

process2.

If one considers interest rate derivatives like caplets/floorlets or swaptions expiring at

time Tm with strike K their payoff function Vm(Tm, K) depends on the discount bonds

D(Tm, Tj ; xt), with j = m + 1 for caplets/floorlets and j > m in the case of swaptions, and

by (2.8) is a function of the numeraire discount bond D(Tm, TN+1; xTm) at time Tm. Thus

Vm(Tm, K) ≡ Vm

(Tm, K, D(Tm, TN+1; xTm)

)and by application of the fundamental theorem

of asset pricing the derivative value at time t is given by

Vm

(t, K; xt

)= D

(t, TN+1; xt

)EQN+1

[Vm


)

D(Tm, TN+1; xTm

)∣∣∣∣xt

]. (2.9)

Hence, if the Markov process xt and thus the conditional probability distribution p(xTm |xt)

is specified, this relation provides a means to extract the functional form of the numeraire

discount bond D(Tm, TN+1; xTm) at time Tm from market observed derivative prices since

the payoff function Vm


)is known. However, in order to proceed

along these lines one has to assume that the discount bonds are monotone functions of the

underlying Markov process.

It has to be emphasized that due to the functional dependence the specified underlying

Markov process xt determines the probability distribution of discount bonds. As the prices

of multi–temporal interest derivatives depend on the joint probability distribution of forward

rates (and thus on the joint distribution of discount bonds) at those times relevant to the

product at hand3, the driving Markov process encodes all information on the correlation

structure. So in designing a Markov functional model for a specific product class the process

xt has to be chosen in such a way as to capture the characteristic product features while

retaining low dimensionality. Referring to the process dimension, the underlying Markov

process should not have more than two brownian drivers. An example of a simple one

dimensional underlying process is considered in the next section where a Libor Markov

functional model is explored.

2For a Markov process xt the relation E[f(xt)

∣∣Ft

]= E

[f(xt)

∣∣xt

]holds.

3E.g., for a bermudan swaption the single exercise dates are the relevant times. Therefore the probabilitydistribution of the process xt only needs to be known on these dates.

9

2.2.2 A Libor Markov Functional Model

We know consider the set of forward Libor rates Fi

(t; Ti, Ti+1

)which reset a times Ti,

i = 1, . . . , N, and specify a Markov functional model. For this we choose the final time

∂S = TN+1 and work in the measure N = QN+1 which is induced by taking the discount

bond D(t, TN+1

)as numeraire.

The underlying Markov process xt is chosen to be a time changed brownian motion.

Assuming that σ(t) is a deterministic, positive real valued function on [T0, TN+1] we define

xt :=

∫ t

T0

σ(s) dWs,

where dWs is a brownian motion under QN+1. Clearly, due to the Markovian character of

the brownian motion xt is a Markov process whose conditional probability distribution is

normal,

p(xs

∣∣xt

)=

1((2π)2

∫ s

tσ(u)2 du

) 12

exp

(−1

2

(xs − xt)2

∫ s

tσ(u)2 du

), s ≥ t. (2.10)

Having specified the underlying process the functional form of the numeraire discount

bond D(t, TN+1; xt

)remains to be determined. This will be done at discrete times Ti (i =

1, . . . , N) according to a recursion scheme in which the functional from will be determined

by calibration to digital caplet prices.

To start with we observe that at time TN we observe that the forward rate FN

(t; TN , TN+1; xTN

)

is a log–normal martingale under QN+1. Thus its dynamics is governed by

dFN

(t; TN , TN+1

)= σ(t)FN

(t; TN , TN+1

)dWt, T0 ≤ t ≤ TN ,

where dWt is a brownian motion under QN+1, which integrates to

FN

(TN ; TN , TN+1; xTN

)= FN

(T0; TN , TN+1

)exp

(−1

2

∫ TN

T0

σ(s)2 ds +

∫ TN

T0

σ(s) dWs

)

= FN

(T0; TN , TN+1

)exp

(−1

2

∫ TN

T0

σ(s)2 ds + xTN

).

With this relation at hand the functional form of forward rate FN at time TN is known.

Because forward rates and discount bond prices are related by

Fm

(Tm; Tm, Tm+1; xTm

)=

1 − D(Tm, Tm+1; xTm)

αm(Tm, Tm+1)D(Tm, Tm+1; xTm), 1 ≤ m ≤ N,

the functional form of D(TN , TN+1; xTN

)unfolds itself as monotone decreasing function of

xTN,

D(TN , TN+1; xTN

)=

1

1 + αN(TN , TN+1)FN

(T0; TN , TN+1

)exp

(−1

2

∫ TN

T0σ(s)2 ds + xTN

) .

(2.11)

10

This result serves as the basis for the recursive calculation of functional forms at earlier

times Tm < TN which will be extracted from market observed digital caplet prices. The

payoff of a digital caplet expiring at time Tm with strike K is given by

Vm

(Tm, K; xTm

)= D

(Tm, Tm+1; xTm

)1Fm(Tm;Tm,Tm+1;xTm )>K ,

and at time T0 < Tm < TN the numeraire rebased derivative value is therefore

Vm

(T0, K; xT0

)

D(T0, TN+1; xT0

) = EQN+1

[Vm


)

D(Tm, TN+1; xTm

)∣∣∣∣xT0

]

= EQN+1

[D

(Tm, Tm+1; xTm

)

D(Tm, TN+1; xTm

) 1Fm(Tm;xTm )>K

∣∣∣∣xT0

]

= EQN+1

[EQN+1

[1

D(Tm+1, TN+1; xTm+1

)∣∣∣∣xTm

]1Fm(Tm;xTm)>K

∣∣∣∣xT0

],

(2.12)

where in the last line the martingale property of numeraire rebased discount bonds (2.8) was

used. Thus if the functional form of the numeraire bond D(Tm+1, TN+1; xTm+1

)is known at

time Tm+1, the expected value on the right hand side can be calculated (2.12) because the

conditional probability distribution of xt is known. Indeed, since the forward rate Fm is a

monotone function of xTm through its dependence on the discount bond D(Tm, Tm+1; xTm

),

there exists a unique value x∗Tm

for which the forward rate matches the digital caplet strike

value K, Fm

(t; Tm, Tm+1; x

∗Tm

)= K 4. Thus (2.12) is equivalent to

Vm

(T0, K; xT0

)

D(T0, TN+1; xT0

) =

∫ ∞

x∗Tm

[∫ ∞

−∞

1

D(Tm+1, TN+1; xTm+1

) p(xTm+1

∣∣xTm

)dxTm+1

]p(xTm

∣∣xT0

)dxT0

,

(2.13)

which provides a relation between x∗Tm

and market derived derivative values. Indeed, be-

cause D(T0, TN+1; xTi

)and Vm

(T0, K; xTi

)can be observed in the market at time T0, the

left hand side of (2.13) is known. With the known functional form D(Tm+1, TN+1; xTm+1

)

the intergrals on the right hand side can be calculated numerically for varying values of

x∗Tm

. That value of x∗Tm

for which the left and right hand side of (2.13) are equal is the

desired target value which satisfies the relation Fm

(Tm; Tm, Tm+1; x

∗Tm

)= K.5 Conducting

this matching procedure for a series of options with different strike values Kj (j = 1, . . .M)

and values Vm

(T0, Kj ; xT0

)results in a set

x∗

Tm,j

∣∣Fm

(Tm; Tm, Tm+1; x

∗Tm,j

)= Kj

=

x∗

Tm,j

∣∣∣∣D(Tm, TN+1; x

∗Tm,j

)=

[(1 + αN(TN , TN+1)Kj

)D(Tm, Tm+1; x

∗Tm,j

)

D(Tm, TN+1; x∗

Tm,j

)]−1

,

(2.14)

4As mentioned above we assume that the discount bonds are monotone functions of xt. Due to theirrelation this behaviour transfers to the forward Libor rate.

5In practice the solution for x∗Tm

is found by a numerical root finding method, e.g., the Brent algorithm.

11

where in the second line the relation

Kj = Fm

(Tm; Tm, Tm+1; x

∗Tm,j

)=

1 − D(Tm, Tm+1; x∗Tm,j)

αm(Tm, Tm+1)D(Tm, Tm+1; x∗Tm,j)

=

1

D(Tm,TN+1;x∗

Tm,j

) − D(Tm,Tm+1;x∗Tm,j)

D(Tm,TN+1;x∗

Tm,j

)

αm(Tm, Tm+1)D(Tm,Tm+1;x∗

Tm,j)

D(Tm,TN+1;x∗

Tm,j

), 1 ≤ m ≤ N,

was used. BecauseD(Tm,Tm+1;x∗

Tm,j)

D(Tm,TN+1;x∗

Tm,j

) corresponds to the inner integral (the bracket term)

of equation (2.13) its value has already been determined in the course of finding x∗Tm,j.

Therefore the set identity (2.14) defines the numeraire discount bond at time Tm as function

of x∗Tm,j. Obviously, because only a finite number of options with different strikes can be

observed in the market the sets in (2.14) are discrete. Therefore continuous functional forms

have to be obtained by interpolation between the single set elements.

Following the above reasoning the recursion scheme starts at time TN−1. Performing

the calibration according to (2.13) the expected value of the inverse of the already known

functional D(TN , TN+1; xTN

)given by (2.11) is calculated. As a result the set (2.14) and

thus the functional forms D(TN−1, TN+1; xTN

)at time TN−1 are obtained. These in turn

serve as input for the calibration at time TN−2 where they enter the inner integral of (2.13).

Pursueing the recursion along these lines until time T1 the functional forms of the numeraire

discount bond D(Ti, TN+1; xTi

)are established for times TN , TN−1, . . . , T1.

2.2.3 Multi–dimensional Markov Functional Models

In the previous section we presented the Libor Markov functional model in which a one–

dimensional Markov process was used as underlying for the discount bond term structure.

However it is also possible to consider higher dimensional underlying processes. As men-

tioned in the previous section the driving process should be selected in such a way that the

essential features of the derivative product for which the model is designed are captured.

As an example for which a two dimensional driving process is required the class of spread

options can be considered. The payout structure of this kind of derivatives can depend on

the level of two different rate types which follow individual dynamics. Therefore a two–

dimensional process is required in order to model the separate rate components. Another

example is the incorporation of stochastic volatility which we focus on in this thesis. In

this context a two dimensional Markov process would encompass a rate and a volatility

component.

As discussed above the specification of the Markov process is only one part in the specif-

cation of a Markov functional model. The second is the determination of the functional

12

forms for the numeraire and the discount bond on the boundary curve. These are ob-

tained by a calibration procedure for which it is essential that the functionals are monotone

functions of the driving Markov process. But when higher dimensional Markov processes

zt are considered the monotonicity of functionals can no longer be maintained because

e.g., in a two dimensional extension of the matching equation (2.13) more than one tuple

zt = (zt,1, zt,2) would be obtained as target value. For an n–dimensional Markov process zt

this problem can be overcome by the introduction of a function

π : R × Rn −→ R, (t, zt) 7−→ π(t, zt) =: xt,

which serves as projector to the one dimensional real axis, comp. [13]. In general, the process

xt = π(t, zt) thereby defined will not be Markovian. However this fact poses no impediment

since the function π merely serves as a means to facilitate the calibration to derivative

prices observed in the market. Following this approach the numeraire discount bond at

time Tm becomes a functional of the multi–dimensional Markov process, D(Tm, TN+1; zTm

),

and in the calibration procedure the expected values are calulated with respect to the

conditional distribution of zt. Therefore the only modification which needs to be applied to

the calibration equation (2.12) is the change from one– to multi–dimensional integrals:

Vm

(T0, K; xT0

= π(T0, zT0))

D(T0, TN+1; xT0

= π(T0, zT0))

= EQN+1

[Vm


)

D(Tm, TN+1; xTm

)∣∣∣∣xT0

]

= EQN+1

[EQN+1

[1

D(Tm+1, TN+1; xTm+1

)∣∣∣∣xTm

]1Fm(Tm;xTm )>K

∣∣∣∣xT0

]

= EQN+1

[EQN+1

[1

D(Tm+1, TN+1; xTm+1

)∣∣∣∣xTm

]1xTm>x∗

Tm

∣∣∣∣xT0

]

= EQN+1

[EQN+1

[1

D(Tm+1, TN+1; zTm+1

)∣∣∣∣zTm

]1π(Tm,zTm

)>x∗Tm

∣∣∣∣zT0

]

=

∫ ∞

x∗Tm

=π(Tm,z∗Tm)

[∫ ∞

−∞

1

D(Tm+1, TN+1; zTm+1

) p(zTm+1

∣∣zTm

)dzTm+1

]p(zTm

∣∣zT0

)dzT0

,

(2.15)

with x∗Tm

= F−1m (Tm; Tm, Tm+1; K). The variable z∗Tm

describes a curve in the n–dimensional

state space for which x∗Tm

= π(Tm, z∗Tm) at time Tm.

Thinking about possible choices for the function π(t, zt) the concept of a pre–model was

devised [14]. It is based on the idea that π can be defined as approximation to a model

which has already been calibrated to market prices, e.g., a Libor Market Model where the

drift terms have been frozen to their initial values. Since the so defined pre–model is an

13

approximation only it is not arbitrage free. However, since the no–arbitrage requirement is

inherent in equation (2.15) a calibration to market quotes via the pre–model will result in

an arbitrage free model.

2.3 A Libor Market Model as pre–model for a Markov Func-

tional Model

In this section an approximation of a forward Libor Market Model (LMM) will be considered

which will then be used as pre–model for a Markov functional model. Working in the

terminal measure QN+1 the forward rate Fi

(t; Ti, Ti+1

)is modelled according to (2.4),

Fi

(t; Ti, Ti+1

)= Fi

(T0; Ti, Ti+1

)

× exp

(∫ t

T0

[µN+1

i (s) − 1

2λi(s)

2|σi(s)|2]

ds +

∫ t

T0

λi(s)σi(s) dW s

),

(2.16a)

µN+1i (t) = −

K∑

l=i+1

αl(Tl, Tl+1)λl(t)ρli(t)Fl

(t; Tl, Tl+1

)

1 + αl(Tl, Tl+1)Fl

(t; Tl, Tl+1

) , (2.16b)

T0 ≤ t ≤ Ti, i = 1, . . . , N, 1 ≤ K ≤ N,

where dW s is a K–dimensional Brownian motion under QN+1 and σi(s) the load vector

which encodes the effect of the individual orthogonal Brownian drivers on forward rate Fi.

From this dynamics it is obvious that the change of forward rate Fi depends on the state

of all forward rates Fl (l > i) at time t which is why the individual processes Fi(t) are not

Markovian6. Therefore one usually resorts to Monte–Carlo methods in numerical evalua-

tions which gets quite expensive as the number of factors increases. However, computations

can be alleviated by referring to an approximation proposed by Rebonato due to which the

forward rates Fl

(t; Tl, Tl+1

)in the drift term (2.16b) are replaced by their time T0 values

Fl

(T0; Tl, Tl+1

), a process which is also denoted as partial freezing [22]. A further simplifi-

cation can be achieved by replacing the Brownian motion terms with normally distributed

variables which exhibit the same mean and variance. Employing these ideas the forward

rate vector can be approximated as

F(t) ≈ F(t) = F(T0) exp(µ0

N+1(t) + M(t) · zt

),

6However, the process for the entire forward rate vector F(t) is Markovian since its change at time t onlydepends on the state of the forward rate vector at time t.

14

with vectors

µ0

N+1(t) =

∫ t

T0

[µ0

N+1i (s) − 1

2λi(s)

2∣∣σi(s)

∣∣2]ds

i=1,...,N

,

where µ0N+1i (t) = −

K∑

l=i+1


(T0; Tl, Tl+1

)

1 + αl(Tl, Tl+1)Fl

(T0; Tl, Tl+1

) ,

zt =Zt,i

i=1,...,K

, where Zt,i ∼ N(0, t − T0

),

and the matrix

M(t) =

[1

t − T0

∫ t

T0

λi(s)2σik(s)

2 ds

] 12

i=1,...,Nk=1,...,K

.

Indeed, the terms Mik(t)Zt,k satisfy

E

[ K∑

k=1

Mik(t)Zt,k

]=

K∑

k=1

Mik(t)E[Zt,k

]= 0 = E

[ K∑

k=1

∫ t

T0

λi(s)σik(s) dWk(s)

],

= E

[∫ t

T0

λi(s)σi(s) dW s

]

because

∫ t

T0

λi(s)σik(s) dWk(s) are Ito integrals,

and var

[ K∑

k=1

Mik(t)Zt,k

]= E

[ K∑

k,l=1

Mik(t)Mil(t)Zt,kZt,l

]=

K∑

k,l=1

Mik(t)Mil(t)E[Zt,kZt,l

]

=K∑

k,l=1

Mik(t)Mil(t) cov[Zt,kZt,l

]︸︷︷︸

=δkl(t−T0)

=K∑

k=1

∫ t

T0

λi(s)2σik(s)

2 ds

=K∑

k=1

E

[∫ t

T0

λi(s)2σik(s)

2 ds

]

=K∑

k=1

E

[(∫ t

T0


)2]by Ito’s isometry

=

K∑

k,l=1

∫ t

T0

λi(s)2σik(s)σil(s) E

[dWk(s)dWl(s)

]︸︷︷︸=cov[dWk(s),dWl(s)]

= var

[ K∑

k=1

∫ t

T0


]

= var

[∫ t

T0

λi(s)σi(s) dW s

], i = 1, . . . , N. (2.17)

Above proxy processes can be related to the original forward rates Fi(t) by introducing

monotone functions gi which act as perturbations on Fi(t, zt). Thus Fi(t) = gi

(Fi(t, zt)

),

15

and one can define projector functions πi := gi Fi by

πi : R × RK −→ R, (t, zt) 7−→ πi(t, zt) := gi

(F i(t, zt)),

with F i(t, zt) = Fi(T0) exp

(µN+1

i (t) +

K∑

k=1

Mik(t)zt,k

), for i = 1, . . . , N,

which were introduced in the previous section. Hence the numeraire discount bond becomes

a functional of the Markov process zTm through its dependence on Fm(Tm) = πm(Tm, zTm) =

gm

(Fm(Tm, zTm)

).

Based on this approach the incorporation of stochastic volatility into a Markov functional

framework will be devised in chapter 4.

16

Chapter 3

Piterbarg’s term structure of skew

forward Libor model

This chapter is dedicated to a survey of Piterbarg’s term structure of skew forward Libor

model (FL–TSS) which was introduced in [19]. As with other forward Libor Market Models

the main motivation is to capture the dynamics of the joint distribution of forward Libors

throughout time. To facilitate this the forward Libor dynamics has to be flexible enough

to capture information on the marginal distributions which is encoded in caplet and/or

swaption prices. Since for these products the market implied volatilities exhibit a skew, i.e.,

the Black implied volatilities appear to be functions of the option strikes, Piterbarg considers

a weighted sum of log–normal and normal dynamics for the forward Libors. Mathematically

this is expressed by the introduction of a skew parameter. Reference to the swaption market

necessitates this parameter to be time–dependent in order to reproduce swaption skews

across expiries and underlying swap maturities. It is this time–dependent skew parameter

which renders the model more flexible over earlier (constant) skew models, e.g., Andersen

and Andreasen [2]. Furthermore, to account for the market observed variability of volatility

levels, similar to the formulation of [2] these are modelled as stochastic CEV processes

whose Brownian components are assumed to be uncorrelated with the stochastic drivers

of the forward rate processes. Hence the FL-TSS model belongs to the class of stochastic

volatility models.

3.1 The forward Libor dynamics

Following the qualitative description of Piterbarg’s model we know set out to specify the

dynamcis of forward Libors Fi

(t; Ti, Ti+1

)resetting at times Ti, i = 1, . . . , N. Working in

the measure QN+1 induced by choosing the terminal discount bond D(t, TN+1

)as numeraire

the forward rates are modelled as

17

dFi(t) =(βi(t)Fi(t) +

(1 − βi

(T0

))Fi

(T0

))λi(t)

√Σt

[µN+1

i (t) dt +K∑

l=1

σi,l(t)dWN+1l (t)

],

(3.1a)


)dt + η

√Σt dV N+1

t , (3.1b)

for times T0 ≤ t < Ti ≤ TN

(i = 1, . . . , N

). All forward rates are driven by K independent

Brownian motions dWN+1l (t) which are assumed to be uncorrelated with the stochastic

volatility driver dV N+1t and therefore

⟨dV N+1

t , dWN+1l (t)

⟩= 0. Their influence on the

forward Libors Fi is mediated by the load factors σi,l(t) which contain information on

forward Libor correlation since the relation∑K

l=1 σi,l(t)σj,l(t) = ρij(t) holds. The drift

terms µN+1i (t)

(i = 1, . . . , N

)result from working in the terminal measure and are given by

expression (2.6). It vanishes for the last forward rate FN

(t; TN , TN+1

)since it is a log–normal

martingale under QN+1. Hence µN+1N (t) = 0.

The model parameters are hence the time–dependent forward skews βi(t) and volatility

levels λi(t) as well as the time–independent volatility of variance η and mean reversion

speed Θ. Piterbarg chooses the latter to be constant which reduces the degrees of freedom

to the set of tuplesβi(t), λi(t)

i=1,...,N

associated with forward Libors Fi

(t; Ti, Ti+1

)for

times T0 ≤ t ≤ Ti ≤ TN and i = 1, . . . , N. These parameters characterise the distribution

of each forward Libor as they are not affected by a change of measure. Reference to (3.1a)

clearly reveals the role of the skews βi(t) as parameters mixing a purely log–normal with a

normal forward Libor dynamics. From there it also is apparent that the parameters λi(t)

determine the level of the stochastic volatility√

Σt which is governed by (3.1b). Together

with correlations ρij(t) among forward rates Fi and Fj these parameters determine the joint

distribution for times T0 ≤ t ≤ TN .

The term structure of forward Libor skews and volatility levels have to be obtained by

calibration to market prices of caplets/floorlets or swaptions. As outlined above the FL-

TSS model is calibrated to the swaption market. Since forward swap rates are the natural

swaption underlyings, it is therefore necessary to formulate a consistent forward swap rate

dynamics and relate the resulting forward swap skews and volatility levels to their forward

Libor counterparts. This is done by requiring that the process governing the forward swap

rates should have the same structure as the forward Libor dynamics. The detailing of this

idea will be presented in the following section.

18

3.2 Swap rate dynamics under the FL-TSS model

To derive a consistent forward swap rate dynamics we first observe that the par rate Smn(t)

of a forward swap starting at time Tm > t ≥ T0 and maturing at Tn > Tm can be expressed

as a weighted sum of its constituent forward rates Fm, . . . Fn−1. :

Smn(t) =D(t, Tm) − D(t, Tn)∑n−1

l=m αl(Tl, Tl+1)D(t, Tl+1)

=n−1∑

l=m

αl(Tl, Tl+1)D(t, Tl+1)∑n−1l=m αl(Tl, Tl+1)D(t, Tl+1)︸︷︷︸

=:wl(t)

Fl

(t; Tl, Tl+1

)

=n−1∑

l=m

wl(t)Fl

(t; Tl, Tl+1

), (3.2)

which follows from the forward rate definition

Fl

(t; Tl, Tl+1

)=

D(t, Tl) − D(t, Tl+1)

αl(Tl, Tl+1)D(t, Tl+1),

where αl(Tl, Tl+1) denotes the year fraction of the period [Tl, Tl+1] and D(t, Tj) stands for

the discount factor corresponding to time Tj .

Thus the forward swap rate is a function of its constituent forward rates, Smn(Fm, . . . , Fn−1),

and a stochastic differential equation is arrived at by application of Ito’s lemma:

dSmn(t) =n−1∑

l=m

∂Smn(t)

∂Fl(t)dFl(t) +

1

2

n−1∑

l,k=0

∂2Smn(t)

∂Fl(t)∂Fk(t)dFl(t)dFk(t)

=n−1∑

l=m

∂Smn(t)

∂Fl(t)

(βl(t)Fl(t) +

(1 − βl

(T0

))Fl

(T0

))

︸︷︷︸=:ϕ(Fl(t))

λl(t)√

Σt

× σT

l (t) ·[µ

(m,n)l dt + dW(m,n)(t)

]

+1

2

n−1∑

l,k=m

∂2Smn(t)

∂Fl(t)∂Fk(t)ϕ(Fl(t))ϕ(Fk(t))ρlk(t)dt

=n−1∑

l=m

∂Smn(t)

∂Fl(t)ϕ(Fl(t))λl(t)

√Σt σT

l (t)dW(m,n)(t) + drift terms. (3.3)

Above expression is formulated in the swap measure Q(m,n) which is induced by us-

ing the present value of a basis point Pmn(t) =∑n−1

l=m αl(Tl, Tl+1)D(t, Tl+1) as numeraire 1.

The K–dimensional drift vectors µ(m,n)l account for the change from the terminal measure

QN+1 to Q(m,n), under which dW(m,n)(t) is a K–dimensional Brownian motion. Additional

1Strictly speaking the quantity Pmn(t) refers to a notional of 1 and therefore represents the present valueof 10,000 basis points.

19

drift terms arise from the non–zero correlations ρlk(t) = σT

l (t)σk(t) =∑K

j=1 σl,j(t)σk,j(t)

between forward rates Fl and Fk. But since covariances and therefore volatilities and corre-

lations remain invariant under a change of measure, one can consider above swap dynamics

under a new measure Q with associated K–dimensional Brownian motion dW(t) in which

the drift terms vanish. Under Q (3.3) transforms into

dSmn(t) =n−1∑

l=m

∂Smn(t)


√Σt σT

l (t)dW(t). (3.4)

As was already mentioned in the introduction we are looking for a forward swap rate

dynamics which has the same structure as the forward rate process. This is equivalent to

requiring a dynamics of the form

dSmn(t) =(βmn(t)Smn(t) +

(1 − βmn

(T0

))Smn

(T0

))

︸︷︷︸=:ϕ(Smn(t))

λmn(t)√

Σt

K∑

l=1

σmn,l(t)dWl(t),

(3.5)

with forward swap rate skews βmn(t), volatility levels λmn(t), and Brownian loadings σmn,l(t)

(l = 1, . . . , K). In order to ensure consistency between the two formulations we set (3.4)

equal to (3.5) and in addition match the slopes of both expression with respect to all forward

rates Fl. This reasoning results in direct relations between swap and forward rate volatility

levels and skews, respectively, which will be presented in the following subsections.

3.2.1 Derivation of the forward swap volatility level

Matching of both expressions for the forward swap rates results in the requirement

n−1∑

l=m

∂Smn(t)


√Σt σT

l (t)dW(t) = ϕ(Smn(t))λmn(t)√

Σt σT

mn(t)dW(t)

from which the forward swap volatility level can be derived:

λmn(t)σT

mn(t) =n−1∑

l=m

∂Smn(t)

∂Fl(t)

ϕ(Fl(t))

ϕ(Smn(t))λl(t)σT

l (t) (3.6)

=⇒ λmn(t) =n−1∑

l=m

∂Smn(t)

∂Fl(t)

ϕ(Fl(t))

ϕ(Smn(t))σT

l (t)σmn(t)λl(t), (3.7)

since σT

mn(t)σmn(t) = 1. Whereas the load factors of the orthogonal forward Libor drivers

can be extracted from the correlation matrix 2 this does not apply to the Brownian motions

2We assume that the correlations between forward Libors Fi and Fj are given in the parametric form

(2.1) propsed by Rebonato [21]: ρij(t) = ρ∞ + (1 − ρ∞) exp(−δ

∣∣(Ti − t)ǫ− (Tj − t)ǫ

∣∣)

20

driving the swap rate. In above case of K > 1 Brownian swap rate drivers we therefore

have to retrieve the swap rate volatility by referring to (3.6) instead of the volatility level

λmn(t) itself. It is obvious that forward skew and volatility parameters as well as their swap

counterparts are intertwined in above expressions, which is due to the presence of the skew

functions ϕ(Fl(t)) and ϕ(Smn(t)). This dependency is resolved when (3.6) is considered at

the money, i.e., at the time T0 forward Libor and swap rate points. In this case ϕ(Fl(t))

and ϕ(Smn(t)) become skew independent, and by further freezing ∂Smn(t)∂Fl(t)

(which are given

by (A.11)) at their initial values (3.6) simplifies to

λmn(t)σT

mn(t) =

n−1∑

l=m

∂Smn(t)

∂Fl(t)

∣∣∣∣t=T0

Fl(T0)

Smn(T0)λl(t)σT

l (t), (3.8)

which is the expression on which calibration will be based.

3.2.2 Derivation of the forward swap skew

The forward swap and Libor skew can be interpreted as slopes of the respective skew

functions ϕ(Smn(t)) and ϕ(Fl(t)) which are linear in the underlying rate variable. Hence

we have

βmn(t) =∂ϕ(Smn(t))

∂Smn(t)=

∂(βmn(t)Smn(t) +

(1 − βmn

(T0

))Smn

(T0

))

∂ϕ(Smn(t)),

βl(t) =∂ϕ(Fl(t))

∂Fl(t)=

∂(βl(t)Fl(t) +

(1 − βl

(T0

))Fl

(T0

))

∂Fl(t)for l = m, . . . , n − 1,

and a relation between both quantities can be established by matching the slope of dSmn(t)

in formulations (3.4) and (3.5) with respect to the forward rates Fk(t). Referring to (3.4)

and assuming that the derivatives ∂Smn(t)∂Fl(t)

(l = m, . . . , n − 1) do not vary significantly as

Fl(t) varies with time, we thus obtain

∂(dSmn(t)

)

∂Fk(t)=

n−1∑

l=m

∂Smn(t)

∂Fl(t)

∂ϕ(Fl(t))

∂Fk(t)+ ϕ(Fl(t)

∂2Smn(t)

∂Fk(t)∂Fl(t)︸︷︷︸≈0

λl(t)√

Σt σT

l (t)dW(t)

≈n−1∑

l=m

∂Smn(t)

∂Fl(t)

∣∣∣∣t=T0

βk(t)δlkλl(t)√

Σt σT

l (t)dW(t)

=∂Smn(t)

∂Fk(t)

∣∣∣∣t=T0

βk(t)λk(t)√

Σt σT

k (t)dW(t), (3.9)

where our assumptions took effect in the second line by freezing ∂Smn(t)∂Fl(t)

at their initial values

and neglecting second derivatives of the swap rate with respect to the forward Libors.

21

Focussing on the formulation in terms of swap volatility levels and skews (3.5), a similar

analysis yields

∂(dSmn(t)

)

∂Fk(t)=

∂ϕ(Smn(t))

∂Smn(t)

∂Smn(t)

∂Fk(t)λmn(t)

√Σt σT

mn(t)dW(t)

≈ βmn(t)∂Smn(t)

∂Fk(t)

∣∣∣∣t=T0

λmn(t)√

Σt σT

mn(t)dW(t). (3.10)

Matching of the forward swap slopes with respect to Libors Fm, . . . , Fn−1 in equations

(3.10) and (3.9) results in a system of n − m equations between forward swap and Libor

skews:

βmn(t)λmn(t)σmn(t) = βk(t)λk(t)σk(t) (k = m, . . . , n − 1),

to which no unique solution exists. Therefore one has to revert to a least squares optimisa-

tion to find an approximate solution. For this sake we consider the functional

J(βmn(t)

)=

n−1∑

k=m

(βmn(t)λmn(t)σmn(t) − βk(t)λk(t)σk(t)

)2,

and obtain the optimal solution for the forward swap skew by requiring the derivative of J

with respect to βmn(t) to vanish:

0 =dJ

dβmn(t)= 2 λmn(t)σT

mn(t)n−1∑

k=m

(βmn(t)λmn(t)σmn(t) − βk(t)λk(t)σk(t)

)

=⇒ βmn(t) =1

n − m

n−1∑

k=m

(λmn(t)σT

mn(t))(

λk(t)σk(t))

(λmn(t)σT

mn(t))(

λmn(t)σmn(t)) βk(t). (3.11)

As the central result of this section we hence have established the relation between for-

ward swap skews and volatilities to their forward Libor counterparts for each swap rate

Smn(t) over time interval T0 ≤ t ≤ Tm. It is this time dependence of the forward Libor

parameters which defines the central idea of a skew term structure in Piterbarg’s FL–TSS

model. As mentioned earlier, in calibrating the FL–TSS model to the swaption market

Piterbarg uses skew and volatility parameters of a Heston model as market input. By

construction Hestons’s model parameters are time–independent which necessitates the in-

troduction of a time-averaging method for the swap rate parameters within the FL–TSS.

This requirement results in the formulation of effective skew and volatility parameters which

are central in paving the way to calibration.

22

3.3 The effective skew and volatility formulation

This section is dedicated to the formulation of swap rate effective skews and volatilities

within Piterbarg’s FL–TSS model. In summary, these facilitate the transition from a swap

rate dynamcis with time–dependent parameters (3.5),


(1 − βmn

(T0

))Smn

(T0

))λmn(t)

√Σt σT

mn(t)dW(t),

to a formulation based on time–independent ones,

dSmn(t) =(βmnSmn(t) +

(1 − βmn

)Smn

(T0

))λmn

√Σt σT

mn(t)dW(t),

where for each swap rate Smn(t) the effective skews βmn and volatilities λmn are in principle

given as weigthed time averages of the time–dependent quantities (3.8) and (3.11). The

detailing of this central concept will be provided in the following, where at first attention

will be paid to the effective skew in subsection 3.3.1 after which effective volatility is covered

in 3.3.2.

3.3.1 The effective forward swap skew

Piterbarg arrives at the effective swap rate skew by considering two diffusion processes where

one has a time–dependent local volatility function and the other a time–independent one.

The latter is defined as a weighted average of its time–dependent counterpart. Focussing on

the weight function w(t) on time interval [0, T ], Piterbarg derives an explicit expression such

that the average of differences between european swaption prices across an infinite range of

strikes, calculated with respect to the respective processes, tends to zero as valuation time

approaches T0. In detail, the following theorem holds:

Theorem 3.3.1 (Piterbarg [19]). For T > 0, let f ∈ C1([0, T ]×R, R+

)be a local volatility

function satisfying the usual growth requirements. Let σ(t), t ∈ [0, T ] be a function of time

only. Fix x0 ∈ R. For any ǫ > 0 define a re–scaled local volatility function

fǫ(t, x) = f(tǫ2, x0 + (x − x0)ǫ

),

and assume without loss of generality

f(t, x0) ≡ 1, t ∈ [0, T ]

which implies

fǫ(t, x0) ≡ 1, t ∈ [0, T ].

23

Let w(t), t ∈ [0, T ] be a weight function satisfying

∫ T

0w(t) dt = 1,

and define an averaged local volatility function

f ǫ(x)2 =

∫ T

0fǫ(t, x)2 w(t) dt. (3.12)

Further define two families of diffusions indexed by ǫ,

dXǫ(t) = fǫ

(t, Xǫ(t)

)λ(t)

√Σ(t)dW (t), (3.13a)

dYǫ(t) = f ǫ

(Yǫ(t)

)λ(t)

√Σ(t)dW (t), (3.13b)

Xǫ(0) = x0, (3.13c)

Yǫ(0) = x0, (3.13d)

for t ∈ [0, T ] with


)dt + η

√Σt dV (t),

⟨dV (t), dW (t)

⟩= 0.

If the weights w(t) are given by the expression

w(t) =v(t)2λ(t)2

∫ T

0 v(t)2λ(t)2 dt, with v(t)2 = E

[Σ(t)

(X0(t) − x0

)2], (3.14)

then

∫ ∞

−∞

(E

[(Yǫ(T ) − K

)+]− E

[(Xǫ(T ) − K

)+])dK = O

(ǫ2

)(3.15)

for ǫ → 0.3

By applying above theorem to the swap rate Smn(t) with local volatility function

f(t, Smn(t)

)= ϕ(Smn(t))

Smn(T0) and replacing time zero with T0, a formulation with time–independent

effective skew parameters is obtain according to


(1 − βmn

(T0

))Smn

(T0

))λmn(t)

√Σt σT

mn(t)dW(t),

= Smn(T0)f1

(t, Smn(t)

)λmn(t)

√Σt σT

mn(t)dW(t),

≈ Smn(T0)f1

(Smn(t)

)λmn(t)

√Σt σT

mn(t)dW(t),

=(βmnSmn(t) +

(1 − βmn

)Smn

(T0

))λmn(t)

√Σt σT

mn(t)dW(t),

3While above theorem provides a relation between the two diffusions, it has to be observed that (3.15)is not formulated in terms of absolute values. Hence the fact that Yǫ(t) complies with relation (3.15) in thelimit ǫ → 0 does not generally ensure convergence to Xǫ(t) in probability.

24

with βmn =∫ Tm

0 βmn(t)w(t) dt. By referring to above theorem one has to bear in mind that

the derived relation between the time–dependent and time averaged re–scaled local volatility

function holds in the limit ǫ → 0. This corresponds to regarding the local volatility function

at time zero, although we are interested in a time–independent proxy for f(t, x) which would

be obtained in the limit ǫ → 1. In this sense the so derived effective skew parameters have

to be regarded as approximate results since their derivation is based on the consideration

of f1(t, x), the re-scaled local volatility function with ǫ = 1. Their derivation is presented

as corollary:

Corollary 3.3.2 (Piterbarg [19]). The effective skew βmn for the equation


(1 − βmn

)Smn

(T0

))λmn(t)

√Σt σT

mn(t)dW(t),

over a time horizon [T0, Tm] is given by

βmn =

∫ Tm

T0

βmn(t)wmn(t) dt,

where the weights w(t) are given by

wmn(t) =vmn(t)2 λmn(t)2

∫ Tm

0 vmn(t)2 λmn(t)2 dt, (3.16)

vmn(t)2 = Σ(T0

)2∫ t

T0

λmn(s)2 ds + Σ(T0

)η2 e−Θ(t−T0)

∫ t

T0

λmn(s)2eΘ(s−T0) − e−Θ(s−T0)

2Θds.

(3.17)

Proof. The proof is provided in section A.4 of Appendix A.

In calibrating the model to the market the swap rates Smn(t) will be considered at

discrete points in time Tk, T0 ≤ Tk < TK = t ≤ Tm, between which the volatility λmn(t)

will be assumed to be constant. For such a piecewise constant swap rate volatility function

25

vmn(t)2 simplifies to:

vmn(t)2 = Σ(T0

)2∫ t

T0

λmn(s)2 ds + Σ(T0

)η2 e−Θ(t−T0)

∫ t

T0


2Θds

= Σ(T0

)2K−1∑

k=0

λmn(Tk)2(Tk+1 − Tk

)

+ Σ(T0

)η2 e−Θ(t−T0)

K−1∑

k=0

λmn(Tk)2

∫ Tk+1

Tk

eΘ(s−T0) − e−Θ(s−T0)

2Θds

= Σ(T0

)2K−1∑

k=0


)

+ Σ(T0

) η2

2Θ2e−Θ(t−T0)

K−1∑

k=0

λmn(Tk)2(eΘ(Tk+1−T0) + e−Θ(Tk+1−T0)

− eΘ(Tk−T0) − e−Θ(Tk−T0)). (3.18)

Above corollary veers towards a formulation in terms of time–independent, effective

parameters in providing a link between time–dependent and effective swap rate skews. As

a result the swap dynamics takes the form


(1 − βmn

)Smn

(T0

))λmn(t)

√Σt σT

mn(t)dW(t), (3.19)

which is time–independent as far as the skews are concerned but which still refers to the

time–dependent swap volatility levels λmn(t). This gap will be closed in the following by

deriving an effective volatility which completes our work towards a swap rate dynamics with

time–independent parameters.

3.3.2 The effective forward swap volatility

We begin by observing that (3.19) represents a displaced diffusion process with displacement

parameter γmn = 1−βmn

βmn

Smn

(T0

), comp. [23]. Indeed (3.19) can be recasted,

dSmn(t) = d(Smn(t) + γmn

)=

(Smn(t) + γmn

)βmnλmn(t)

√Σt σT

mn(t)dW(t), (3.20)

and because the Brownian driver of the variance process, dV (t), is independent of dW(t)

it is obvious that for a given volatility level√

Σt the shifted swap rate follows a log–normal

process. Therefore the terminal variance of the swap rate Smn

(Tm

)conditioned on a par-

ticular variance pathΣt

Tm

T0is given by

σ(Tm)2 = β2mn

∫ Tm

T0

λmn(s)2 Σs ds. (3.21)

26

Also the mentioned independence of Brownian drivers enables us to value european options

with strike K by integrating the well known Black76-formula against the distribution of the

stochastic terminal swap rate variance. Focussing on expected values only and working in

the measure Q we have

E[(

Smn

(Tm

)− K

))+]= E

[E

[(Smn

(Tm

)− K

)+∣∣∣σ(Tm)2

]],

where the inner expectation directly yields the displaced Black76 formula with volatility

σ(Tm) [3]. For at the money options it becomes especially simple and, omitting the nu-

meraire once again, the option value becomes

E[(

Smn

(Tm

)− Smn

(T0

))+]= E

[E

[(Smn

(Tm

)− Smn

(T0

))+∣∣∣σ(Tm)2

]]

= E[g(σ(Tm)2

)], (3.22)

with the displaced ’at the money’ Black76 function

g(σ(Tm)2

)=

(Smn

(T0

)+ γmn

)[2 Φ

(1

2

[β

2mn

∫ Tm

T0

λmn(s)2 Σs ds

] 12)− 1

]

=Smn

(T0

)

βmn

[2 Φ

(1

2

√σ(Tm)2

)− 1

].

To arrive at an effective volatility λmn Piterbarg considers the swap variance (3.21)

σ(Tm)2 = β2mn λ

2mn

∫ Tm

T0

Σs ds, (3.23)

and again formulates values of european at the money options as integrals against the

probability distribution of σ(Tm)2, following the same analysis which lead to (3.22). Since

the option values calculated either way have to match, this requirement results in a defining

equation for the effective swap rate volatility:

E[g(σ(Tm)2

)]= E

[g(σ(Tm)2

)]

⇐⇒ E

[g

(β

2mn

∫ Tm

T0

λmn(s)2 Σs ds

)]= E

[g

(β

2mn λ

2mn

∫ Tm

T0

Σs ds

)](3.24)

Both sides of equation (3.24) involve expected values of the cumulative normal distri-

bution which makes it difficult to solve for the effective volatility. The situation can be

alleviated by considering an approximation to (3.24) obtained through the formulation of

an equivalent matching condition. It is based on using the analytic function

h(x) = amn + bmne−cmnx

27

as approximation for g(x) around the expected value of the stochastic variable umn(Tm) :=(σ(Tm)

βmn

)2. From (3.21) the expected value of the latter reads

umn := E[umn(Tm)] = E

[∫ Tm

T0

λmn(s)2 Σs ds

]=

∫ Tm

T0

λmn(s)2 E[Σs

]ds

= ΣT0

∫ Tm

T0

λmn(s)2 ds.

With this at hand the coefficients amn, bmn, and cmn are determined as solutions of the

system of matching equations,

h(umn) = amn + bmne−cmnumn = g(umn),

dh

dx

∣∣∣x=umn

= (−bmncmn)e−cmnumn =dg

dx

∣∣∣x=umn

,

d2h

dx2

∣∣∣x=umn

= (bmnc2mn)e−cmnumn =

d2g

dx2

∣∣∣x=umn

,

=⇒ cmn = −(

d2g

dx2

∣∣∣x=umn

)/(dg

dx

∣∣∣x=umn

)=

1

8+

1

2(Tm − T0

)β

2mn λ

2mn

, (3.25)

where the derivation of coefficient cmn is provided in Appendix A (comp. expression (A.13)).

For values x around the expected variance level umn the relation

h(x) ≃ g(x)

holds. The approximate matching condition for the expected values (3.24) then becomes:

E[h(σ(Tm)2

)]= E

[h(σ(Tm)2

)]

⇐⇒ amn + bmn E

[exp

(−cmn β

2mn

∫ Tm

T0

λmn(s)2 Σs ds

)]

= amn + bmn E

[exp

(−cmn β

2mn λ

2mn

∫ Tm

T0

Σs ds

)]

=⇒ E

[exp

(−cmn β

2mn

∫ Tm

T0

λmn(s)2 Σs ds

)]= E

[exp

(−cmn β

2mn λ

2mn

∫ Tm

T0

Σs ds

)].

(3.26)

The expected values on either side of equation (3.26) can be represented in terms of the

function

ϕ(µ∣∣Σ0

)= E

[exp

(−µ

∫ Tm

T0

f(s)Σs ds

)∣∣∣∣Σ0

], (3.27)

if µ and f(s) are defined as µ = cmn β2mn, f(s) = λmn(s)2 and µ = cmn β

2mnλ

2mn, f(s) ≡ 1 =

const., respectively.

28

Since the processΣt

t≥T0

is affine, i.e., the drift and variance terms in (3.1b) are affine

functions of Σt, ϕ(µ∣∣Σ0

)can be expressed as

ϕ(µ∣∣Σ0

)= exp

(Aµ,f

(T0, Tm

)+ Bµ,f

(T0, Tm

)Σ0

), (3.28)

where the functions Aµ,f

(t, Tm

)and Bµ,f

(t, Tm

)obey the Riccati system of ordinary differ-

ential equations

dAµ,f

dt

(t, Tm

)= −Θ Σ0 Bµ,f

(t, Tm

),

dBµ,f

dt

(t, Tm

)= µ f(t) + ΘBµ,f

(t, Tm

)− η2

2Bµ,f

(t, Tm

)2,

Aµ,f

(Tm, Tm

)= Bµ,f

(Tm, Tm

)= 0, (3.29)

for times T0 ≤ t ≤ Tm. Obviously, the coefficients A and B depend on the function f(t) over

the considered time interval which is the reason why the system does not possess an analytic

solution. Therefore one has to refer to numerical methods, e.g., a Runge-Kutta scheme, or

define a discretization of the considered time span [T0, Tm] in sub intervals [Ti, Ti+1] on

each of which the function f(t) is assumed to be constant. Since in the case of a constant

function f(t) above system can be solved analytically, time T0 solutions Aµ,f

(T0, Tm

)and

Bµ,f

(T0, Tm

)can be obtained by a recursion scheme in which the analytical solutions of

adjacent sub intervals are concatenated.

With (3.28) the matching condition defining the effective forward swap volatility be-

comes

exp(A

cmn β2mn,λmn(t)2

(T0, Tm

)+ B

cmn β2mn,λmn(t)2

(T0, Tm

)Σ0

)

= exp(A

cmn β2mnλ

2mn,1

(T0, Tm

)+ B

cmn β2mnλ

2mn,1

(T0, Tm

)Σ0

)

=⇒ Acmn β

2mn,λmn(t)2

(T0, Tm

)+ B

cmn β2mn,λmn(t)2

(T0, Tm

)

= Acmn β

2mnλ

2mn,1

(T0, Tm

)+ B

cmn β2mnλ

2mn,1

(T0, Tm

), (3.30)

because Σ0 = 1. The left hand side involves the time–dependent forward swap volatility

λmn(t), whereas the coefficients on the right hand side depend on λmn, the effective forward

swap volatility which is constant over the entire time interval [T0, Tm]. Therefore the right

hand side coefficients Ac β

2mnλ

2mn,1

(T0, Tm

)and B

c β2mnλ

2mn,1

(T0, Tm

)are given by analytic

expressions (comp. (A.40) and (A.41) in Appendix A).

In the calibration process the time–dependent forward swap volatility λmn(t) entering

the left hand side of (3.30) will be determined iteratively as described above in order to

match a market derived effective volatility level. As the former in turn depend on the for-

ward swap rate volatilities through (3.8), equation (3.30) provides a means to calibrate these

29

to given market parameters. The detailing of the calibration procedure will be presented

next.

3.4 Calibration of the FL-TSS model

In calibrating the FL–TSS model Piterbarg assumes that the market information is encoded

in a set of time–independent skew and volatility parameters of a Heston model which already

is calibrated to a set of swaptions across expiries and underlying swap maturities. Thus

calibration of the FL-TSS model is done in an indirect fashion by considering the Heston

parameters as primary market input rather than market implied swaption prices.

So let us assume that the Heston parameters (λHmn, βH

mn) m=1,...,Nn=m+1,...,2N

are known. Each

tuple encodes the market prices of swaptions on the underlying swap rate Smn expiring at

time Tm for a range of strikes (comp. Appendix B), and is thus considered to be constant on

time interval [T0, Tm]. Calibration is then based on minimisation of the squared differences

between effective and market derived (Heston) swap rate volatilities and skews. Because

the effective volatilities are derived at the money (comp. (3.22)) they depend on the swap

rate skews only weakly. Therefore calibration of swap rate volatilities and skews can be

carried out separately. Specifically, the quantity

M = M1 + M2, (3.31a)

with M1 =N∑

m=1

m+N∑

n=m+1

(λmn − λH

mn

)2(3.31b)

and M2 =N∑

m=1

m+N∑

n=m+1

(βmn − βH

mn

)2, (3.31c)

will be minimised by an iteration scheme in which the skews are kept constant during the

volatility calibration step and vice versa. Hence the process rests on a sequential minimi-

sation of M1 and M2 that will be repeated until convergence of M is established.

The aim of above calibration scheme is to extract forward rate volatilities and skews out

of swaption market data. As we are assuming knowledge of Heston parameters (λHmn, βH

mn)

for m = 1, . . . , N and n = m + 1, . . . , m + N as market data input, TN represents the latest

time for which information on forward rate volatilities and skews can be obtained. Indeed, as

the swap rates consist of underlying forward rates, i.e., Smn ≡ Smn(Fm, Fm+1, . . . , Fn−1),

the provided market data only allow us to obtain volatilities and skews of forward rates

F1, . . . , F2N−1 for times T0 ≤ t ≤ TN . Further market data for swaption expiries TN+1, . . . , T2N

would be needed to extract forward rate volatilities and skews for times TN < t ≤ T2N .

In the following sections we provide a detailed explanation of the forward rate volatility

and skew calibration process which is concluded by a presentation of results.

30

3.4.1 Forward rate volatility calibration

Focussing on the calibration of forward rate volatilities, the process can be broken down

into two steps:

1. The calibration of (time–dependent) forward rate volatilities to (time–independent)

effective swap rate volatilities.

2. Variation of forward rate volatilities (and thus via 1. effective swap rate volatilities)

to minimise M1 for fixed forward swap skews.

Starting with 1., we recall that the forward swap volatility is given by (3.8),

λmn(t)σT

mn(t) =n−1∑

l=m

∂Smn(t)

∂Fl(t)

∣∣∣∣t=T0

Fl(T0)

Smn(T0)︸︷︷︸=:dmn,l

λl(t)σT

l (t), (3.32)

where the vector σT

l (t) is obtained by a singular value decomposition of the known corre-

lation matrix ρmn(t). Also, the time T0 values of the swap rate derivatives and hence dmn,l

are known. Hence the forward swap rate volatility level λmn(t) is a function of forward rate

volatility levels λm(t), λm+1(t), . . . , λn−1(t).

The fitting of time–dependent forward rate volatilities to effective swap rate volatilities

is then achieved by imposing the matching condition

exp(A

cmn β2mn,λmn(t)2

(T0, Tm

)+ B

cmn β2mn,λmn(t)2

(T0, Tm

)Σ0

)

= exp(A

cmn β2mnλ

2mn,1

(T0, Tm

)+ B

cmn β2mnλ

2mn,1

(T0, Tm

)Σ0

)

=⇒ Acmn β

2mn,λmn(t)2

(T0, Tm

)+ B

cmn β2mn,λmn(t)2

(T0, Tm

)

= Acmn β

2mnλ

2mn,1

(T0, Tm

)+ B

cmn β2mnλ

2mn,1

(T0, Tm

), Σ0 = 1,

(3.33)

for times T0 ≤ t ≤ Tm and m = 1, . . . , N, n = m + 1, . . . , m + N, where TN corresponds

to the latest swaption expiry. Here the left hand side involves the time–dependent forward

swap volatility λmn(t), whereas the coefficients on the right hand side depend on λmn, the

effective forward swap volatility which is constant over the entire time interval [T0, Tm].

The problem (3.33) is discretised with piecewise constant forward rate volatility levels and

skews4. Specifically, time interval [T0, Tm] is split into sub intervals Ii := [Ti, Ti+1] for

4As mentioned above, the forward rate skews are kept constant during the volatility calibration.

31

i = 0, . . . , m − 1. On each Ii the forward rate volatilities are then assumed to take on

constant values λm(Ti), λm+1(Ti), . . . , λn−1(Ti), and enter coefficients A(i), B(i).

The calibration of forward rate volatilities to effective forward swap rate volatilities

is performed according to the following iteration, in the course of which λm(Ti) will be

determined for m = 1, . . . , 2N − 1 and times T0 ≤ Ti < Tm < TN . Hence for a fixed m,

the forward rate volatility λm(t) is obtained on intervals I0, I1, . . . , Im−1 where the values

λm(T0), λm(T1), . . . , λm(Tm−1) are attained. At the beginning of the iteration, all forward

rate volatilities λm (m = 1, . . . , 2N) are set to a constant value, e.g., the average of all

Heston volatility parameters λHmn.

On interval I0 the volatilities are determined as follows:

1. λ1(T0) is obtained by fitting λ12(T0)σT

12(T0) = d12,1 λ1(T0)σT1 (T0) to λ12. This is done

by a one–dimensional root finding method, e.g., Brent’s algorithm.

2. λ2(T0) is obtained by a fitting to λ13, since λ13(T0)σT

13(T0) = d13,1 λ1(T0)σT1 (T0) +

d13,2 λ2(T0)σT2 (T0), and λ1(T0) was obtained in the first iteration step. Hence the

swap rate volatility only depends on λ2(T0) and the optimisation problem only is

one–dimensional.

3. Assuming that λ1(T0), λ2(T0), . . . , λn−2(T0) have already been determined, λn−1(T0) is

derived from a calibration of λ1n(T0)σT

1n(T0) =∑n−2

l=1 d1n,l λl(T0)σTl (T0) + d1n,n−1

×λn−1(T0)σTn−1(T0) to λ1n. Because the sum term is known, the swap rate volatility

only depends on λn−1(T0) and again the optimisation problem is one–dimensional.

Therefore a one–dimensional root finding algorithm can be applied.

As a result of performing this iterative calibration at time T0 for underlying swap matu-

rities T2, . . . , T2N−1, forward rate volatilities λ1(T0), . . . , λ2N−1(T0) are obtained which are

calibrated to the effective volatilities λ12, . . . , λ1,2N . These results will be used in further

iteration steps.

On interval I1 = [T1, T2] the goal is to obtain volatilities λ2(T1), . . . , λ2N−1(T1).5 This is

done as follows:

1. The interval [T0, T2] is split into sub–intervals I0 and I1. From the iteration procedure

on I0 the time T0 volatilities are already known. Hence, λ1(T1) is obtained by ap-

plication of matching condition (3.33) at time T1 with respect to the forward swap

rate λ23(T1). Due to the relation λ23(T1)σT

23(T1) = d23,2 λ2(T1)σT2 (T1), the forward rate

5As forward rate F1 resets at time T1, λ1 only lives for times T0 ≤ t < T1.

32

volatility level λ2(T1) can be obtained by calibration to the effective volatility λ23. The

matching condition results in equations

A(0)

c23 β223,λ23(T0)2

(T1

)= A

(1)

c23 β223,λ23(T1)2

(T1

),

B(0)

c23 β223,λ23(T0)2

(T1

)= B

(1)

c23 β223,λ23(T1)2

(T1

),

with terminal condition

A(1)

c23 β223(T1)2,λ23(T1)2

(T2

)= B

(1)

c23 β223(T1)2,λ23(T1)2

(T2

)= 0,

and initial condition

A(0)

c23 β223,λ23(T0)2

(T0

)= A

c23 β223,λ

223

(T0

),

B(0)

c23 β223,λ23(T0)2

(T0

)= B

c23 β223,λ

223

(T0

),

where the coefficients A(i), B(i) for i = 0, 1 are given by expressions (A.40) and (A.41)

derived in Appendix A.6. Since the volatility λ2(T0) and hence λ23(T0) are known from

the previous iteration step, coefficients A(0)(T1), B(0)(T1) are given as well. Therefore

the matching condition together with the initial and terminal conditions enable us

to determine coefficients A(1)(T1), B(1)(T1) which depend on λ23(T1) and thus λ2(T1)

by a one–dimensional optimisation algorithm. Through this matching of coefficients,

λ2(T1) is obtained by a fitting to λ23.

2. With λ2(T1) at hand, λ3(T1) is obtained by a fitting to effective volatility λ24. Again

this is done by a matching of coefficients at time T1, however here forward swap

rate λ24(T1) is serving as reference function. Due to the relation λ24(T1)σT

24(T1) =

d24,2 λ2(T1)σT2 (T1) + d24,3 λ3(T1)σT

3 (T1) and the fact that λ2(T1) has been obtained

previously, the forward swap rate solely depends on λ3(T1). Hence, the calibration to

λ24 via a matching of coefficients yields the desired forward rate volatility λ3(T1).

3. Assuming that λ2(T1), λ3(T1), . . . , λn−2(T1) have already been determined, λn−1(T1)

is derived by a matching of coefficients at time T1 with respect to λ2,n(T1)σT

2,n(T1) =∑n−2

l=1 d2n,l λl(T1)σTl (T1)+d2n,n−1 λn−1(T1)σT

n−1(T1) and calibration to λ2n. As the sum

term is known from previous iteration steps, the forward swap rate solely depends on

λn−1(T1). Hence, the calibration to λ2n via a matching of coefficients yields the desired

forward rate volatility λn−1(T1).

As a result of performing this iterative calibration to the effective forward swap volatilities

λ23, . . . , λ2,2N at time T1 for underlying swap maturities T3, . . . , T2N−1, forward rate volatili-

ties λ2(T1), . . . , λ2N−1(T1) are obtained. These are referenced in further calibration steps at

later times Ti > T1.

33

Assuming now that the forward rate volatilities have already been determined at times

T1, . . . , Tm−2 by a sequential application of matching condition (3.33) at times T1, . . . , Tm−3,

on interval Im−1 = [Tm−1, Tm] the goal is to obtain volatilities λm(Tm−1), . . . , λ2N−1(Tm−1).6

This is done as follows:

1. The interval [T0, Tm] is split into sub–intervals I0, I0, . . . , Im−1. From the iteration pro-

cedure on interval Im−2 the time Tm−2 volatilities are already known. Hence, λm(Tm−1)

is obtained by application of matching condition (3.33) at time Tm−1 with respect to

the forward swap rate λm,m+1(Tm−1). Due to the relation λm,m+1(Tm−1)σT

m,m+1(Tm−1) =

d(m,m+1),m λm(Tm−1)σTm(Tm−1), the forward rate volatility level λm(Tm−1) can be ob-

tained by calibration to the effective volatility λm,m+1. The matching condition is a

special case of the general formulation which is given under number 3..

2. Having determined λm(Tm−1) in the previous step, λm+1(Tm−1) is obtained by a fitting

to effective forward swap rate volatility λm,m+2. Here matching condition (3.33) is

applied at time Tm−1 with respect to λm,m+2(Tm−1)σT

m,m+2(Tm−1) = d(m,m+2),m λm(Tm−1)

×σTm(Tm−1)+d(m,m+2),m+1 λm+1(Tm−1)σT

m+1(Tm−1), which only depends on λm+1(Tm−1)

since the first term is known from 1.. Hence a one–dimensional optimisation with

λm,m+2 as target variable yields the desired forward rate volatility λm+1(Tm−1). The

matching condition is the special case n = m+2 of the general formulation presented

next.

3. Assuming that λm(Tm−1), λm+1(Tm−1), . . . , λn−2(Tm−1) have already been determined,

λn−1(Tm−1) follows by application of matching condition (3.33) at time Tm−1 with re-

spect to forward swap rate λm,n(Tm−1)σT

m,n(Tm−1) =∑n−2

l=m d(m,n),l λl(Tm−1)σTl (Tm−1)

+ d(m,n),n−1 λn−1(Tm−1)σTn−1(Tm−1). Because the sum term is already known, λn−1(Tm−1)

can be obtained by calibration to the effective swap rate volatility λm,n. The matching

condition results in equations

A(m−2)

cmn β2mn,λm,n(Tm−2)2

(Tm−1

)= A

(m−1)


(Tm−1

), (3.34a)

B(m−2)


(Tm−1

)= B

(m−1)


(Tm−1

), (3.34b)

with terminal condition

A(m−1)

cmn β2mn(Tm−1)2,λm,n(Tm−1)2

(Tm

)= B

(m−1)

cmn β2mn(Tm−1)2,λm,n(Tm−1)2

(Tm

)= 0,

6As forward rates Fm−1 reset at time Tm−1, the λm−1 only live for times T0 ≤ t < Tm−1.

34

and initial condition

A(0)

cmn β2mn,λm,n(T0)2

(T0

)= A

cmn β2m,n,λ

2m,n

(T0

),

B(0)

cmn β2mn,λm,n(T0)2

(T0

)= B

cmn β2m,n,λ

2m,n

(T0

),

where the coefficients A(i), B(i) for i = 0, . . . , m − 1 are given by expressions (A.40)

and (A.41) derived in Appendix A.6. Since all forward rates are known at time

Tm−2 the forward swap rate λm,n(Tm−2) and hence the coefficients A(m−2), B(m−2)

are known as well. Furthermore, the coefficients A(m−1), B(m−1) on the right hand

side of the matching conditions (3.34a), (3.34b) only depend on λn−1(Tm−1) because

all λj(Tm−1) with j < n − 1 are already known. Therefore the matching condition

together with the initial and terminal conditions enable us to determine coefficients

A(m−1)(Tm−1), B(m−1)(Tm−1) by a one–dimensional optimisation algorithm. Through

this matching of coefficients, λn−1(Tm−1) is obtained by a fitting to λmn. Performing

this iterative calibration for n = m + 3, . . . , 2N results in calibrated forward rate

volatilities λm+2(Tm−1), . . . , λm+2(T2N−1).

Above iterative calibration scheme enables us to extract forward rate volatilities λm(Tm−1),

. . . , λ2N−1(Tm−1) out of effective forward swap volatilities λm,m+1, . . . , λm,2N for m = 1, . . . , N .

By identifiying the effective volatilities with market derived Heston volatilities λHmn for

n = m + 1, . . . , 2N and n = 1, . . . , N , expression M1 (comp. (3.31b)) is automatically min-

imised by performing the one–dimensional optimisations at times T1, . . . , TN−1 as detailed

above.

During the entire forward volatility calibration the forward (and hence the swap rate)

skews are kept constant as their calibration can be conducted separately. With respect to

this, the detailing is provided in the following section.

3.4.2 Forward rate skew calibration

As with the forward rate volatilities, the calibration process for forward rate skews can be

sub–divided into two steps:

1. The calibration of (time–dependent) forward rate skews to (time–independent) effec-

tive swap rate skews.

2. Variation of forward rate skews (and thus via 1. effective swap rate skews) to minimise

M2 for fixed forward swap volatilities.

35

Starting with 1. and assuming that forward rate skews are piecewise constant, from

(3.11) we known that the swap skews are related to their forward rate counterparts by

βmn(Tj) =n−1∑

k=m

1

n − m

(λmn(Tj)σ

T

mn(Tj))(

λk(Tj)σk(Tj))

(λmn(Tj)σ

T

mn(Tj))(

λmn(Tj)σmn(Tj))

︸︷︷︸=:gmn,k(Tj)

βk(Tj) =n−1∑

k=m

gmn,k(Tj)βk(Tj),

for times T0 ≤ Tj < Tm, where the gmn,k are known from the preceding volatility calibration.

From corollary 3.3.2 the effective forward swap skew is given by

βmn =

∫ Tm

T0

βmn(t)wmn(t) dt,

with weights w(t) of the form

wmn(t) =vmn(t)2 λmn(t)2

∫ Tm

0 vmn(t)2 λmn(t)2 dt,

vmn(t)2 = Σ(T0

)2∫ t

T0

λmn(s)2 ds

+ Σ(T0

)η2 e−Θ(t−T0)

∫ t

T0


2Θds,

for m = 1, . . . , N and n = m+1, . . . , m+N. Since the forward rate skews are assumed to be

piecewise constant, the following discretised expression for the effective skew is considered

in the calibration process:

βmn =m−1∑

l=0

βmn(Tl)wmn(Tl)(Tl+1 − Tl

),

wmn(TK−1) =vmn(TK−1)

2 λmn(TK−1)2

∑K−1k=0 vmn(Tl)2 λmn(Tl)2 (Tl+1 − Tl)

,

vmn(TK−1)2 = Σ

(T0

)2K−1∑

k=0


)

+ Σ(T0

) η2

2Θ2e−Θ(t−T0)

K−1∑

k=0

λmn(Tk)2(eΘ(Tk+1−T0) + e−Θ(Tk+1−T0)

− eΘ(Tk−T0) − e−Θ(Tk−T0)), (3.35)

for T0 ≤ TK−1 < Tm, where the expression for vmn(TK−1)2 follows from (3.18). As the

forward swap rate volatilities are known from the volatility calibration, the weights wmn(t)

and vmn(t) can be evaluated straight away.

The calibration of forward rate to effective forward swap rate skews is performed ac-

cording to the following iteration, in the course of which βm(Ti) will be determined for

m = 1, . . . , 2N−1 and times T0 ≤ Ti < Tm < TN . Hence for a fixed m, the forward rate skew

36

βm(t) is obtained on intervals I0, I1, . . . , Im−1 where the values βm(T0), βm(T1), . . . , βm(Tm−1)

are attained. At the beginning of the iteration, all forward rate skews βm (m = 1, . . . , 2N)

are set to a constant value, e.g., the average of all market derived Heston skew parameters

βHmn.

On interval I0 the skews are determined as follows:

1. β1(T0) is obtained by fitting β12 = β12(T0)w12(T0)(T1 − T0

)= g12,1(T0)β1(T0)w12(T0)

×(T1−T0

)to βH

12 . This is done by a one–dimensional root finding method, e.g., Brent’s

algorithm.

2. With β1(T0) at hand, β2(T0) is obtained by fitting β13 = β13(T0)w13(T0)(T1 − T0

)=

(g13,1(T0)β1(T0) + g13,2(T0)β2(T0)

)w12(T0)

(T1 − T0

)to βH

13 . Since the weights and

β1(T0) is already known from 1., β13(T0) only depends on β2(T0), and the calibration

to the market input βH13 is performend by a one-dimensional root finding algorithm.

3. Assuming that β1(T0), β2(T0), . . . , βn−2(T0) have already been determined, βn−1(T0)

follows by fitting β1n = β1n(T0)w1n(T0)(T1−T0

)=

(∑n−2l=1 g13,l(T0)βl(T0)+g13,n−1(T0)

×βn−1(T0

)w1n(T0)

(T1 − T0

)to βH

1n. Because the sum term is already known, β1n(T0)

only depends on βn−1(T0) and the forward rate skew is obtained by a one–dimensional

root finding with βH1n as target variable.

As a result of performing this iterative calibration at time T0 for underlying swap maturities

T2, . . . , T2N−1, forward rate skews β1(T0), . . . , β2N−1(T0) are obtained which are calibrated to

the Heston skews βH12 , . . . , β

H1,2N . These results will be used in further iteration steps.

On interval I1 = [T1, T2] the goal is to obtain forward rate skews β2(T1), . . . , β2N−1(T1).7

This is done as follows:

1. β2(T1) is obtained by fitting β23 = β23(T0)w23(T0)(T1−T0

)+ β23(T1)w23(T1)

(T2−T1

)=

(g23,2(T0)β2(T0)

)w23(T0)

(T1 −T0

)+

(g23,2(T1)β2(T1)

)w23(T1)

(T2 −T1

)to βH

23 . Because

β2(T0) is known form the iteration at time T0, this expression solely depends on β2(T1)

and calibration to βH23 is performed with a one–dimensional root finding method.

2. With β2(T1) at hand, β3(T1) is obtained by fitting β24 = β24(T0)w24(T0)(T1 − T0

)

+β24(T1)w24(T1)(T2 − T1

)=

(g24,2(T0)β2(T0) + g24,3(T0)β3(T0)

)w24(T0)

(T1 − T0

)+

(g24,2(T1)β2(T1) + g24,3(T1)β3(T1)

)w24(T1)

(T2 − T1

)to βH

24 . In this expression β3(T1)

is the only unknown which is why a one–dimensional optimisation with βH24 as target

variable yields the desired forward rate skew.

7As forward rate F1 resets at time T1, β1 only lives for times T0 ≤ t < T1.

37

3. Assuming that β2(T1), β3(T1), . . . , βn−2(T1) have already been determined, βn−1(T1)

follows by fitting β2n = β2n(T0)w2n(T0)(T1 − T0

)+ β2n(T1)w2n(T1)

(T2 − T1

)

=(∑n−2

l=1 g(2,n),l(T0)βl(T0)+g(2,n−1),l(T0)βn−1(T0))w2n(T0)

(T1−T0

)+

(∑n−2l=1 g(2,n),l(T1)

βl(T1) + g(2,n−1),l(T1)βn−1(T1))w2n(T1)

(T2 − T1

)to βH

2n. Since all weight factors and

skews at time T0 are known, and βn−1(T1) is the only unknown at time T1, a one–

dimensional optimisation with βH2n as target variable yields the desired forward rate

skew.

As a result of performing this iterative calibration to the Heston skews βH23 , . . . , β

H2,2N at time

T1 for underlying swap maturities T3, . . . , T2N−1, forward rate skews β2(T1), . . . , β2N−1(T1)

are obtained. These are referenced in further calibration steps at later times Ti > T1.

Assuming now that the forward rate skews have already been determined at times

T1, . . . , Tm−2, on interval Im−1 = [Tm−1, Tm] the goal is to obtain skews βm(Tm−1), . . . ,

β2N−1(Tm−1).8 This is done as follows:

1. The interval [T0, Tm] is split into sub–intervals I0, I0, . . . , Im−1. From the iteration pro-

cedure on interval Im−2 the time Tm−2 skews are already known. Hence, βm(Tm−1) is

obtained by fitting βm,m+1 =∑m−1

l=0 βm,m+1(Tl)wm,m+1(Tl)(Tl+1 − Tl

)

=∑m−2

l=0

(g(m,m+1),m(Tl)βm(Tl)

)wm,m+1(Tl)

(Tl+1 −Tl

)+

(g(m,m+1),m(Tm−1)βm(Tm−1)

)

×wm,m+1(Tm−1)(Tm − Tm−1

)to βH

m,m+1. By assumption βm(Tl) is known for l =

1, . . . , m − 2. As βm,m+1 only depends on the unknown skew βm(Tm−1), latter is ob-

tained by a one–dimensional optimisation with βHm,m+1. as target variable.

2. With βm(Tm−1) at hand, βm+1(Tm−1) is obtained by fitting βm,m+2 =∑m−1

l=0 βm,m+2(Tl)

wm,m+2(Tl)(Tl+1 − Tl

)=

∑m−2l=0

(g(m,m+2),m(Tl)βm(Tl) + g(m,m+2),m+1(Tl)βm+1(Tl)

)

wm,m+1(Tl)(Tl+1−Tl

)+

(g(m,m+2),m(Tm−1)βm(Tm−1)+g(m,m+2),m+1(Tm−1)βm+1(Tm−1)

)

wm,m+1(Tm−1)(Tm − Tm−1

)to βH

m,m+2. Because the skews βm(Tl) are known for l =

1, . . . , m− 1, and βm+1(Tl) is given for previous times corresponding to l = 1, . . . , m−2, βm,m+2 only depends on βm+1(Tm−1). Hence the latter is obtained by a one–

dimensional optimisation with βHm,m+2 as target variable.

3. Assuming that βm(Tm−1), βm+1(Tm−1), . . . , βn−2(Tm−1) have already been determined,

βn−1(Tm−1) follows by fitting βm,n =∑m−1

l=0 βm,n(Tl)wm,n(Tl)(Tl+1 − Tl

)

=∑m−1

l=0

∑n−2k=m g(m,n),k(Tl)βk(Tl)wm,n(Tl)

(Tl+1 − Tl

)+

∑m−2l=0 g(m,n),n−1(Tl)βn−1(Tl)

wm,n(Tl)(Tl+1−Tl

)+g(m,n),n−1(Tm−1)βn−1(Tm−1)wm,n(Tm−1)

(Tm−Tm−1

)to βH

m,n. By

assumption the first term is known. Also, as all forward rates have been determined

at earlier times Ti < Tm−1, the second sum is known as well. Hence βm,n only depends

8As forward rates Fm−1 reset at time Tm−1, the βm−1 only live for times T0 ≤ t < Tm−1.

38

on βn−1(Tm−1), which is then obtained by a one–dimensional optimisation with βHm,n

as target variable.

Above iterative calibration scheme enables us to extract forward rate skews βm(Tm−1),

. . . , β2N−1(Tm−1) out of market derived Heston skews βHm,m+1, . . . , β

Hm,2N for m = 1, . . . , N .

In doing so expression M2 (comp. (3.31c)) is automatically minimised by performing the

one–dimensional optimisations at times T1, . . . , TN−1 as detailed.

As mentioned in the introduction, the forward rate volatility and skew calibration are

carried out sequentially. Hence the error measures M1 (comp. 3.31b) and M2 (comp.

3.31c) are minimised separately with respect to a given volatility (for M2) or skew level

(for M1). The resulting volatility and skew vectors are then used in the next iteration step.

This process is repeated until the error measure M (comp. 3.31a) is minimised. Results

obtained by this calibration procedure are presented in the next section.

3.4.3 Calibration results

In the following we present results of the calibration to a set Heston parameters (λHm,n, βH

m,n)

corresponding to swaptions with expiries Tm and underlying swap maturities Tn which was

performed according to the iteration schemes outlined in the previous sections. The Heston

parameters were obtained by calibration to Euro swaption prices of May 7th, 2008, on

underlying swaps of a one year tenor with maturities between one and ten years. Swaption

expiries ranged from one to ten years. Hence Heston parameters were available for m =

1, . . . , 10 and n = m + 1, . . . , m + 10.

With these data at hand, the Piterbarg skew and volatility term structure were obtained

for forward reset times T1, . . . , T10 with T0–forward rates taken from Table C.2. For the

calibration the forward rate skews were initialised with a constant value of 0.5. Furthermore

the volatility of variance and mean reversion speed of the variance process (3.1b) were set

to η = 0.59, Θ = 0.15, respectively. Forward rate volatilities were initially set to 0.02.

Table (C.3) presents βi(t) and λi(t) for times T0 = 0 y ≤ t < Ti = i y and i = 1, . . . , 10.

The following figure displays the skew and volatility of forward rate F10 over time period

[0 y, 10 y]. Here it becomes apparent that in Piterbarg’s FL–TSS model the skew can become

negative.

39

0 2 4 6 8 10

t (years)

-0.4

-0.2

0.0

0.2

0.4

0101)t(

β,)t(λ

Forward rate volatility and skew

01 )t(λ

01 )t(β

Figure 3.1: Volatility level λ10(t) and skew β10(t) of forwardrate F10(t) for times T0 = 0 y ≤ t < T10 = 10 y.

40

Chapter 4

A Markov functional model with

stochastic volatility

This chapter is dedicated to the formulation of a Markov functional model with stochas-

tic volatility. Following the ideas presented in section 2.2.3 of chapter 2 where multi–

dimensional Markov functional models were discussed, a two–dimensional Libor Markov

functional model is devised which is calibrated to the (digital) caplet market. With a model

of this kind the forward rate dynamics can be described by reference to the distribution of a

two–dimensional Markov process. As swap rates are functions of their constituent forward

rates, the pricing of derivatives on underlying swaps, e.g., Bermudan swaptions, can be

conducted more efficiently. Our formulation is based on an approximation to Piterbarg’s

FL–TSS Libor Market Model which is used as pre–model. That is, the numeraire discount

bonds are expressed as functionals of pre–model processes. Since the latter have a stochas-

tic volatility component, this approach enables us to incorporate the concept of stochastic

volatility into a Markov functional framework.

The outline of this chapter is as follows: Based on the FL–TSS model, two–dimensional

proxy processes for the forward rates are devised in sections 4.1 to 4.3. This is done in three

steps, which can be summarised as

1. Approximate forward rates Fi(t) by displaced diffusion processes Fi(t) with partially

frozen drift terms in section 4.1

2. Find a two–dimensional formulation of Fi(t), i.e., a formulation with two stochastic

components in section 4.2

3. Replace the two–dimensional proxies to Fi(t) with processes Fi(t, zt) exhibiting the

same mean and variance, which are functions of a two–dimensional Brownian motion

zt with independent components in section 4.3

41

The forward rates themselves are then formulated as functionals of the processes Fi(t).

Thus Fi(t) = gi

(Fi(t, zt)

)with monotone functions gi, and the functional dependence of

the forward rates on the two–dimensional Markov process is only through the pre–model

process Fi(t, zt

). The latter allows for a non–zero correlation between the Brownian drivers

of the forward rate and variance components.

The construction and calibration of a two–dimensional Libor Markov functional model

is demonstrated in section 4.4. Due to the dependence on independent Brownian motions,

the calibration involves Gaussian integrals which are easier to handle in numerical imple-

mentations.

4.1 Piterbarg’s FL–TSS Libor Market Model as pre–model

The incorporation of stochastic volatility into MFMs will be based on a calibrated Piterbarg

FL–TSS model,

dFi(t) =(βi(t)Fi(t) +

(1 − βi

(T0

))Fi

(T0

))

︸︷︷︸=:ϕi(Fi(t))

λi(t)√

Σt

[µN+1


t

],

(4.1a)

µN+1i (t) = −

√Σt

N∑

l=i+1

αl λl(t) ρil(t)ϕl

(Fl

)

1 + αlFl(t), µN+1

N (t) = 0, (4.1b)


)dt + η

√Σt dV N+1

t , (4.1c)

for times T0 ≤ t < Ti ≤ TN

(i = 1, . . . , N

). By calibrated we mean that the skew and

volatility parameters βi(t), λi(t) are known on the considered time interval.

Following the ideas presented in 2.3 an approximation to (4.1) will be introduced which

is used as pre–model. As such it will serve as prerequisite for the derivation of numeraire

discount bond functional forms from market observed derivative prices.

First we observe that the FL–TSS model can be formulated as displaced diffusion model

with time dependent displacement parameter γi(t), comp. [23],

dFi(t) =(Fi(t) + γi(t)

)βi(t)λi(t)

√Σt

[µN+1


t

], (4.2a)


)dt + η

√Σt dV N+1

t , (4.2b)

γi(t) =1 − βi

(T0

)

βi(t)Fi

(T0

), i = 1, . . . , N, T0 ≤ t < Ti ≤ TN . (4.2c)

Introducing the displaced forward rate Fi(t) by defining Fi(t) := Fi(t) + γi(t), by reference

42

to (4.2) its dynamics is given by

dFi(t) = d(Fi(t) + γi(t)

)= dFi(t) −

(γi(t)

βi(t)

dβi(t)

dt

)dt

= Fi(t)βi(t)λi(t)√

Σt

[µN+1


t

]−

(γi(t)

βi(t)

dβi(t)

dt

)dt,

because dγi(t)dt

= − 1−βi(T0)

βi(t)2

dβi(t)dt

Fi

(T0

)= − γi(t)

βi(t)dβi(t)

dt. Now we approximate the dynamics of

the process Fi(t) by freezing the local volatility functions ϕl(Fl(t)) at their time T0 values,

and omitting the additional drift which is due to the time dependence of the displacement

parameter γi(t). Hence ϕl(Fl(T0)) = Fl(T0), and we can define the approximate displaced

diffusion dynamics

dFi(t) := Fi(t)βi(t)λi(t)√

Σt

[µ0

N+1i (t) dt + σi(t)

TdWN+1t

], (4.3a)

µ0N+1i (t) = −

√Σt

K∑

l=i+1


(T0; Tl, Tl+1

)

1 + αl(Tl, Tl+1)Fl

(T0; Tl, Tl+1

) , µ0N+1N (t) = 0, (4.3b)

Fi(t) ≈ Fi(t) + γi(t), γi(t) =1 − βi

(T0

)

βi(t)Fi

(T0

), (4.3c)


)dt + η

√Σt dV N+1

t , (4.3d)

for i = 1, . . . , N and times T0 ≤ t < Ti ≤ TN , which serves as starting point for a two–

dimensional pre–model.

4.2 The pre–model with two Brownian drivers

As already discussed above, a key motivation of Markov functional models is to formulate

the numeraire discount bond process in terms of a low dimensional driving process. Here we

want to define a pre-model with only two Brownian drivers which is based on the displaced

diffusion approximation (4.3).

For this we consider the one dimensional Brownian drivers of the rate and variance

processes, dWN+1t and dV N+1

t , and assume that their correlation is expressed by a function

Γ(t). Hence 0 ≤ Γ(t) ≤ 1, and⟨dWN+1

t , dV N+1t

⟩= Γ(t) dt. By introducing the independent

Brownian driver dZN+1t the process dWN+1

t can be decomposed into

dWN+1t = Γ(t) dV N+1

t +√

1 − Γ(t)2 dZN+1t , (4.4)

and the correlation⟨dWN+1

t , dV N+1t

⟩= Γ(t)

⟨dV N+1

t , dV N+1t

⟩= Γ(t) dt is recovered since

⟨dZN+1

t , dV N+1t

⟩= 0. Also,

⟨dWN+1

t , dWN+1t

⟩=

(Γ(t)2+

(1−Γ(t)2

))dt = dt. By reference

to (4.3d) dV N+1t can be expressed in terms of the variance level,

dV N+1t = −Θ

(Σ0 − Σt

)

η√

Σt

dt +1

η√

Σt

dΣN+1t . (4.5)

43

Applying above expressions to the process (4.3a) with only one Brownian driver one obtains

dFi(t) = Fi(t)βi(t)λi(t)√

Σt

[µ0

N+1i (t) dt + dWN+1

t

]

= Fi(t)βi(t)λi(t)√

Σt

[µ0

N+1i (t) dt + Γ(t) dV N+1

t +√

1 − Γ(t)2 dZN+1t

]

= Fi(t)βi(t)λi(t)[(√

Σt µ0N+1i (t) − Γ(t)

Θ

η

(Σ0 − Σt

))dt

+Γ(t)

ηdΣN+1

t +√

Σt

√1 − Γ(t)2 dZN+1

t

], (4.6)

where the dynamics of Fi(t) is formulated in terms of the variance process dΣN+1t and an

independent Brownian motion dZN+1t . By Ito’s lemma (4.6) integrates to

Fi(t) = Fi(T0) exp

(∫ t

T0

βi(s)λi(s)[µ0

N+1i (s) − 1

2(βi(s)λi(s))

]Σs ds

− Θ

η

∫ t

T0

Γ(s)βi(s)λi(s)(Σ0 − Σs

)ds

+1

η

∫ t

T0

Γ(s)βi(s)λi(s) dΣN+1s +

∫ t

T0

√1 − Γ(s)2βi(s)λi(s)

√Σs dZN+1

s

)

(4.7)

with

µ0N+1i (t) :=

µ0N+1i (t)√

Σt

= −N∑

l=i+1

αl(Tl, Tl+1)λl(t)Fl

(T0; Tl, Tl+1

)

1 + αl(Tl, Tl+1)Fl

(T0; Tl, Tl+1

) , µ0N+1N (t) = 0. (4.8)

Note that in this one dimensional setting the forward rates Fl(t) are perfectly correlated

and thus ρli(t) ≡ 1 for all l, i = 1, . . . , N.

The processes Fi(t) involve integrals of the stochastic variables Σt,√

Σt and dZN+1t . In

order to obtain a pre–model which is of practical use these expressions have to be simplified.

This will be done in the following section where the stochastic entities will be replaced by

appropriate approximations.

4.3 A simplification of the pre–model process

The aim of this section is to replace the stochastic integrals in (4.7) with simpler stochastic

variables which can be controlled more easily, thereby defining a process Fi(t) as proxy to

Fi(t). Specifically, we intend to find approximations in terms of variables whose probability

density is given by a standard normal distribution, comp. [12].

For this we consider the integrals

Ii,2(t) :=

∫ t

T0

gi(s) dΣN+1s and Ii,3(t) :=

∫ t

T0

hi(s)√

Σs dZN+1s , (4.9)

44

with deterministic functions gi(s), hi(s) and calculate their expected values and variances.

Starting with Ii,2 we have

E

[∫ t

T0

gi(s) dΣN+1s

]= E

[Θ

∫ t

T0

gi(s)(Σ0 − Σs

)ds

]

︸︷︷︸=0

+ E

[η

∫ t

T0

√Σs dV N+1

s

]

︸︷︷︸=0

= 0,

because E[Σs

]= E

[ΣT0

]≡ E

[Σ0

]= 1, and Mt :=

∫ t

0

√Σs dV N+1

s is a martingale(E

[Σ(s)

]<

∞)

which is why E[Mt − MT0

|ΣT0

]= 0. Furthermore,

var

[∫ t

T0

gi(s) dΣN+1s

]= E

[(∫ t

T0

gi(s) dΣN+1s

)2]− E

[∫ t

T0

gi(s) dΣN+1s

]

︸︷︷︸=0

2

= E

[η2

∫ t

T0

gi(s)2Σs ds

]by Ito’s isometry,

= η2

∫ t

T0

gi(s)2 E

[Σs

]ds = Σ0︸︷︷︸

=1

η2

∫ t

T0

gi(s)2 ds = η2

∫ t

T0

gi(s)2 ds.

Turning to Ii,3 we obtain

E

[∫ t

T0

hi(s)√

Σs dZN+1s

]= 0,

because the integral is a martingale(E

[hi(s)

2Σ(s)]

< ∞). Secondly,

var

[∫ t

T0

hi(s)√

Σs dZN+1s

]= E

[(∫ t

T0

hi(s)√

Σs dZN+1s

)2]− E

[∫ t

T0

hi(s)√

Σs dZN+1s

]

︸︷︷︸=0

2

= E

[∫ t

T0

hi(s)2Σs ds

]by Ito’s isometry,

=

∫ t

T0

hi(s)2 E

[Σs

]ds = Σ0︸︷︷︸

=1

∫ t

T0

hi(s)2 ds =

∫ t

T0

hi(s)2 ds.

Hence the stochastic integrals Ii,2 and Ii,3 can be approximated as

Ii,2(t) ≃ var[Ii,2(t)

]U =

[η2

∫ t

T0

gi(s)2 ds

]U and Ii,3(t) ≃ var

[Ii,3(t)

]Z =

[∫ t

T0

hi(s)2 ds

]Z,

with independent standard normal variables U ,Z ∼ N (0, 1). In particular⟨U ,Z

⟩= 0, and

the independence of Ii,2 and Ii,3, i.e.,⟨Ii,2, Ii,3

⟩= 0, is preserved.

The desired approximation to Fi(t) is arrived at by applying these results to expression

(4.7) where the functions gi(s) and hi(s) are defined by

gi(s) :=Γ(s)

ηβi(s)λi(s) and hi(s) :=

√1 − Γ(s)2βi(s)λi(s). (4.10)

45

Furthermore the stochastic drift terms will be replaced by their expected values, i.e.,

∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1

2(βi(s)λi(s)) + Γ(s)

Θ

η

]Σs − Γ(s)

Θ

ηΣ0

ds

−→ E

[∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1


Θ

η

]Σs − Γ(s)

Θ

ηΣ0

ds

]

=

∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1


Θ

η

]E

[Σs

]︸︷︷︸

=Σ0

−Γ(s)Θ

ηΣ0

ds

= Σ0︸︷︷︸=1

∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1


Θ

η

]− Γ(s)

Θ

η

ds

=

∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1


Θ

η

]− Γ(s)

Θ

η

ds.

Utilising (4.10), the final result for the proxy Fi(t) ≃ Fi(t) can be summarised as

Fi(t) = Fi(T0) exp

(∫ t

T0

βi(s)λi(s)[

µ0N+1i (s) − 1


Θ

η

]− Γ(s)

Θ

η

ds

+

[∫ t

T0

Γ(s)2 (βi(s)λi(s))2 ds

]U

︸︷︷︸=:W1,σ1(t)

+

[∫ t

T0

(1 − Γ(s)2

)(βi(s)λi(s))

2 ds

]Z

︸︷︷︸=:W2,σ2(t)

],

(4.11)

with Brownian motions W1,σ1(t) and W2,σ2(t) of variance

σ1(t)2 :=

∫ t

T0

Γ(s)2 (βi(s)λi(s))2 ds and σ2(t)

2 :=

∫ t

T0

(1 − Γ(s)2

)(βi(s)λi(s))

2 ds, (4.12)

for i = 1, . . . , N and times T0 ≤ t ≤ Ti. At time T0 the forward rates Fi(T0) are known and

we have Fi(T0) = Fi(T0) + γi(T0) = Fi(T0)βi(T0) for i = 1, . . . , N.

According to the discussion of multi–dimensional Markov functional models in 2.2.3,

Fi(t) defines a projection function for the two–dimensional Markov process zt :=(W1,σ1(t), W2,σ2(t)

)T

in the terminal measure QN+1. Hence Fm(t, zt) is of Markov functional form and defines

a one dimensional stochastic process. Process (4.11) is general enough to account for a

correlation function Γ(s) between the Brownian drivers of the rate and volatility processes.

For the zero correlation case Γ(s) ≡ 0, and the proxy process Fm becomes one–dimensional

(comp. Figure 4.1).

One can now define functional relationships between forward rates Fi(t) and their ap-

proximations Fi(t, zt) by introducing monotone functions gi for which Fi(t) = gi

(Fi(t, zt)

).

The gi can be interpreted as perturbation functions which act on the proxy processes Fi

so that they fall in line with the arbitrage free Fi for i = 1, . . . , N, comp. [14]. Re-

calling that forward discount bonds are functions of their constituent forward rates, i.e.,

46

0 1 2 3 4 5 62z

0

1

2

3

4

5

6

01012)z,

T(F

Forward rate functional form

01 01 2)z,T(F

Figure 4.1: Proxy forward rate F10(T10) (4.11) as functionof zt =

(0, z2) at reset time T10 = 10y. This corresponds to

the zero correlation case, Γ(s) ≡ 0.

D(Tm, Tn) ≡ D(Tm, Tn; Fm(Tm), . . . , Fn−1(Tm)), we are now in a position to consider the

forward discount bond functional

D(Tm, Tn; gm

(Fm(Tm, zTm)

), Fm+1(Tm), . . . , Fn−1(Tm)),

which is of Markov functional form through its dependence on the Markov process zt at time

Tm. This will be utilised in constructing a Libor Markov functional model in the following

section.

4.4 Construction of a two–dimensional Libor Markov func-

tional model

In section 2.2.2 we have already discussed the construction of a multi–dimensional Libor

Markov functional model and its calibration to digital caplets. We know want to apply these

ideas and demonstrate how a two–dimensional model can be formulated in terms of the one

dimensional pre–model processes (4.11). The latter are based on Piterbarg’s FL–TSS model

and depend on a Markov process which is given by a two–dimensional Brownian motion

whose variance is entirely determined by the pre–model skew and volatility parameters. As

47

such, the pre–model processes incorporate the correlation structure of a Piterbarg Libor

Market Model with stochastic volatility. The forward rates themselves are formulated as

monotone functionals of the pre–model process

Fi(t) = gi

(Fi(t, zt)

), (4.13)

with monotone functions gi and zt =(W1,σ1(t), W2,σ2(t)

)T. Thus they depend on the two–

dimensional Markov process only via the pre–model forward rates Fi(t, zt).

4.4.1 A two–dimensional Libor Markov functional model in the terminal

measure

The construction of a two–dimensional Libor Markov functional model will be based on the

calibration to digital caplets. A digital caplet expiring at time Tm with strike K has the

payout profile

Vm(Tm; K) = D(Tm, Tm+1)1Fm(Tm;Tm,Tm+1)>K . (4.14)

Assuming that the forward rates Fm(t) follow a log–normal process in their individual

martingale measures Qm+1 (m = 1, . . . , N), the digital caplet values at time T are given by

the Black76 formula,

Vm(T0) = D(T0, Tm+1)Φ(d−m), (4.15)

d−m =log

(Fm(T0;Tm,Tm+1)

K

)

σ(Tm)√

Tm − T0

− 1

2σ(Tm)

√Tm − T0,

with constant volatility σ(Tm) and cumulative normal distribution function Φ(x). Based on

this assumption market quotes for digital caplets are provided in terms of Black76 implied

volatilities.

As the Piterbarg FL–TSS and the pre–model process (4.11) are formulated in the ter-

minal measure which is induced by the numeraire discount bond process D(t, TN+1) we

construct the Libor Markov functional model under QN+1. By the fundamental theorem

of asset pricing the numeraire rebased digital caplet process is a martingale. Referencing

(4.14), its value at time T0 is thus given by

Vm(T0; K)

D(T0, TN+1)= E

[Vm(Tm; K)

D(Tm, TN+1)

∣∣∣∣FT0

]= E

[D(Tm, Tm+1)

D(Tm, TN+1)1Fm(Tm)>K

∣∣∣∣FT0

]

= E

[1Fm(Tm)>KE

[1

D(Tm+1, TN+1)

∣∣∣∣FTm

]∣∣∣∣FT0

].

As discussed in the introduction, according to (4.13) the forward rates are monotone func-

tionals of the pre–model processes Fm(t, zt) for m = 1, . . . , N. This transfers to the discount

bond functionals with

D(Tm, Tm+1; gm

(Fm(Tm, zTm)

)),

48

and hence

D(Tm, TN+1;

gm

(Fm(Tm, zTm)

), Fm+1(Tm), . . . , FN(Tm)

)

depending on Fm(Tm, zTm) as driving variable, which is of Markov functional form due

to the Markov property of zt. For clarity we will suppress the functional relationship

gm

(Fm(Tm, zTm)

)in the discount bond processes and directly refer to Fm(Tm, zTm) instead.

Thus the digital caplet value can be expressed in terms of the process zt whose probability

density is known:

Vm(T0; K)

D(T0, TN+1)= E

[1Fm(Tm)>K E

[1

D(Tm+1, TN+1)

∣∣∣∣FTm

]∣∣∣∣FT0

]

= E

[1gm(Fm(Tm,zTm ))>K

× E

[1

D(Tm+1, TN+1; Fm+1(Tm+1, zTm+1), Fi(Tm+1)i)

∣∣∣∣zTm

]∣∣∣∣zT0

]

= E

[1Fm(Tm,zTm )>x∗

m

× E

[1


∣∣∣∣zTm

]∣∣∣∣zT0

]

=

∫

x∗m

dzTm p(zTm

∣∣zT0

)

×[∫

R2

p(zTm+1

∣∣zTm

) 1


dzTm+1

],

(4.16)

where x∗m describes a boundary curve in two–dimensional state space,

F−1m (Tm, x∗

m) =zTm ∈ R2

∣∣Fm(Tm, zTm) = x∗m

,

and satisifies gm(x∗m) = K since gm is monotone. Details on the evaluations of the inner and

outer expected values are given in Appendix A, where the two–dimensional integals above

are given by expressions (A.45) and (A.46).

In the course of the calibration this functional equation is solved for a series of strikes

Km1 , . . . , Km

NmS

, where NmS stands for the number of caplet strikes available at expiry time

Tm. As a result we obtain the set

Sm =x∗

m,j

∣∣gm(x∗m,j) = Km

j for j = 1, . . . , NmS

,

which defines the desired forward rate functional form Fm(Tm, xm) at reset time Tm. Indeed,

since Fm = gm Fm all xm,j ∈ Sm satisfy Fm(Tm, xm,j) = Kmj .

The functional forms of the numeraire discount bond are determined recursively starting

at time TN . At each forward reset date Tm ≤ TN equation (4.16) is solved numerically

49

by application of Brents’s algorithm for a set of digital caplets with strikes Kmj for j =

1, . . . , NmS . Because

1

D(Tm, TN+1; Fm(Tm), Fi(Tm)i=m+1,...,N)

=(1 + αm gm

(Fm(Tm)

)) D(Tm, Tm+1; Fm(Tm))

D(Tm, TN+1; Fm(Tm), Fi(Tm)i=m+1,...,N), (4.17)

the time Tm numeraire discount bond is related to its functional form at time Tm+1 which

was determined in the previous calibration step. By this fitting to market quotes the sets

Sm are recovered for m = N, . . . , 1. Obviously, due to relation (4.17) this applies to the

numeraire discount bond as well.

4.4.2 Calibration results

Here we present results of the construction of a two–dimensional Libor Markov functional

model for ten forward rates F1, . . . , F10 on time grid T0 = 0 y < 1 y ≤ Tj ≤ 10 y for

j = 1, . . . , 10. The considered underlying forward rate tenor was one year, and the model was

calibrated to the Euro caplet market on May 7th, 2008. Caplet volatilities were available for

strikes ranging from ATM−0.025 to ATM+0.025 with a step size of 0.0025 and a one–year

spacing between caplet expiries. Hence the final time of the model was TN+1 = T11 = 11 y,

and the numeraire discount bond given by D(t, T11). Digital caplet prices needed for the

calibration of the model were calculated according to (4.15) with implied volatilities taken

from the caplet market1.

In the implementation, at each time step Ti a grid of 10 nodes is set up in the x– and

y dimensions of the two–dimensional Markov process zt = (xt, yt)T. Hereby the respective

maximum nodes are placed at values Mσ1(Ti) and Mσ2(Ti), where the variances σ1(Ti)2

and σ2(Ti)2 are given by (4.12) and refer to the x, y–dimension, respectively. In our imple-

mentation we set the node spacing to three standard deviations, i.e., M = 3. Furthermore

a constant correlation function Γ(s) ≡ 0.1 was assumed. Working backwards from time T10

the inner expected values were calculated at each pair of nodes of the respective time slice

Ti. The numeraire discount bond functional forms D(Ti, T11; F (Ti, zTi)) were then obtained

by a cubic spline interpolation. Hence the calculation of the outer expected values in (4.16)

involved the integration of third order polynomials which can be performed efficiently (SALI

method, comp. [9]).

Plots of the numeraire discount bond functional forms at various forward reset times

can be found in Appendix C (Figures C.1 – C.4). Figure 4.2 displays the functional form of

the numeraire discount bond at forward reset time T10 obtained on a grid of 10× 10 nodes.

1This is justified since under the assumption of log–normality VDig. Caplet(K) = −∂VCaplet(K)

∂K.

50

0.030 0.035 0.040 0.045 0.050 0.055 0.060

01 01 )T(F

0.0

0.2

0.4

0.6

0.8

1.001

1101

01))

T(F;

T,T(

D

Figure 4.2: The numeraire discount bond as functional ofF10(T10, zT10): D(T10, T11; F10(T10, zT10)).

51

Chapter 5

Conclusion

Within this study the incorporation of stochastic volatility into Markov functional models

was investigated. It was explained that this can be achieved by introducing the idea of a

pre–model which serves as driver of the numeraire discount bond process. For this sake

Piterbarg’s forward Libor term structure of skew model (FL–TSS LMM) was discussed

and calibrated to the swaption market. An approximation of this LMM was then used as

pre–model in constructing a two–dimensional Libor Markov functional model.

Following the introduction, the concept of market models was presented in chapter 2.

The Libor Market Model was reviewed and generalisations to non log–normal dynamics were

detailed. Special attention was paid to the derivation of drift terms under consideration

of a stochastic volatility component. Subsequently the Markov Functional framework was

introduced. After a formal definition it was shown how a model of this kind is calibrated

to the digital caplet market. Furthermore the case of multi–dimensional Markov processes

as drivers of the term structure was discussed.

Because Piterbarg’s FL–TSS LMM was used as pre–model for a Libor Markov functional

model, chapter three was dedicated to its detailed exposure. As this stochastic volatility

Libor Market model is calibrated in the parameter domain, the fitting to a grid of Hes-

ton parameters which were derived form the Euro swaption market was demonstrated. A

calibration algorithm was implemented and the resulting one–year forward rate skew and

volatility term structure presented on an equally spaced time grid over a period of ten years.

Chapter 4 focussed on the construction of a two–dimensional Libor Markov functional

model. This was achieved by introducing approximations to the forward rate processes

induced by Piterbarg’s FL–TSS dynamics. These were used as drivers of the numeraire

discount bond process under the terminal measure. As the proxy processes were functions

of a two–dimensional time–changed brownian motion, calibration involved the integration

with respect to a Gaussian probability density.

53

As result of our work an implementation of a Libor Markov functional model is available

which possesses the correlation structure of Piterbarg’s FL–TSS Libor Market Model. A

such it can be used as an efficient tool for the pricing of exotic interest rate derivatives.

Future research should encompass a comparison between Piterbarg’s FL–TSS LMM and

the described pre–model based MFMs for a range of exotic products. Here the specific

MFMs have to be calibrated to instruments which are relevant to the exotic product under

consideration. In doing so also non–constant forms of the correlation function Γ(s) should

be investigated. Although the interplay between pre–model choice and derivative pricing

still offers much room for future reseach, our study illustrates that pre–model based MFMs

are a promising alternative to Libor Market Models.

54

Appendix A

Mathematical details

A.1 The drift term in the Libor Market Model

In the following we derive an expression for the drift in equation (2.5) by referring to the fact

that forward rate agreements (FRAs) are tradable and thus martingales by the fundamental

theorem of asset pricing. In the following we work in the terminal measure QN+1 which

corresponds to using the terminal bond D(t, TN+1) as numeraire. At time t the value of a

FRA for period [Ti, Ti+1] is given by

FRA(t) = D(t, TN+1

)EQN+1

[α(Ti, Ti+1

)D(Ti, Ti+1

)(Fi

(Ti; Ti, Ti+1

)− K

)

D(Ti, TN+1

)], (A.1)

where K denotes the agreed rate and α(Ti, Ti+1

)the accrual factor for period [Ti, Ti+1].

Since the numeraire rebased value of the FRA is a martingale it follows from the above

that EQN+1

[Fi

(Ti;Ti,Ti+1

)D(Ti,Ti+1

)

D(Ti,TN+1

)]

must be a martingale as well. Therefore the expression

d( D(t,Ti+1)

D(t,TN+1) Fi(t))

must be driftless. As discount bonds are tradable and therefore martin-

gales under QN+1 this also applies to the expression d(

D(t,Ti+1)D(t,TN+1

). Accordingly the relation

d

(D(t, Ti+1)

D(t, TN+1)Fi(t)

)

︸︷︷︸driftless

=D(t, Ti+1)

D(t, TN+1)dFi(t) + d

(D(t, Ti+1)

D(t, TN+1)

)

︸︷︷︸driftless

Fi(t)

+

⟨d

(D(t, Ti+1)

D(t, TN+1)

), dFi(t)

⟩,

provides a means to determine an expression for the drift term because from the above it

becomes apparent that the drift which results from the correlation term must cancel the

forward Libor drift,

drift ofD(t, Ti+1)

D(t, TN+1)dFi(t) = −

⟨d

(D(t, Ti+1)

D(t, TN+1)

), dFi(t)

⟩. (A.2)

Before we derive the drift term following (A.2) we reformulate the forward rate process in

terms of independent Brownian drivers. Because the Brownian motion driving the variance

55

process is correlated with each Brownian forward Libor driver this will be achieved by

introducing the K–dimensional vector of independent Brownian motions dZN+1t which in

addition is independent of the variance process. Therefore we have⟨dZN+1

k (t), dV N+1t

⟩= 0,

⟨dZN+1

l , dZN+1j

⟩= δlj dt and

⟨dV N+1

t , dWN+1k (t)

⟩= Γk(t) dt for k = 1, . . . , K. Using these

relations the Brownian motions dWN+1k can be decomposed,

dWN+1k (t) = Γk(t) dV N+1(t) +

√1 − Γk(t)2 dZN+1

k (t). (A.3)

Obviously hereby the relations

⟨dV N+1

t , dWN+1k (t)

⟩= Γk(t)

⟨dV N+1

t , dV N+1t

⟩︸︷︷︸

=dt

+√

1 − Γk(t)2⟨dV N+1

t , dZN+1k (t)

⟩︸︷︷︸

=0

= Γk(t) dt,⟨dWN+1

l (t), dWN+1j (t)

⟩= Γl(t)Γj(t)

⟨dV N+1

t , dV N+1t

⟩︸︷︷︸

=dt

+√

1 − Γl(t)2√

1 − Γj(t)2⟨dZN+1

l (t), dZN+1j (t)

⟩︸︷︷︸

=δlj dt

=(Γl(t)Γj(t) +

√1 − Γl(t)2

√1 − Γj(t)2δlj

)dt,

and⟨dWN+1

l (t), dWN+1l (t)

⟩= dt, l, j = 1, . . . , K, (A.4)

are recovered.

Following the decomposition (A.3) of dWN+1t into uncorrelated Brownian motions dZN+1

t

and dV N+1t the forward Libor process (2.5) becomes

dFi

(t; Ti, Ti+1

)= ϕ

(Fi

)λi(t)

√Σt

[µN+1


t

]

= ϕ(Fi

)λi(t)

√Σt

[µN+1

i (t) dt + σi(t)TΩ(t)T dZN+1

t

+ σi(t)TΓ(t) dV N+1

t

], (A.5a)


)dt + η

√Σt dV N+1

t , (A.5b)⟨dV N+1(t), dZN+1

k (t)⟩

= 0, T0 ≤ t < Ti ≤ TN , i, k = 1, . . . , N,

where the matrix Ω(t) =√

1 − Γk(t)2 δkj

k=1,...,Kj=1,...,K

, and the vector Γ(t) = Γk(t)k=1,...,K

56

were defined. Hence the covariance between forward rates changes dFi and dFj is given by

⟨dFi

(t; Ti, Ti+1

), dFj

(t; Tj , Tj+1

)⟩

= ϕ(Fi

)ϕ(Fj

)λi(t)λj(t)Σt

×K∑

k,l=1

σi,k(t)σj,l(t)[√

1 − Γk(t)2√

1 − Γk(t)2δkl + Γk(t)Γl(t]dt

= ϕ(Fi

)ϕ(Fj

)λi(t)λj(t)Σt

×[ K∑

k=1

σi,k(t)σj,k(t)(1 − Γk(t)

2)

+

K∑

k,l=1

σi,k(t)σj,l(t)Γk(t)Γl(t)]dt

= ϕ(Fi

)ϕ(Fj

)λi(t)λj(t)Σt

×[(

σi(t)TΩ(t)T

)(Ω(t)σj(t)

)+

(σi(t)Γ(t)

)(σj(t)Γ(t)

)]dt. (A.6)

Focussing now on the right hand side of (A.2) we first recall that the numeraire rebased

discount bonds are related to the forward Libors by

D(t, Ti+1)

D(t, TN+1)=

N∏

l=i+1

(1 + αl(Tl, Tl+1)Fl(t)

)Fl(t, Tl, Tl+1),

which leads to

d

(D(t, Ti+1)

D(t, TN+1)

)=

D(t, Ti+1)

D(t, TN+1)

N∑

l=i+1

αl

1 + αlFl(t)dFl(t) + drift terms,

where the time reference in the accrual factors αl has been omitted for simplicity1. In

calculating

⟨d

(D(t,Ti+1)D(t,TN+1)

), dFi(t)

⟩only terms proportional to < dV N+1(t), dV N+1(t) >=

dt and < dZN+1i (t), dZN+1

j (t) >= δij dt contribute. Hence

⟨d

(D(t, Ti+1)

D(t, TN+1)

), dFi(t)

⟩=

D(t, Ti+1)

D(t, TN+1)

N∑

l=i+1

αl

1 + αlFl(t)

⟨dFl(t), dFi(t)

⟩

=D(t, Ti+1)

D(t, TN+1)Σt λi(t)ϕ

(Fi

) N∑

l=i+1

αl λl(t)ϕ(Fl

)

1 + αlFl(t)

×[(

σl(t)TΩ(t)T

)(Ω(t)σi(t)

)+

(σl(t)Γ(t)

)(σi(t)Γ(t)

)]dt,

(A.7)

where (A.6) was used in the second line. By reference to (A.5) the drift of D(t,Ti+1)D(t,TN+1) dFi(t)

is given by

drift ofD(t, Ti+1)

D(t, TN+1)dFi(t) =

D(t, Ti+1)

D(t, TN+1)ϕ(Fi

)λi(t)

√Σt σi(t)

T µN+1i (t) dt,

1SinceD(t,Ti+1)

D(t,TN+1)is a martingale the drift terms cancel those drifts which are contained in the individual

forward Libor changes dFl.

57

and employing (A.7) the drift now follows from (A.2):

µN+1i (t) := σi(t)

TµN+1i (t) = −

√Σt

N∑

l=i+1

αl λl(t)ϕ(Fl

)

1 + αlFl(t)

×[(

σl(t)TΩ(t)T

)(Ω(t)σi(t)

)+

(σl(t)Γ(t)

)(σi(t)Γ(t)

)]

(A.8)

In a deterministic volatility setting we have Σt ≡ 1, Ω(t) = 1K and Γk = 0 for all

k = 1, . . . , K. In this case the decomposition (A.3) collapses to dWN+1k (t) = dZN+1

k (t) and

the drift vector (A.8) becomes

µN+1i (t) = −

N∑

l=i+1

αl λl(t)ϕ(Fl

)

1 + αlFl(t)σl(t)

T)σi(t)︸︷︷︸=ρli(t)=ρil(t)

= −N∑

l=i+1


)

1 + αlFl(t). (A.9)

If one chooses ϕ to be the identity function, ϕ(Fl

)= Fl (l = 1, . . . , N), a lognormal dynamics

is recovered for forward Libor FN and the drift vector becomes

µN+1i (t) = −

N∑

l=i+1

αl λl(t) ρil(t)Fl

1 + αlFl(t). (A.10)

A.2 The derivative of the forward swap rate w.r.t the for-

ward Libor rates

In this section we derive the derivative of the forward swap rate Smn(t) with respect to the

forward Libor rates Fk(t).

The swap rate is given by

Smn(t) =D(t, Tm) − D(t, Tn)

Pmn(t)=

D(t, Tm) − D(t, Tn)∑n−1l=m αl(Tl, Tl+1)D(t, Tl+1)

where αl(Tl, Tl+1) denotes the year fraction of the period [Tl, Tl+1] and D(t, Tk) stands for

the discount factor corresponding to time Tk. Obviously the forward swap rate Smn(t) is a

function of the discount bonds D(t, Tk) which in turn depend on the forward Libor rates

Fl

(t; Tl, Tl+1

)because

D(t, Tk) =k−1∏

l=0

1

1 + αl(Tl, Tl+1)Fl

(t; Tl, Tl+1

) .

58

Thus we arrive at the relations

∂Pmn(t)

∂D(t, Tk)= αk−1(Tk−1, Tk)1m<k<n+1,

∂D(t, Tk)

∂Fj(t)= − αj(Tj, Tj+1)D(t, Tk)

1 + αj(Tj, Tj+1)Fj

(t; Tj , Tj+1

) 10≤j≤k−1

= −αj(Tj, Tj+1)D(t, Tj+1)

D(t, Tj)D(t, Tk)10≤j≤k−1,

and∂Smn(t)

∂D(t, Tk)=

(δmk − δnk)Pmn(t) −(D(t, Tm) − D(t, Tn)

)αk−1(Tk−1, Tk)1m<k<n+1

Pmn(t)2,

from where the derivatives for follow as

∂Smn(t)

∂Fj(t)=

n∑

k=m

∂Smn(t)

∂D(t, Tk)

∂D(t, Tk)

∂Fj(t)

= −αj(Tj, Tj+1)

Pmn(t)

D(t, Tj+1)

D(t, Tj)D(t, Tm)10≤j≤m−1 +

αj(Tj, Tj+1)

Pmn(t)

D(t, Tj+1)

D(t, Tj)D(t, Tn)10≤j≤n−1

+D(t, Tm) − D(t, Tn)

Pmn(t)2αj(Tj, Tj+1)

D(t, Tj+1)

D(t, Tj)

×[ n∑

k=m

αk−1(Tk−1, Tk)D(t, Tk)10≤j≤k−11m<k<n+1

]

︸︷︷︸=

∑nk=m+1 αk−1(Tk−1,Tk)D(t,Tk)10≤j≤k−1

= −αj(Tj, Tj+1)

Pmn(t)

D(t, Tj+1)

D(t, Tj)D(t, Tm)10≤j≤m−1 +

αj(Tj, Tj+1)

Pmn(t)

D(t, Tj+1)

D(t, Tj)D(t, Tn)10≤j≤n−1

+D(t, Tm) − D(t, Tn)

Pmn(t)2αj(Tj, Tj+1)

D(t, Tj+1)

D(t, Tj)

×[ n∑

k=m+1

αk−1(Tk−1, Tk)D(t, Tk)10≤j≤k−1

], (A.11)

for j = 1, . . . . Specifically,

∂Smn(t)

∂Fj(t)= 0 for j < m, since terms cancel,

and∂Smn(t)

∂Fj(t)= 0 for j ≥ n, because in this case all indicator functions are zero.

A.3 Derivation of the coefficient cmn

Here we provide the derivation of coefficient cmn in (3.25) used in the matching condition

(3.26) for the effective volatility. We have

g(x)

=Smn

(T0

)

βmn

[2 Φ

(1

2

√x)− 1

],

59

where Φ(z) =∫ z

−∞12π

exp(−1

2x2)dx denotes the cumulative normal distribution. Taking

derivatives with respect to x we obtain:

dg

dx=

Smn

(T0

)

βmn

2[ 1

2πexp

(−1

2

[1

2

√x]2)] 1

4x− 1

2 =1

2

Smn

(T0

)

βmn

[ 1

2πexp

(−1

8x)]

x− 12 ,

=⇒ d2g

dx2=

1

2

Smn

(T0

)

βmn

[ 1

2πexp

(−1

8x)]((

−1

8

)x− 1

2 − 1

2x− 3

2

),

=⇒(

d2g

dx2

)/(dg

dx

)=

(−1

8

)x− 1

2 − 12x− 3

2

x− 12

= −1

8− 1

2 x.

With umn(Tm) :=(

σ(Tm)

βmn

)2which has an expected value of

umn := E[umn(Tm)] = E

[∫ Tm

T0

λmn(s)2 Σs ds

]=

∫ Tm

T0

λmn(s)2 E[Σs

]ds

= ΣT0

∫ Tm

T0

λmn(s)2 ds,

the coefficient cmn becomes

cmn = −(

d2g

dx2

∣∣∣x=umn

)/(dg

dx

∣∣∣x=umn

)=

1

8+

1

2 umn

. (A.12)

The expected value umn also satisifies

umn ≃(

σ(Tm)

βmn

)2

= ΣT0

(Tm − T0

)β

2mn λ

2mn =

(Tm − T0

)β

2mn λ

2mn,

since σ(Tm) = β2mn λ

2mn

∫ Tm

T0Σs ds and ΣT0 = 1. Inserting this into (A.12) we arrive at

cmn =1

8+

1

2(Tm − T0

)β

2mn λ

2mn

, (A.13)

which will be used in the process of calibrating the forward rate volatilities.

A.4 Proof of corollary 3.3.2

Identify X0(t) from theorem 3.3.1 with the swap rate Smn(t) and define the local volatility

function

f(t, Smn(t)) =1

Smn

(T0

)(βmn(t)Smn(t) +

(1 − βmn(t)

)Smn

(T0

)).

Consider further the re-scaled local volatility function of theorem 3.3.1 with parameter ǫ

set to 1, ǫ = 1. Then we have

f1(t, x) = f(t, x), and f(t, x0) = f(t, Smn

(T0

))= 1,

60

so that from theorem 3.3.1 the time–independent effective volatility function becomes

f1(Smn(t)) =1

Smn

(T0

)[∫ Tm

T0

(βmn(t)Smn(t) +

(1 − βmn(t)

)Smn

(T0

))2wmn(t) dt

] 12

.

Approximating f1(Smn(t)) with a linear function g(Smn(t)) matching the effective local

volatility function and its slope at Smn(t) = Smn

(T0

),

f1(Smn(t)) ≈ g(Smn(t)

)

= g(Smn

(T0

))+

dg(Smn(t))

dSmn(t)

∣∣∣∣Smn(t)=Smn(T0)

(Smn(t) − Smn

(T0

))

= f1

(Smn

(T0

))︸︷︷︸

=1

+df1(Smn(t))

dSmn(t)


(Smn(t) − Smn

(T0

)),

and observing that

df1(Smn(t))

dSmn(t)=

1

Smn

(T0

)∫ Tm

T0

[βmn(t)Smn(t) +

(1 − βmn(t)

)Smn

(T0

)]βmn(t)wmn(t) dt

[∫ Tm

T0

(βmn(t)Smn(t) +

(1 − βmn(t)

)Smn

(T0

))2wmn(t) dt

] 12

=⇒ df1(Smn(t))

dSmn(t)


=1

Smn(T0)

∫ Tm

T0

βmn(t)wmn(t) dt, since

∫ Tm

T0

wmn(t) dt = 1,

we arrive at

f1(Smn(t)) ≈ 1 +Smn(t) − Smn(T0)

Smn(T0)

∫ Tm

T0

βmn(t)wmn(t) dt

=1

Smn

(T0

)(βmnSmn(t) +

(1 − βmn

)Smn

(T0

)),

with

βmn =

∫ Tm

T0

βmn(t)wmn(t) dt,

as asserted.

Turning to the function v(t) we can write, using conditional independence of X0(t)2 and

Σ(t),

v(t)2 = E

[Σ(t)

(X0(t) − x0

)2]

by definition (3.14),

= E

[Σ(t) E

[(X0(t) − x0

)2∣∣∣Σ(t)

]]by conditional independence,

= E

[Σ(t) E

[(∫ t

T0

λ(s)√

Σ(s)dW (s)

)2∣∣∣∣Σ(t)

]]by (3.13a) since f0

(t, X0(t)

)= 1,

= E

[Σ(t) E

[∫ t

T0

λ(s)2Σ(s) ds

∣∣∣∣Σ(t)

]]by Ito’s isometry,

=

∫ t

T0

λ(s)2E[Σ(t)Σ(s)

]ds by linearity. (A.14)

2Since X0(t) is identified with swap rate Smn(t) its volatility λ(t) corresponds to the swap rate volatility

λmn(t). For clarity we suppress the subscripts in the following.

61

Now, defining the process U(t) = eΘtΣ(t), its dynamics is given by

dUt = ΘeΘtΣ(t) + eΘtdΣt = Θ Σ(T0)eΘt dt + η

√Σ(t)eΘt dV (t)

=⇒ Ut = U0 + Θ Σ(T0)

∫ t

T0

eΘs ds + η

∫ t

T0

eΘs√

Σ(s) dV (s)

︸︷︷︸=:Mt

.

Since E

[(ηeΘs

√Σ(s)

)2]

< ∞, the term denoted by Mt is a martingale and therefore we

have for s ≤ t :

E[Us Ut

]= E

[Us

(Us + Θ Σ(T0)

∫ t

s

eΘu du +(Mt − Ms

))]

= E

[U2

s + Us Θ Σ(T0)

∫ t

s

eΘu du

]

= E[U2

s

]+ E

[Us

]Θ Σ(T0)

∫ t

s

eΘu du.

From this we deduce that for s ≤ t :

E[Σ(s)Σ(t)

]= e−Θs e−Θt E

[Us Ut

]

= e−Θs e−Θt E[e2ΘsΣ(s)2

]+ e−Θs e−Θt E

[eΘsΣ(s)

]Θ Σ(T0)

∫ t

s

eΘu du

= e−Θ(t−s) E[Σ(s)2

]+ Σ(T0)

2 Θ e−Θt

∫ t

s

eΘu du, because E[Σ(s)

]= Σ(T0),

= e−Θ(t−s) E[Σ(s)2

]+ Σ(T0)

2(1 − e−Θ(t−s)

)(A.15)

To arrive at an analytic expression for v(t)2 the first term of (A.15) needs to be further

analysed. For this we first consider the quantity Σ(t)2 and derive its dynamics:

d(Σ(t)2

)= 2Σ(t) dΣ(t) + dΣ(t)2

= 2Σ(t)Θ(Σ(T0) − Σ(t)

)dt + 2ηΣ(t)

32 dV (t) + η2Σ(t)dV (t)

=⇒ Σ(t)2 − Σ(T0

)2=

∫ t

T0

(2Θ Σ(T0

)+ η2

)Σ(s) ds − 2Θ

∫ t

T0

Σ(s)2 ds

+ 2η

∫ t

T0

Σ(s)32 dV (s),

and since the last term is a martingale we obtain, upon taking expectations and bearing in

mind that E[Σ(t)

]= Σ

(T0

),

E[Σ(t)2

]= Σ

(T0

)2+

(2Θ Σ

(T0

)2+ η2Σ

(T0

))(t − T0) − 2Θ

∫ t

T0

E[Σ(s)2

]ds. (A.16)

Defining

u(t) = E[Σ(t)2

],

62

it follows from (A.16) that u(t) solves the inhomogeneous differential equation

du(t)

dt= 2Θ Σ

(T0

)2+ η2 Σ

(T0

)− 2Θ u(t),

where the solution can be expressed as

u(t) = g(t) + Σ(T0

)2, (A.17a)

dg(t)

dt= η2 Σ

(T0

)− 2Θ g(t). (A.17b)

Equation (A.17b) is solved by

g(t) = η2 Σ(T0

)e−2Θ t

∫ t

T0

e2Θ u du,

so that the expectation of the squared volatility becomes:

u(t) = E[Σ(t)2

]= Σ

(T0

)2+ η2 Σ

(T0

)e−2Θ t

∫ t

T0

e2Θ u du,

= Σ(T0

)2+ η2 Σ

(T0

) 1 − e−2Θ (t−T0)

2Θ. (A.18)

Finally an expression for v(t) can be derived by inserting expression (A.18) into (A.15)

and carrying out the integration in (A.14):

v(t)2 =

∫ t

T0

λ(s)2 E

[Σ(t)Σ(s)

]ds

=

∫ t

T0

λ(s)2[Σ

(T0

)2e−Θ(t−s) + Σ

(T0

)2(1 − e−Θ(t−s)

)

+ η2 Σ(T0

) 1 − e−2Θ (s−T0)

2Θe−Θ(t−s)

]ds

= Σ(T0

)2∫ t

T0

λ(s)2 ds + η2 Σ(T0

)e−Θ (t−T0)

∫ t

T0

λ(s)2eΘ (s−T0) − e−Θ(s−T0)

2Θds,

which is, upon replacing λ(s) with λmn(s), the result (3.17) we wanted to derive.

A.5 A recursion scheme for a system of time dependent Ric-

cati equations

In this section we consider the system of Riccati equations

dA

dt

(t, TN

)= α(t)D

(t, TN

),

dD

dt

(t, TN

)= f(t) + Θ D

(t, TN

)+ D

(t, TN

)2,

A(TN , TN

)= D

(TN , TN

)= 0, (A.19)

63

for times t ∈ [T0, TN ], where α : [T0, TN ] → R is a real valued differentiable function, f(t) a

differentiable complex valued function, f : [T0, TN ] → C, and Θ a complex valued constant,

Θ ∈ C. Due to the time dependence of f, the system does not possess an analytic solution

on [T0, TN ]. However, above problem can be approximated by dividing the considered time

span into sub intervals [Ti, Ti+1], i = 0, . . . , N − 1, on each of which the functions α and f

are assumed to take on constant values α(Ti

)and f

(Ti

). As in this case each sub problem

does allow for an analytic solution, the overall solution on time interval [T0, TN ] is obtained

by concatenating the individuals solutions of adjacent sub intervals by imposing separate

matching conditions on coefficients Ai and Bi, respectively. The so defined discretisation

of system (A.19) can be expressed as

dAi

dt

(t, Ti+1

)= α(Ti)Di

(t, Ti+1

), (A.20a)

dDi

dt

(t, Ti+1

)= f(Ti) + Θ Di

(t, Ti+1

)+ Di

(t, Ti+1

)2, (A.20b)

Ai

(Ti+1, Ti+1

)= Ai+1

(Ti+1, Ti+1

), Di

(Ti+1, Ti+1

)= Di+1

(Ti+1, Ti+1

), (A.20c)

AN−1

(TN , TN

)= DN−1

(TN , TN

)= 0, (A.20d)

for t ∈ [Ti, Ti+1] and i = 0, . . . , N − 1 with αi := α(Ti) ≡ const., fi := f(Ti) ≡ const.. We

know derive analytic solutions for functions Ai

(t, Ti+1

)and Di

(t, Ti+1

)on each sub interval.

In doing so we start with equation (A.20b).

A.5.1 An analytic solution for Di

(t, Ti+1

)

In solving (A.20b) on time interval [Ti, Ti+1] we first define the function g(t) by

g(t) = Θ2 − 4 f(t),

and abbreviate gi = g(Ti) ≡ const., since f(Ti) ≡ const.. Furthermore, we introduce the

complex valued constants

d±i =1

2

(Θ ±√

gi

),

and observe the relations

dDi

dt

(t, Ti+1

)= Di

(t, Ti+1

)2+ Θ Di

(t, Ti+1

)+ fi

= Di

(t, Ti+1

)2+ Θ Di

(t, Ti+1

)+

1

4

(Θ2 − gi

)

= Di

(t, Ti+1

)2 −(d+

i + d−i)Di

(t, Ti+1

)+ d+

i d−i

=(Di

(t, Ti+1

)− d+

i

)(Di

(t, Ti+1

)− d−i

), (A.21)

and d+i − d−i =

√gi. (A.22)

Since the derivation of Di

(t, Ti+1

)on time interval [Ti, Ti+1] for non–zero values of gi is

different to the gi = 0 case these two scenarios will be separated.

64

A.5.1.1 The case gi 6= 0

Now, if gi 6= 0 one can consider the total differential√

gi ds and use above formulation to

arrive at the following decomposition,

√gi ds =

√gi

ds

dDidDi =

√gi(

dDi

ds

) dDi =

√gi(

Di − d+i

)(Di − d−i

) dDi by (A.21),

=

(Di − d−i

)−

(Di − d+

i

)(Di − d+

i

)(Di − d−i

) dDi

=

(1

Di − d+i

− 1

Di − d−i

)dDi, (A.23)

which will now be used in integrating over time interval [t, Ti+1] :

√gi

(Ti+1 − t

)=

∫ Ti+1

t

√gi ds

=

∫ Di(Ti+1,Ti+1)

Di(t,Ti+1)

dDi

Di − d+i

−∫ Di(Ti+1,Ti+1)

Di(t,Ti+1)

dDi

Di − d−iby (A.23),

= ln

(Di

(Ti+1, Ti+1

)− d+

i

Di

(t, Ti+1

)− d+

i

)− ln

(Di

(Ti+1, Ti+1

)− d−i

Di

(t, Ti+1

)− d−i

)

= ln

(Di

(Ti+1, Ti+1

)− d+

i

Di

(Ti+1, Ti+1

)− d−i

Di

(t, Ti+1

)− d−i

Di

(t, Ti+1

)− d+

i

). (A.24)

From this result the solution Di

(t, Ti+1

)is obtained. (A.24) is equivalent to

Di

(Ti+1, Ti+1

)− d+

i

Di

(Ti+1, Ti+1

)− d−i

Di

(t, Ti+1

)− d−i

Di

(t, Ti+1

)− d+

i

= e√

gi

(Ti+1−t

)

=⇒ Di

(t, Ti+1

) [1 − Di

(Ti+1, Ti+1

)− d+

i

Di

(Ti+1, Ti+1

)− d−i

e−√

gi

(Ti+1−t

)]

= d+i − d−i

Di

(Ti+1, Ti+1

)− d+

i

Di

(Ti+1, Ti+1

)− d−i

e−√

gi

(Ti+1−t

)

⇐⇒ Di

(t, Ti+1

)=

d+i − d−i

Di

(Ti+1,Ti+1

)−d+

i

Di

(Ti+1,Ti+1

)−d−i

e−√

gi

(Ti+1−t

)

1 − Di

(Ti+1,Ti+1

)−d+

i

Di

(Ti+1,Ti+1

)−d−i

e−√

gi

(Ti+1−t

) ,

=d+

i

(Di

(Ti+1, Ti+1

)− d−i

)− d−i

(Di

(Ti+1, Ti+1

)− d+

i

)e−

√gi (Ti+1−t)

(Di

(Ti+1, Ti+1

)− d−i

)−

(Di

(Ti+1, Ti+1

)− d+

i

)e−

√gi (Ti+1−t)

=d+

i

(Di

(Ti+1, Ti+1

)− d+

i +√

gi

)− (d+

i −√gi)

(Di

(Ti+1, Ti+1

)− d+

i

)e−

√gi (Ti+1−t)

(Di

(Ti+1, Ti+1

)− d+

i

) (1 − e−

√gi (Ti+1−t)

)+√

gi

=d+

i

(Di

(Ti+1, Ti+1

)− d+

i

) (1 −

[1 −

√gi

d+i

]e−

√gi

(Ti+1−t

))+ d+

i

√gi

(Di

(Ti+1, Ti+1

)− d+

i

) (1 − e−

√gi (Ti+1−t)

)+

√gi

,

(A.25)

since d+i − d−i =

√gi, which is the solution for Di

(t, Ti+1

)on interval [Ti, Ti+1] if gi 6= 0.

65

A.5.1.2 The case gi = 0

In the case gi = 0, we have d+i = d−i and (A.21) becomes

dDi

dt

(t, Ti+1

)=

(Di

(t, Ti+1

)− d+

i

)2. (A.26)

Again Di

(t, Ti+1

)is obtained by transforming a time integral into an integral with respect

to D :

(Ti+1 − t

)=

∫ Ti+1

t

ds =

∫ Di(Ti+1,Ti+1)

Di(t,Ti+1)

ds

dDidDi

=

∫ Di(Ti+1,Ti+1)

Di(t,Ti+1)

dDi(Di

(t, Ti+1

)− d+

i

)2 by (A.26)

=1

Di

(t, Ti+1

)− d+

i

− 1

Di

(Ti+1, Ti+1

)− d+

i

=⇒ 1

Di

(t, Ti+1

)− d+

i

=

(Ti+1 − t

) [Di

(Ti+1, Ti+1

)− d+

i

]+ 1

Di

(Ti+1, Ti+1

)− d+

i

, (A.27)

from where Di

(t, Ti+1

)follows:

Di

(t, Ti+1

)= d+

i +Di

(Ti+1, Ti+1

)− d+

i

1 +(Ti+1 − t

)[Di

(Ti+1, Ti+1

)− d+

i

]

=Di

(Ti+1, Ti+1

)+ d+

i

[Di

(Ti+1, Ti+1

)− d+

i

](Ti+1 − t

)

1 +[Di

(Ti+1, Ti+1

)− d+

i

](Ti+1 − t

) , (A.28)

for times t ∈ [Ti, Ti+1].

A.5.2 An analytic solution for Ai

(t, Ti+1

)

As in the derivation of Di

(t, Ti+1

)on time interval [Ti, Ti+1], the derivation of Ai

(t, Ti+1

)

for gi 6= 0 follows different lines as the gi = 0 case which is why these two scenarios will be

treated separately again.

A.5.2.1 The case gi 6= 0

In solving (A.20a) for gi 6= 0 we first introduce the function

mi(t) =Di

(Ti+1, Ti+1

)− d+

i

Di

(Ti+1, Ti+1

)− d−i

e−√

gi

(Ti+1−t

), (A.29)

with which the solution for Di

(t, Ti+1

)can be abbreviated as

Di

(t, Ti+1

)=

d+i − d−i mi(t)

1 − mi(t)(A.30)

66

and which satisfies

dmi(t)

dt=

√gi mi(t). (A.31)

Now the total differential Di

(s, Ti+1

)ds can be written as,

Di

(s, Ti+1

)ds =

d+i − d−i mi(s)

1 − mi(s)

( ds

dmi

)dmi

=d+

i − d−i mi(s)

1 − mi(s)

1√gi mi(s)

dmi by (A.30) and (A.31)

=dmi√

gi

[d+

i

mi(s)(1 − mi(s)

) − d−i1 − mi(s)

]

=dmi√

gi

[d+

i

1 − mi(s) + mi(s)

mi(s)(1 − mi(s)

) − d−i1 − mi(s)

]

=dmi√

gi

[d+

i

1

mi(s)+

d+i − d−i

1 − mi(s)

]

=dmi√

gi

[d+

i

1

mi(s)+

√gi

1 − mi(s)

]from (A.22),

which integrates to

∫ Ti+1

t

Di

(s, Ti+1

)ds =

d+i√gi

∫ mi(Ti+1)

mi(t)

dmi

mi+

∫ mi(Ti+1)

mi(t)

dmi

1 − mi

=d+

i√gi

ln

(mi(Ti+1)

mi(t)

)− ln

(mi(Ti+1) − 1

mi(t) − 1

). (A.32)

But this result leads us to the solution for Ai

(t, Ti+1

), since from (A.20a)

Ai

(t, Ti+1

)= Ai

(Ti+1, Ti+1

)−

∫ Ti+1

t

dAi

(s, Ti+1

)

dsds

= Ai

(Ti+1, Ti+1

)− αi

∫ Ti+1

t

Di

(s, Ti+1

)ds

= Ai

(Ti+1, Ti+1

)− αi

[d+

i√gi

ln

(mi(Ti+1)

mi(t)

)− ln

(mi(Ti+1) − 1

mi(t) − 1

)]

= Ai

(Ti+1, Ti+1

)− αi

[d+

i

(Ti+1 − t

)− ln

(mi(Ti+1) − 1

mi(t) − 1

)], (A.33)

where in the last step the relation mi(Ti+1)mi(t)

= e√

gi(Ti+1−t) was inserted, comp. (A.29).

67

Since

mi(Ti+1) − 1

mi(t) − 1=

Di

(Ti+1,Ti+1

)−d+

i

Di

(Ti+1,Ti+1

)−d−i

− 1

Di

(Ti+1,Ti+1

)−d+

i

Di

(Ti+1,Ti+1

)−d−i

e−√

gi (Ti+1−t) − 1

=Di

(Ti+1, Ti+1

)− d+

i −(Di

(Ti+1, Ti+1

)− d−i

)(Di

(Ti+1, Ti+1

)− d+

i

)e−

√gi (Ti+1−t) −

(Di

(Ti+1, Ti+1

)− d−i

)

=d+

i − d−i

Di

(Ti+1, Ti+1

)(1 − e−

√gi (Ti+1−t

)) +

[d+

i e−√

gi (Ti+1−t) − d+i + d+

i − d−i

]

=

√gi

Di

(Ti+1, Ti+1

)(1 − e−

√gi (Ti+1−t)

)− d+

i

(1 − e−

√gi (Ti+1−t)

)+√

gi

,

and ln(

1x

)= − ln(x), (A.33) simplifies to

Ai

(t, Ti+1

)= Ai

(Ti+1, Ti+1

)− αi

[d+

i

(Ti+1 − t

)

+ ln

(1 +

1√gi

[Di

(Ti+1, Ti+1

)− d+

i

][1 − e−

√gi (Ti+1−t)

])](A.34)

A.5.2.2 The case gi = 0

In solving (A.20a) for gi = 0 we re–define the function mi(t),

mi(t) =(Di

(Ti+1, Ti+1

)− d+

i

) (Ti+1 − t

), (A.35)

with which the solution for Di

(t, Ti+1

)(A.28) can be abbreviated as

Di

(t, Ti+1

)=

Di

(Ti+1, Ti+1

)+ d+

i mi(t)

1 + mi(t)(A.36)

and which satisfies

dmi(t)

dt= d+

i − Di

(Ti+1, Ti+1

). (A.37)

68

Therefore the integral of Di

(t, Ti+1

)over time can be expressed as

∫ Ti+1

t

Di

(s, Ti+1

)ds =

∫ Ti+1

t

Di

(Ti+1, Ti+1

)+ d+

i mi(s)

1 + mi(s)ds

=

∫ mi(Ti+1)

mi(t)

Di

(Ti+1, Ti+1

)+ d+

i mi

1 + mi

ds

dmidmi

=

∫ mi(Ti+1)

mi(t)

Di

(Ti+1, Ti+1

)+ d+

i mi

1 + mi

dmi

d+i − Di

(Ti+1, Ti+1

)

=Di

(Ti+1, Ti+1

)

d+i − Di

(Ti+1, Ti+1

)∫ mi(Ti+1)

mi(t)

dmi

1 + mi

+d+

i

d+i − Di

(Ti+1, Ti+1

)∫ mi(Ti+1)

mi(t)

1 + mi − 1

1 + midmi

=Di

(Ti+1, Ti+1

)

d+i − Di

(Ti+1, Ti+1

) ln

(1 + mi(Ti+1)

1 + mi(t)

)

+d+

i

d+i − Di

(Ti+1, Ti+1

)[mi

(Ti+1

)− mi(t) − ln

(1 + mi(Ti+1)

1 + mi(t)

)]

=Di

(Ti+1, Ti+1

)− d+

i

d+i − Di

(Ti+1, Ti+1

)︸︷︷︸

=(−1)

ln

(1 + mi(Ti+1)

1 + mi(t)

)

︸︷︷︸=ln

(1

1+mi(t)

), since mi(Ti+1) = 0

+d+

i

d+i − Di

(Ti+1, Ti+1

)[

mi(Ti+1)︸︷︷︸=0, from (A.35)

−mi(t)]

= ln(1 + mi(t)

)− d+

i

d+i − Di

(Ti+1, Ti+1

) mi(t)

= ln(1 +

(Di

(Ti+1, Ti+1

)− d+

i

) (Ti+1 − t

))+ d+

i

(Ti+1 − t

), (A.38)

where in the last line the definition of mi(t) was inserted. From this Ai

(t, Ti+1

)in the case

gi = 0 follows by integration of (A.20a):

Ai

(t, Ti+1

)= Ai

(Ti+1, Ti+1

)− αi

∫ Ti+1

t

Di

(s, Ti+1

)ds

= Ai

(Ti+1, Ti+1

)− αi

[ln

(1 +

(Di

(Ti+1, Ti+1

)− d+

i

) (Ti+1 − t

))+ d+

i

(Ti+1 − t

)].

(A.39)

69

A.5.3 Summary of the solution

The system of time dependent Riccati equations (A.20) is solved by

Ai

(t, Ti+1

)= Ai

(Ti+1, Ti+1

)

− αi

d+i

(Ti+1 − t

)+ ln

(1 + 1√

gi

[Di

(Ti+1, Ti+1

)− d+

i

][1 − e−

√gi (Ti+1−t)

])if gi 6= 0,

ln(1 +

(Di

(Ti+1, Ti+1

)− d+

i

) (Ti+1 − t

))+ d+

i

(Ti+1 − t

)if gi = 0,

(A.40)

and

Di

(t, Ti+1

)=

d+i

(Di

(Ti+1,Ti+1

)−d+

i

)(1−

[1−

√gi

d+i

]e−√

gi

(Ti+1−t

))+√

gi

(Di

(Ti+1,Ti+1

)−d+

i

)(1−e−

√gi (Ti+1−t)

)+√

gi

if gi 6= 0,

Di

(Ti+1,Ti+1

)+d+

i

[Di

(Ti+1,Ti+1

)−d+

i

](Ti+1−t

)

1+[Di

(Ti+1,Ti+1

)−d+

i

](Ti+1−t

) if gi = 0,

(A.41)

for times t ∈ [Ti, Ti+1] with i = 1, . . . , N − 1, and complex valued constants

gi = g(Ti

)= Θ2 − 4 f

(Ti

),

d±i =1

2

(Θ ±√

gi

),

=⇒ d+i − d−i =

√gi.

If gi = 0, the relation d+i = d−i = Θ

2 holds. Furthermore,

AN−1

(TN , TN

)= DN−1

(TN , TN

)= 0.

A.6 Derivation of relation (3.28)

In the following we provide the derivation of the result

ϕ(µ∣∣Σt

)= E

[exp

(−µ

∫ Tm

t

f(s)Σs ds

)∣∣∣∣Σt

]= exp

(Aµ,f

(t, Tm

)+ Bµ,f

(t, Tm

)Σt

)

for T0 ≤ t ≤ Tm where f(s) is a real valued function and the coefficients A, B satisfy the

Riccati system (3.29), which is a special case of a relation provided by Duffie et al. in [6]

and for time t = T0 corresponds to expression (3.27).

We begin by taking the total derivative of ϕ(µ∣∣Σ0

)with respect to time. Observing that

the derivative of the conditional probability distribution of the process Σt, p(Σt

∣∣Σ0

), with

respect to time vanishes, i.e.,d p

(Σt

∣∣Σ0

)dt

= 0, the differential operator can be drawn into the

70

expectation and for T0 ≤ t ≤ Tm one obtains the ordinary differential equation (ODE)

dϕ(µ∣∣Σt

)

dt= E

[d

dt

(exp

(−µ

∫ Tm

t

f(s)Σs ds

))∣∣∣∣Σt

]

= E

[(−µ f(t)Σt

)exp

(−µ

∫ Tm

t

f(s)Σs ds

)∣∣∣∣Σt

]

=(−µ f(t)Σt

)ϕ(µ∣∣Σt

),

ϕ(µ∣∣Σ

(Tm

))= 1.

Considering the process


)dt + η

√Σt dVt,

it is obvious that the drift and variance coefficients, Θ(Σ0 − Σt

)and η2 Σt respectively,

are linear in Σt and thus affine. Furthermore, by Ito’s lemma the total differential of any

function of Σt, F(Σt

), is given by

dF =∂F

∂t+

∂F

∂ΣtdΣt +

1

2

∂2F

∂Σ2t

dΣ2t

=

(∂F

∂t+ Θ

(Σ0 − Σt

) ∂F

∂Σt+

η2

2Σt

∂2F

∂Σ2t

)dt + η

√Σt

∂F

∂ΣtdVt.

Applying the obtained differential operator d(·)dt

to above ODE, we see that it is equivalent

to

∂ϕ

∂t+ Θ

(Σ0 − Σt

) ∂ϕ

∂Σt+

η2

2Σt

∂2ϕ

∂Σ2t

+ µf(t)Σt ϕ = 0,

the general solution of which can be written in the form

ϕ(µ∣∣Σt

)= exp

(A

(t, Tm

)+ B

(t, Tm

)Σt

). (A.42)

Upon inserting this ’ansatz’ we arrive at

dA

dt+

dB

dtΣt + Θ

(Σ0 − Σt

)B

(t, Tm

)+

η2

2Σt B

(t, Tm

)2+ µf(t)Σt = 0, (A.43)

which is equivalent to the Riccati system of ordinary differential equations

dA

dt

(t, Tm

)= −Θ Σ0 B

(t, Tm

),

dB

dt

(t, Tm

)= µ f(t) + ΘB

(t, Tm

)− η2

2B

(t, Tm

)2,

A(Tm, Tm

)= B

(Tm, Tm

)= 0. (A.44)

This system of ODEs is exactly the one stated in (3.29), where the dependence of A and B

on µ and f(t) was made explicit.

71

Obviously the system (A.44) corresponds to expression (A.19) with the replacements

TN −→ Tm,

α(t) −→ 2

η2ΘΣ0 =

2

η2Θ, because Σ0 = 1,

Θ −→ Θ,

D(t, Tm) −→ −η2

2B(t, Tm),

f(t) −→ −η2

2µf(t),

and can be discretised analogously to (A.20). Hence its solution on sub intervals [Ti, Ti+1],

i = 1, . . . , m − 1, is given by (A.40) and (A.41), where

gi = g(Ti

)= Θ2 + 2 η2µf

(Ti

),

d±i =1

2

(Θ ±√

gi

),

=⇒ d+i − d−i =

√gi.

A.7 2d–Markov functional integration

Defining zTm :=(xTm , yTm

)Twith zT0

:=(0, 0

)T, the two–dimensional integrals are evalu-

ated. The inner expected value is hence given by

E(zTm) :=

∫

R2

p(zTm+1

∣∣zTm

) 1


dzTm+1

=

∫ ∞

−∞dxTm+1

px

(xTm+1

|xTm

) ∫ ∞

−∞

1

D(Tm+1, TN+1; xTm+1, yTm+1

)py

(yTm+1

|yTm

)dyTm+1

.

(A.45)

The outer expected value is then calculated as∫

x∗p(zTm

∣∣zT0

)E(zTm) dzTm

= 1 −∫ z∗

−∞dxTm px

(xTm |0

) ∫ z∗−xTm

−∞E(xTm , yTm) py

(yTm |0

)dyTm , (A.46)

where the boundary value z∗ is defined by

z∗ = Fm(Tm; (xTm , 0))−1∣∣x∗= ln

( x∗

Fm(T0)e−

∫ TmT0

µm(s) ds)

with the integrand of expression (4.11) as drift term µm(s). The conditional probability

densities are given by Gaussian normal distributions:

px

(xTm+1

∣∣xTm

)=

1√2πσx(Tm, Tm+1)

exp

(−1

2

(xTm+1− xTm)2

σx(Tm, Tm+1)2

),

py

(yTm+1

∣∣yTm

)=

1√2πσy(Tm, Tm+1)

exp

(−1

2

(yTm+1− yTm)2

σy(Tm, Tm+1)2

),

72

with variances

σx(Tm, Tm+1)2 =

∫ Tm+1

Tm

Γ(s)2 (βm+1(s)λm+1(s))2 ds, (A.47)

σy(Tm, Tm+1)2 =

∫ Tm+1

Tm

(1 − Γ(s)2

)(βm+1(s)λm+1(s))

2 ds (A.48)

from expression (4.12).

73

Appendix B

The Heston Model

This chapter is dedicated to Hestons’s stochastic volatility model [11]. Although the model

was originally introduced in the context of option pricing for equities, we present it as a

model for forward swap rates. The main reason for this is that Piterbarg’s Libor Market

Model (FL–TSS) is calibrated to market derived Heston parameters for european swaptions.

Thus in the context of Piterbarg FL–TSS model calibration takes place in the parameter

domain and for this the existence of a calibrated Heston model is a prerequisite.

B.1 Specification of the model dynamics

Let S(t) denote the time t ≥ T0 par rate of a (forward) swap maturing at time T > t, the

final time. Then the swap rate is modelled as a displaced diffusion process with stochastic

volatility,

dS(t) =(β S(t) + (1 − β)S(T0)

)λ√

Σt dWt, (B.1a)


)dt + η

√Σt dVt, (B.1b)

⟨dWt, dVt

⟩= ρ dt, (B.1c)

for T0 ≤ t ≤ T, where the constants β and λ denote the skew paramter and volatility

λ. The dynamics of the volatility level Σt is given by a CEV process with start value

Σ0 = Σ(T0

)= 1. Whereas above model specificaton allows five degrees of freedom, namely

the swap rate skew β, the swap rate volatility λ, the mean reversion speed Θ, the volatility of

variance η and the correlation between the Brownian drivers1 of the swap rate and variance

processes ρ, we restrict the parameter space to the set of tuples (β, λ). Thus in calibrating

1Both Brownian motions dWt and dVt are considered under the swap measure induced by taking thepresent value of a basis point as numeraire.

75

the model to european swaption prices the mean reversion and volatility of variance as well

as the correlation will be kept constant across swaption strikes and expiries2.

The model dynamics (B.1) is equivalent to

dS(t) = d(S(t) + γ

)=

(S(t) + γ

)βλ

√Σt dWt, (B.2a)


)dt + η

√Σt dVt, (B.2b)

⟨dWt, dVt

⟩= ρ dt, (B.2c)

γ =1 − β

βS(T0), (B.2d)

for T0 ≤ t ≤ T, where γ denotes the constant displacement parameter, comp. [23]. From

(B.2) it is obvious that the stochastic variable S(t) := S(t) + γ follows a lognormal process

with stochastic volatility βλ√

Σt.

The value of any derivative product under this model depends on the conditional dis-

tribution of the stochastic process (S(t), Σt)t≥T0 . However, for valuation purposes it is

more convenient to consider the stochastic process x(t)t≥T0 := (ln S(t), Σt)t≥T0 instead

of (S(t), Σt)t≥T0 itself. Since x(t) can be cast into the form

dx(t) =

[(0

ΘΣ0

)−

(12(λβ)2

Θ

)Σt

]

︸︷︷︸=:µ(xt)

dt +√

Σt

(λβρ λβ

√1 − ρ2

η 0

)

︸︷︷︸=:σ(xt)

·(

dVt

dZt

), (B.3)

where dZt is a Brownian motion orthogonal to dVt which facilitates the decomposition

dWt = ρ dVt +√

1 − ρ2 dZt

(→ dW 2

t = dt and⟨dWt, dVt

⟩= ρ dt, since

⟨dVt, dZt

⟩= 0

), it is

an affine process because µ(xt) and σ(xt)Tσ(xt) are affine functions of xt.

Expression (B.3) involves the dynamics of ln S(t) which is obtained by applying Ito’s

lemma,

d ln S(t) =∂ ln S(t)

∂S(t)dS(t) +

1

2

∂2 ln S(t)

∂S(t)2dS(t)2

=1

S(t)

[S(t)βλ

√Σt dWt

]− 1

2

1

S(t)2

[S(t)βλ

√Σt dWt

]2

= −1

2(λβ)2Σt dt + λβ

√Σt dWt, (B.4)

since dW 2t = dt. Furthermore we observe that

⟨dΣt, d ln S(t)

⟩= (λβη)Σt ρ dt, (B.5)

since dWt dt = 0 = dVt dt = 0 = dt2. This will now be used to derive a differential equation

for the characteristic function of the Heston model.

2The restriction to the mentioned parameter space corresponds to Piterbarg’s approach in [19] and [20].There also a zero correlation between the driving Brownian motions is assumed. Nonetheless, we includethe case of non–zero correlation for completeness of exposure.

76

B.2 The characteristic function

The characteristic function is defined as the fourier transform of the conditional probability

density. For the stochastic process xt = (ln St, Σt) the conditional probability distribution is

denoted as p(xu|xt), T0 ≤ t < u ≤ T, and the characteristic function at time u conditioned

on xt is written as

ϕtu(s|xt) =

∫

R2

eis·xup(xu|xt) dxu = E[eis·xu

∣∣xt

], (B.6)

where in the last step the interpretation of ϕtu(s|xt) as conditional expected value of the

function eis·xu was established. An application of the tower law for conditional expectations

shows that

ϕtu(s|xt) = E[eis·xu

∣∣xt

]= E

[E

[eis·xu

∣∣xv

]∣∣xt

]= E

[ϕvu(s|xv)

∣∣xt

], T0 ≤ t ≤ v ≤ u ≤ T,

(B.7)

which indicates that ϕtu(s|xt) as stochastic function of xt is a martingale. Therefore it has

to be driftless and the relation dϕtu(s|xt)dt

= 0 must hold. From this requirement an ordinary

differential equation for the characteristic function is derived by application of Ito’s lemma,

dϕtu(s|xt) =∂ϕtu

∂tdt +

∂ϕtu

∂ΣtdΣt +

∂ϕtu

∂ ln St

d(ln St)

+1

2

(∂2ϕtu

∂Σ2t

(dΣt)2 + 2

∂2ϕtu

∂Σt∂(ln St)(dΣt)(d(ln St)) +

∂2ϕtu

∂(ln St)2(d(ln St))

2

)

=

[∂ϕtu

∂t+ Θ

(Σ0 − Σt

)∂ϕtu

∂Σt− 1

2(λβ)2Σt

∂ϕtu

∂ ln St

+1

2η2Σt

∂2ϕtu

∂Σ2t

+ (λβη)Σt ρ∂2ϕtu

∂Σt∂(ln St)+

1

2(λβ)2Σt

∂2ϕtu

∂(ln St)2

]dt

+ Brownian motion terms,

where relations (B.4) and (B.5) were used, and from which the ODE follows:

0 =∂ϕtu

∂t+ Θ

(Σ0 − Σt

)∂ϕtu

∂Σt− 1

2(λβ)2Σt

∂ϕtu

∂ ln St

+1

2η2Σt

∂2ϕtu

∂Σ2t

+ (λβη)Σt ρ∂2ϕtu

∂Σt ∂(ln St)+

1

2(λβ)2Σt

∂2ϕtu

∂(ln St)2

(T0 ≤ t ≤ T

)

(B.8)

From the analysis in section B.1 it is apparent that the process xt is affine. The general

form of the characteristic function for processes of this kind at final time T is

ϕtT (s|xt) = exp(As(t, T ) + Ms(t, T ) · xu

), (B.9)

with time dependent functions A : R2×[T0, T ] → R, (s, t) 7→ As(t, T ), and M : R2×[T0, T ] →R2, (s, t) 7→ Ms(t, T ), comp. [6], which will be further explored in the next section where

the Heston ODE (B.8) is solved.

77

B.3 The solution of the Heston ODE

In accordance with (B.9) we define M(s) = Ms :=(Bs, Cs

)and make the ’ansatz’

ϕtT (s|xt) = exp(As(t, T ) + Ms(t, T ) · xt

)= exp

(As(t, T ) + Bs(t, T ) ln St + Cs(t, T )Σt

),

(B.10)

where the scalar product was expanded in the second step. Upon insertion into (B.8) we

obtain

dAs

dt+

dBs

dtln St +

dCs

dtΣt + Θ

(Σ0 − Σt

)Cs(t, T ) − 1

2(λβ)2Σt Bs(t, T )

+1

2η2Σt Cs(t, T )2 + (λβη)Σt ρ Bs(t, T )Cs(t, T ) +

1

2(λβ)2Σt Bs(t, T )2 = 0

⇐⇒dAs

dt+ ΘΣ0 Cs(t, T )

+

[dCs

dt− Θ Cs(t, T ) +

η2

2Cs(t, T )2 +

1

2(λβ)2

(Bs(t, T )2 − Bs(t, T )

)

+ (λβη)ρ Bs(t, T )Cs(t, T )

]Σt

+dBs

dtln St = 0,

from which it is obvious that above relation can only be fulfilled for all times t ∈ [T0, T ] and(ln St, Σt

)if each row vanishes separately. This results in the following system of ordinary

differential equations,

dAs

dt= −ΘΣ0 Cs(t, T ), (B.11a)

dCs

dt= ΘCs(t, T ) − η2

2Cs(t, T )2 − 1

2(λβ)2

(Bs(t, T )2 − Bs(t, T )

)

− (λβη)ρ Bs(t, T )Cs(t, T ), (B.11b)

dBs

dt= 0, (B.11c)

for t ∈ [T0, T ] and from (B.11c) it is clear that Bs(t, T ) ≡ const..

B.3.1 Boundary conditions

From the definition of the characteristic function we know that at final time T

ϕTT (s|xT ) = E[eis·xT

∣∣xT

]= eis·xT = eis1 ln ST +is2ΣT , where s := (s1, s2)

T,

and ϕTT (s|xT ) = exp(As(T, T ) + Bs(T, T ) ln ST + Cs(T, T )ΣT

)by (B.10).

78

Thus matching of exponents results in

As(T, T ) = 0, (B.12a)

Bs(T, T ) = is1, (B.12b)

Cs(T, T ) = is2, (B.12c)

and since Bs(t, T ) ≡ const. on [T0, T ] it follows from (B.12b) that

Bs(t, T ) = is1 for all t ∈ [T0, T ]. (B.13)

B.3.2 A system of Riccati ODEs

Hence the characterisitc function of the Heston model at time t is given by

ϕtT (s1, s2|xt) = exp(A(s1,s2)(t, T ) + is1 ln St + C(s1,s2)(t, T )Σt

), (B.14)

and the system (B.11) transforms into

dA(s1,s2)

dt= −ΘΣ0 C(s1,s2)(t, T ), (B.15a)

dC(s1,s2)

dt=

(Θ − is1(λβη)ρ

)C(s1,s2)(t, T ) − η2

2C(s1,s2)(t, T )2 +

1

2(λβ)2

(s2

1 + is1

), (B.15b)

B(s1,0) = is1. (B.15c)

But this systems corresponds to the Riccati equations (A.19) discussed in appendix A

with the following correspondents,

T → TN ,

D(t, T

)→ −η2

2C(s1,s2)(t, T ),

α(t) → 2

η2ΘΣ0 ≡ const.,

f(t) → −η2

4(λβ)2

(s2

1 + is1

)≡ const.,

Θ → Θ − is1(λβη)ρ,

for t ∈ [T0, T ]. We therefore obtain A(s1,s2)

(t, T

)and C(s1,s2)

(t, T

)for times T0 ≤ t ≤ T by

reference to (A.40) and (A.41) and application of these discrete solutions to time interval

[T0, TN = T ]. As there are no sub intervals in this case, we have i = 0 and the replace-

ment Ti+1 → TN results in the correspondents A0

(t, TN

)→ A(s1,s2)

(t, T

), D0

(t, TN

)→

79

D(s1,s2)

(t, T

)= −η2

2 C(s1,s2)

(t, T

). Recalling that A(T, T ) = 0 and substituting parameters

as described above we thus obtain,

A(s1,s2)

(t, T

)=

− 2η2 ΘΣ0

[d+

0 (s1)(T − t

)+ ln

(1 + 1√

g0(s1)

[−iη2

2 s2 − d+0 (s1)

][1 − e−

√g0(s1) (T−t)

])]

if g0(s1) 6= 0,

− 2η2 ΘΣ0

[ln

(1 +

(−iη2

2 s2 − d+0 (s1)

) (T − t

))+ d+

0 (s1)(T − t

)]

if g0(s1) = 0,

(B.16)

and

C(s1,s2)

(t, T

)=

− 2η2 d+

0 (s1)

(−i η2

2s2−d+

0 (s1))(

1−[1−

√g0(s1)

d+0 (s1)

]e−√

g0(s1)

(T−t

))+√

g0(s1)

(−i η2

2s2−d+

0 (s1))(

1−e−√

g0(s1) (T−t)

)+√

g0(s1)

if g0(s1) 6= 0,

− 2η2

−i η2

2s2+d+

0 (s1)[−i η2

2s2−d+

0 (s1)](

T−t)

1+[−i η2

2s2−d+

0 (s1)](

T−t)

if g0(s1) = 0,

(B.17)

for times T0 ≤ t ≤ T and complex valued constants

g0(s1) = gs1

(T0

)= Θ2

s1− 4 fs1

(T0

)=


)2+ η2(λβ)2

(s2

1 + is1

), (B.18)

d±0 (s1) =1

2

(Θs1 ±

√g0(s1)

)=

1

2

(Θ − is1(λβη)ρ ±

√g0(s1)

). (B.19)

If g0(s1) = 0, the relation d+0 (s1) = d−0 (s1) =

Θs12 holds.

With above solutions the characeristic function of the Heston model is

ϕtT (s1, s2|xt) = exp(A(s1,s2)(t, T ) + is1 ln St + C(s1,s2)(t, T )Σt

), (B.20)

for T0 ≤ t ≤ T.

B.4 Option pricing by transformation techniques

In this section we want to present the valuation of european swaptions within the Heston

model. In turns out that prices of european options can be obtained by evaluation of an

integral over the product of the characteristic function with the fourier transformed payoff.

Now, if f(xT ) denotes a payoff function at time T with respect to the stochastic variable

xT = (ln ST , ΣT ) its fourier transform f(s), s ∈ R2, is defined by

f(s) =

∫

R2

ei s·xT f(xT ) dxT , (B.21a)

f(xT ) =1

(2π)2

∫

R2

e−i s·xT f(s) ds. (B.21b)

80

Moreover, since the characteristic function (B.6) is the fourier transform of the probabiliy

density function p(xT |xt), the relation

p(xT |xt) =1

(2π)2

∫

R2

e−i s·xT ϕtT (s|xt) ds (B.22)

holds.

With the necessary tools at hand we can proceed to calculate the expected value of the

payoff at time T,

E[f(xT )

∣∣xt

]=

∫

R2

f(xT ) p(xT |xt) dxT

=

∫

R2

f(xT )

[1

(2π)2

∫

R2

e−i s·xT ϕtT (s|xt) ds

]dxT

=

∫

R2

[1

(2π)2

∫

R2

e−i s·xT f(xT ) dxT

]ϕtT (s|xt) ds

=1

(2π)2

∫

R2

f(−s)ϕtT (s|xt) ds, (B.23)

where the right hand side can obviously be interpreted as fourier inversion of the function

f(−s)ϕtT (s|xt) ei s·xT .

We now turn to the valuation of european (payer) swaptions of strike K which certify

the right to enter a (payer) swap at time T whose par rate is then given by S(T ). Hence

the swaption payoff at time T is defined as

max(S(T ) − K, 0

)P (T ) ≡

(S(T ) − K

)+P (T )

=((S(T ) − γ) − (K − γ)︸︷︷︸

=:K

)+P (T )

=(S(T ) − K

)+P (T ) by definition of the displaced diffusion

=(eln S(T ) − eln K

)+P (T ), (B.24)

where P (T ) denotes the present value of a basis point of the underlying swap at time T

which will be used as numeraire.

By the fundamental theorem of asset pricing, in a complete economy all numeraire

rebased assets are martingales and therefore the value of a derivative V (t) with respect to

the numeraire P (t) is given by, comp. [10],

V (t)

P (t)= E

[V (T )

P (T )

∣∣∣∣Ft

].

Since at option expiry T the derivative value equals the payoff, application of the funda-

mental theorem of asset pricing to an european swaption results in the relation

V(ln St, ln K

):= V (t) = P (t) E

[(eln S(T ) − eln K

)1

ln ST >ln K

∣∣∣ln S(t)], (B.25)

81

since (B.2) is a Markov process and because the payoff function is f((

ln St, 0)T)

= f(xT ) ≡V (T ) =

(eln S(T ) − eln K

)P (T ).

So in order to calculate the derivative value one has to evaluate the expected value of

the numeraire rebased payoff function which is calculated by integrating over the product

of its fourier transform and the characteristic function according to (B.23). We therefore

calculate the fourier transform f(s) of the numeraire rebased payoff function f(xT ) := f(xT )P (T )

and in doing so we split the latter in an asset or nothing and a cash or nothing component,

f(xT ) =(eln ST − eln K

)1

ln ST >ln K

= eln ST 1ln ST >ln K

− eln K1ln ST >ln K

:= fAN(xT ) − fCN(xT ).

Starting with the asset or nothing component we have

fAN(s) =

∫

R2

ei s·xT fAN(xT ) dxT

=

∫

R

dΣT

∫

R

eis1 ln ST +is2ΣT eln ST 1ln ST >ln K

d(ln ST

)

=

∫

R

eis2ΣT dΣT

∫ ∞

ln K

e(is1+1) ln ST d(ln ST

)

=

∫

R

eis2ΣT dΣT

︸︷︷︸=2π δ(s2)

[∫ ∞

−∞ei(s1−i) ln ST d

(ln ST

)

︸︷︷︸=2π δ(s1−i)

−∫ ln K

−∞e(is1+1) ln ST d

(ln ST

)]

= 2π δ(s2)

[2π δ(s1 − i) − e(is1+1) ln K

is1 + 1

],

since e(is1+1) ln ST → 0 for ln ST → −∞, and by definition of the delta distribution, δ(x) =12π

∫R

eikx dk for x ∈ C.

Now we turn to the fourier transform of the cash or nothing component. Observing that

∫ ln K

∞eis1 ln St d(ln St) = lim

ǫ→0+

[∫ ln K

∞e(is1+ǫ) ln St d(ln St)

]

= limǫ→0+

[e(is1+ǫ) ln St

is1 + ǫ

∣∣∣∣ln K

−∞

]

= limǫ→0+

[e(is1+ǫ) ln K

is1 + ǫ− lim

ln St→−∞

(e(is1+ǫ) ln St

is1 + ǫ

)

︸︷︷︸=0

]

=eis1 ln K

is1,

82

we obtain

fCN(s) =

∫

R2

ei s·xT fCN(xT ) dxT

= eln K

∫

R

dΣT

∫

R

eis1 ln ST +is2ΣT 1ln ST >ln K

d(ln ST

)

= eln K

∫

R

eis2ΣT dΣT

︸︷︷︸=2π δ(s2)

[∫ ∞

−∞eis1 ln ST d

(ln ST

)

︸︷︷︸=2π δ(s1)

−∫ ln K

−∞eis1 ln ST d

(ln ST

)]

= 2π eln K δ(s2)

[2π δ(s1) −

eis1 ln K

is1

].

With the fourier transforms at hand we calculate the swaption value at time t according

to (B.23):

Vt

(ln St, ln K; T

)

P (t)= E

[(eln S(T ) − eln K

)1

ln ST >ln K

∣∣∣ln S(t)]

=1

(2π)2

∫

R2

(fAN(−s) − fCN(−s)

)ϕtT (s|xt) ds

=1

2π

∫ ∞

−∞ds1

∫ ∞

−∞ds2 δ(−s2)

[2π δ(−s1 − i) − e(1−is1) ln K

1 − is1

]ϕtT (s1, s2|xt)

− 1

2π

∫ ∞

−∞ds1

∫ ∞

−∞ds2 eln K δ(−s2)

[2π δ(−s1) +

e−is1 ln K

is1

]ϕtT (s1, s2|xt)

=(−1) eln K

[ϕtT (0, 0|xt)

+1

2π

∫ ∞

−∞ds1 e−is1 ln K

[1

1 − is1+

1

is1

]ϕtT (s1, 0|xt)

]

=(−1) eln K

[ϕtT (0, 0|xt)

+1

2π

∫ ∞

−∞ds1 e−is1 ln K

[1

s21 + 1

− is1

s21

(1 + s2

1

)]

ϕtT (s1, 0|xt)

],

(B.26)

where the term involving the delta distribution δ(−s1 − i) vanishes because integration is

done over the real axis.

From (B.18) we observe that

g0(s1) =(Θ − is1(λβη)ρ

)2+ η2(λβ)2

(s2

1 + is1

)

=[Θ2 + (λβη)2

(1 − ρ2

)s2

1

]︸︷︷︸

=:R(s1)

+i (λβη)((λβη) − 2Θρ

)

︸︷︷︸=:I

s1

= R(s1) + iIs1 =√

R(s1)2 + I2 eiϕ(s1) with ϕ(s1) = arctan

(Is1

R(s1)

).

83

Since R(s1) = R(−s1) and arctan (−x) = − arctan (x) for all x ∈ R, it follows that

g0(s1)∗ = g0(−s1),

and(√

g0(s1))∗

=√

R(s1)2 + I2 e−iϕ(s1)

2 =√

R(−s1)2 + I2 eiϕ(−s1)

2 =√

g0(−s1).

Similarly by reference to (B.19),

d±0 (s1) =1

2


√g0(s1)

)

=1

2

[Θ ±

√R(s1)2 + I2 cos

(ϕ(s1)

2

)]− i

1

2

[(λβη)ρ s1 ∓

√R(s1)2 + I2 sin

(ϕ(s1)

2

)],

and because R(s1) = R(−s1), cos (−s1) = cos (s1), sin(ϕ(−s1)

2

)= sin

(−ϕ(s1)

2

)= − sin

(ϕ(s1)2

),

we have

ℜ(d±0 (s1)

)= ℜ

(d±0 (−s1)

), ℑ

(d±0 (s1)

)= −ℑ

(d±0 (−s1)

),

d±0 (s1)∗ = d±0 (−s1). (B.27)

Thus g0(s1),√

g0(s1) and d±0 (s1) all have symmetric real and antisymmetric imaginary parts.

Furthermore for any pair of complex numbers with symmetric real and antisymmetric

imaginary parts, E(s1) = |E(s1)|eiϕ1(s1) and F (s1) = |F (s1)|eiϕ2(s1), the relations

E(s1)∗ = E(−s1), F (s1)

∗ = F (−s1),(E(s1)

F (s1)

)∗=

|E(s1)||F (s1)|

e−i(ϕ1(s1)−ϕ2(s1)) =|E(−s1)||F (−s1)|

ei(ϕ1(−s1)−ϕ2(−s1)) =E(−s1)

F (−s1),

(lnE(s1)

)∗= ln

(|E(s1)|

)− iϕ1(s1) = ln

(|E(−s1)|

)+ iϕ1(−s1) = lnE(−s1)

hold. Thus(

d+0 (s1)√g0(s1)

)∗=

d+0 (−s1)√g0(−s1)

,

and(e−

√g0(s1) (T−t)

)∗= e−

(√g0(s1)

)∗(T−t) = e−

√g0(−s1) (T−t).

With these relations at hand we observe that the coefficients (B.16) and (B.16) evaluated

at s2 = 0,

A(s1,0)

(t, T

)=

− 2η2 ΘΣ0

[d+

0 (s1)(T − t

)+ ln

(1 − 1√

g0(s1)d+

0 (s1)[1 − e−

√g0(s1) (T−t)

])]

if g0(s1) 6= 0,

− 2η2 ΘΣ0

[ln

(1 − d+

0 (s1)(T − t))

+ d+0 (s1)(T − t)

]

if g0(s1) = 0,

84

and

C(s1,0)

(t, T

)=

− 2η2 d+

0 (s1)d+0 (s1)

(1−

[1−

√g0(s1)

d+0 (s1)

]e−√

g0(s1)

(T−t

))−√

g0(s1)

d+0 (s1)

(1−e−

√g0(s1) (T−t)

)−√

g0(s1)

if g0(s1) 6= 0,2η2

d+0 (s1)2(T−t)

1−d+0 (s1)(T−t)

if g0(s1) = 0,

,

for times T0 ≤ t ≤ T and complex valued constants

g0(s1) = gs1

(T0

)= Θ2

s1− 4 fs1

(T0

)=


)2+ η2(λβ)2

(s2

1 + is1

), (B.28)

d±0 (s1) =1

2

(Θs1 ±

√g0(s1)

)=

1

2


√g0(s1)

), (B.29)

satisfy

A(s1,0)

(t, T

)∗= A(−s1,0)

(t, T

), C(s1,0)

(t, T

)∗= C(−s1,0)

(t, T

),

which is a behaviour that transfers to the characteristic function (B.20) evaluated at s2 = 0,

ϕtT (s1, 0|xt)∗ = exp

(A(s1,0)(t, T ) + is1 ln St + C(s1,0)(t, T )Σt

)∗

= exp(A(s1,0)(t, T )∗ − is1 ln St + C(s1,0)(t, T )∗ Σt

)

= exp(A(−s1,0)(t, T ) + i(−s1) ln St + C(−s1,0)(t, T )Σt

)

= ϕtT (−s1, 0|xt).

As a consequence, for any complex valued function f(s1) with symmetric real and antisym-

metric imaginary part, i.e., f(s1)∗ = f(−s1),

∫ ∞

−∞ϕtT (s1, 0|xt)f(s1) ds1 =

∫ ∞

0

[ϕtT (s1, 0|xt)

∗f(s1)∗ + ϕtT (s1, 0|xt)f(s1)

]ds1

= 2

∫ ∞

0ℜ

(ϕtT (s1, 0|xt)f(s1)

)ds1.

Applying this result to equation (B.26) for the numeraire rebased swaption value one

obtains

Vt

(ln St, ln K; T

)

P (t)=

(−1) eln K

[ϕtT (0, 0|xt)

+1

2π

∫ ∞

−∞e−is1 ln K

[1

s21 + 1

− is1

s21

(1 + s2

1

)]

ϕtT (s1, 0|xt) ds1

],

=(−1) eln KϕtT (0, 0|xt)

− 1

πeln K

∫ ∞

0ℜ

(e−is1 ln K

[1

s21 + 1

− is1

s21

(1 + s2

1

)]

ϕtT (s1, 0|xt)

)ds1.

(B.30)

This equation establishes the link between the characteristic function and swaption prices.

In computations the integral in (B.30) has to be evaluated numerically.

85

B.5 Calibration of the Heston Model

As mentioned at the beginning we restrict the parameter space of the time dependent Heston

model to the set of tuples (β, λ) while keeping the mean reversion Θ, the volatility of

variance η and the correlation ρ of the Brownian drivers at fixed values. Hence for each

maturity T the swaption values given by (B.30) depend on the tuples (βT , λT ), namely

the swaption skew and volatility level through the variables g0(s1) and d+0 (s1), which both

enter the characteristic function. Thus the value of a swaption expiring at Ti with strike K

can be expressed as

Vt

(ln St, ln K; Ti

)≡ Vt

(ln St, ln K; Ti, (βTi

, λTi))

= Vt

(ln(St + γTi

), ln(K + γTi); Ti, (βTi

, λTi)),

and the tuple (βTi, λTi

) can be obtained by fitting expression (B.30) with T = Ti to swaption

market prices for a series of strikes Kij , VMarket(K

ij ; Ti). For a number of N market quotes

corresponding to strikes Ki1, . . . , K

iN this is done by minimising

Mi =N∑

j=1

w(Kij)

(VMarket(K

ij ; Ti) − Vt

(ln(St + γTi

), ln(Kij + γTi

); Ti, (βTi, λTi

)))2

, (B.31)

where the weight function w(Kij) is chosen as a monotone decreasing function of the mon-

eyness. We will use an exponential weight function of the form

w(Kij) = exp

(−ξ (St − Ki

j)2), ξ > 0,

from which it is obvious that strikes Kij which are further away from the ATM-point St =

S(t) = Kij are assigned lower weights. Thereby it becomes apparent that for each swaption

maturity Ti the tuple (βTi, λTi

) captures the Black76–volatility smile which is encoded in

the strikes Kij , j = 1, . . . , N.

Based on this fitting procedure the calibration of the model is performed for the single

swaption expiries Ti according to a bootstrapping algorithm, comp. [8]. Starting with

swaptions expiring at T1 the tuple (βT1, λT1

) is determined by minimisation of M1 according

to (B.31). These values are then used in the next time step T2 where they enter the

recursion scheme for the calculation of the characteristic function ϕtT2on time interval

[T0, T2] according to (A.40) and (A.41). There market data of swaptions expiring at T2

for various strikes K2j , j = 1, . . . , N, are needed, and the tuple (βT2

, λT2) is obtained by

minimisation of M2. Continuing along these lines for swaption expiries T3, . . . , TN results in

a calibrated Heston model on time interval [T0, TN ].

86

Appendix C

Tables and figures

Table C.1: EUR term structure at T0 = May 7th, 2008 for maturities Ti = i y.

Maturity Discount Factor

T1 0.95958686

T2 0.92140904

T3 0.88521045

T4 0.85033692

T5 0.81669553

T6 0.78418277

T7 0.75260437

T8 0.72171881

T9 0.69149311

T10 0.66207251

T11 0.63345619

Table C.2: Euribor–1y forward rates Fi(T0; Ti, Ti+1) at T0 = May 7th, 2008 for reset timesTi = i y and i = 1, . . . , 10.

Forward rate Forward rate value

F1 0.04194355

F2 0.04169943

F3 0.04146102

F4 0.0412333

F5 0.04105982

F6 0.04095103

F7 0.04089707

F8 0.04118552

F9 0.0409216

F10 0.04087711

87

Table C.3: Piecewise constant skew and volatility term structure of forward rates Fi(t) fortimes Tj with T0 = 0 y ≤ Tj < Ti = i y with i = 1, . . . , 10 (rounded to three decimal places).Initially the skew/volatility was set to 0.5/0.02.

Skew/Time T0 T1 T2 T3 T4 T5 T6 T7 T8 T9

β1 0.503 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β2 0.456 0.515 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β3 0.396 0.439 0.545 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β4 0.369 0.346 0.436 0.577 0.5 0.5 0.5 0.5 0.5 0.5

β5 0.315 0.298 0.301 0.448 0.653 0.5 0.5 0.5 0.5 0.5

β6 0.275 0.203 0.226 0.284 0.457 0.659 0.5 0.5 0.5 0.5

β7 0.232 0.235 0.094 0.147 0.244 0.549 0.66 0.5 0.5 0.5

β8 0.262 0.191 0.206 -0.06 0.194 0.311 0.501 0.744 0.5 0.5

β9 0.288 0.211 0.145 0.132 -0.314 0.26 0.372 0.511 0.73 0.5

β10 0.167 0.197 0.137 0.066 0.137 -0.417 0.262 0.436 0.521 0.752

Volatility/Time T0 T1 T2 T3 T4 T5 T6 T7 T8 T9

λ1 0.113 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ2 0.132 0.119 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ3 0.133 0.138 0.12 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ4 0.138 0.133 0.14 0.123 0.02 0.02 0.02 0.02 0.02 0.02

λ5 0.144 0.133 0.132 0.142 0.13 0.02 0.02 0.02 0.02 0.02

λ6 0.123 0.158 0.129 0.133 0.14 0.127 0.02 0.02 0.02 0.02

λ7 0.124 0.124 0.16 0.126 0.13 0.13 0.128 0.02 0.02 0.02

λ8 0.11 0.129 0.131 0.157 0.119 0.128 0.128 0.126 0.02 0.02

λ9 0.115 0.121 0.123 0.132 0.143 0.123 0.124 0.13 0.125 0.02

λ10 0.112 0.122 0.12 0.122 0.129 0.135 0.117 0.121 0.135 0.124

88

0.042 0.043 0.044 0.045 0.046 0.047

1 1)T(F

0.0

0.2

0.4

0.6

0.8

1.0

111

11))

T(F;

T,T(

D

Figure C.1: The numeraire discount bond asfunctional of F1(T1, zT1): D(T1, T11; F1(T1, zT1)).

0.042 0.044 0.046 0.048 0.050 0.052 0.054

3 3)T(F

0.0

0.2

0.4

0.6

0.8

1.0

311

33))

T(F;

T,T(

D


89

0.042 0.044 0.046 0.048 0.050

5 5)T(F

0.0

0.2

0.4

0.6

0.8

1.0

511

55))

T(F;

T,T(

D


0.040 0.045 0.050 0.055

7 7)T(F

0.0

0.2

0.4

0.6

0.8

1.0

711

77))

T(F;

T,T(

D

Figure C.4: The numeraire discountbond as functional of F10(T7, zT7):D(T7, T11; F7(T7, zT7)).

90

Table C.4: Digital caplet (on Euribor–1y) implied volatilities and (derived) numeraire re-

based prices V (T0;Ti,K)D(T0,T11) for expiries Ti = i y and absolute moneyness Mi = Fi(T0; Ti, Ti+1)−K

for i = 1, . . . , 10. Implied volatilities were observed on T0 = May 7th, 2008.

Expiry T1 T2 T3 T4 T5

Mi Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price

-0.025 0.189 1.455 0.199 1.396 0.203 1.333 0.206 1.263 0.21 1.19

-0.023 0.175 1.455 0.185 1.394 0.189 1.325 0.192 1.249 0.196 1.169

-0.02 0.163 1.455 0.173 1.39 0.177 1.312 0.18 1.227 0.183 1.142

-0.018 0.153 1.454 0.162 1.381 0.166 1.29 0.169 1.197 0.172 1.107

-0.015 0.143 1.453 0.152 1.362 0.156 1.257 0.159 1.155 0.162 1.062

-0.013 0.135 1.447 0.144 1.328 0.148 1.207 0.151 1.099 0.154 1.005

-0.01 0.128 1.426 0.137 1.267 0.14 1.134 0.143 1.026 0.146 0.935

-0.008 0.122 1.366 0.131 1.168 0.134 1.035 0.137 0.934 0.14 0.851

-0.005 0.118 1.231 0.126 1.027 0.129 0.91 0.132 0.825 0.135 0.756

-0.003 0.114 0.996 0.122 0.848 0.125 0.766 0.128 0.704 0.131 0.653

0.0 0.112 0.695 0.12 0.651 0.123 0.614 0.125 0.58 0.128 0.549

0.003 0.111 0.41 0.119 0.466 0.121 0.471 0.124 0.463 0.126 0.449

0.005 0.111 0.206 0.118 0.312 0.121 0.346 0.123 0.358 0.125 0.36

0.008 0.111 0.09 0.118 0.198 0.121 0.247 0.123 0.272 0.125 0.284

0.01 0.112 0.036 0.119 0.121 0.121 0.172 0.123 0.203 0.125 0.222

0.013 0.113 0.013 0.12 0.073 0.122 0.119 0.124 0.151 0.126 0.173

0.015 0.115 0.005 0.122 0.043 0.124 0.082 0.126 0.112 0.127 0.134

0.018 0.117 0.002 0.124 0.025 0.125 0.056 0.127 0.083 0.129 0.104

0.02 0.119 0.001 0.125 0.015 0.127 0.039 0.129 0.062 0.131 0.082

0.023 0.121 0.0 0.127 0.009 0.129 0.027 0.131 0.047 0.132 0.064

0.025 0.123 0.0 0.129 0.005 0.131 0.019 0.133 0.035 0.134 0.051

Expiry T6 T7 T8 T9 T10

Mi Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price

-0.025 0.209 1.121 0.208 1.054 0.205 0.99 0.203 0.934 0.201 0.879

-0.023 0.195 1.098 0.194 1.029 0.191 0.964 0.19 0.908 0.188 0.853

-0.02 0.182 1.068 0.182 0.999 0.179 0.934 0.177 0.878 0.176 0.824

-0.018 0.171 1.032 0.171 0.962 0.168 0.898 0.167 0.843 0.165 0.791

-0.015 0.161 0.987 0.161 0.918 0.158 0.857 0.157 0.804 0.155 0.753

-0.013 0.153 0.932 0.152 0.866 0.149 0.808 0.148 0.758 0.147 0.711

-0.01 0.145 0.867 0.144 0.805 0.142 0.752 0.14 0.706 0.139 0.662

-0.008 0.139 0.79 0.138 0.736 0.135 0.688 0.134 0.647 0.133 0.608

-0.005 0.133 0.705 0.132 0.659 0.13 0.619 0.129 0.583 0.127 0.549

-0.003 0.129 0.613 0.128 0.577 0.126 0.545 0.125 0.515 0.123 0.487

0.0 0.126 0.521 0.125 0.495 0.123 0.47 0.122 0.447 0.12 0.425

0.003 0.124 0.433 0.123 0.416 0.121 0.399 0.12 0.382 0.118 0.365

0.005 0.123 0.353 0.122 0.344 0.12 0.334 0.119 0.322 0.117 0.31

0.008 0.123 0.285 0.122 0.282 0.12 0.277 0.119 0.269 0.117 0.261

0.01 0.124 0.227 0.123 0.229 0.12 0.229 0.119 0.224 0.118 0.219

0.013 0.125 0.181 0.123 0.186 0.121 0.188 0.12 0.187 0.119 0.184

0.015 0.126 0.145 0.125 0.152 0.123 0.156 0.121 0.156 0.12 0.155

0.018 0.127 0.116 0.126 0.124 0.124 0.129 0.123 0.13 0.121 0.131

0.02 0.129 0.093 0.128 0.101 0.126 0.107 0.124 0.11 0.123 0.112

0.023 0.131 0.075 0.13 0.084 0.127 0.09 0.126 0.093 0.125 0.095

0.025 0.132 0.061 0.131 0.069 0.129 0.076 0.128 0.079 0.127 0.082

91

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