L´evy Processes in Finance: L´evy ... - Allan- · PDF fileMSc Business Administration & Management Science Master Thesis Department of Finance Copenhagen Business School 2008 Author:

MSc Business Administration & Management Science

Master Thesis

Department of Finance

Copenhagen Business School

2008

Author: Allan Sall Tang Andersen180682-XXXX

Supervisor: Bjarne Astrup Jensen

Levy Processes in Finance:

Levy Heath-Jarrow-Morton Models

Handed in: January 14th, 2008

Dansk Resume

Dette speciale beskæftiger sig med finansiel modellering vha. Levy processer.Mere præcist betragter det, som i Eberlein & Raible (1999), rente modelleringi et Heath-Jarrow-Morton setup baseret pa Levy processer.

Vi begynder ved at introducere Levy processer, deres grundlæggende defi-nition, stokastiske integraler med hensyn til Levy processer og malskift forLevy processer. Vi beskriver ogsa prisfastsættelse af derivater i modeller,hvor man ikke kan opna et lukket udtryk for optionspræmien. Vi benytterFourier inversions teknikker som beskrevet i Heston (1993) og videre forfineti Carr & Madan (1999) ved at bruge Fast Fourier Transform-algoritmen.

Specialet beskriver Heath-Jarrow-Morton modeller baseret pa Levy processer,for eksempel Heath-Jarrow-Morton driftbetingelsen (som bliver reformulerettil at anvende mere generelle processer vha. den karakteristiske funktion) ogNumeraire-skift. Ydermere beskriver vi, hvordan vi ved at benytte Heath-Jarrow-Morton modeller baseret pa Levy processer og de førnævnte nu-meriske metoder, kan prisfastsætte standard LIBOR derivater, nemlig capsog floors.

Endelig implementerer vi modellen i C++, samt kalibrerer modellen til datafra en specifik handelsdag. Mere præcist bestar vores data af caps tilknyttet6 maneder EURIBOR renter.

Vi finder, naturligvis, at modeller baseret pa Levy processer klarer sig bedreen Gaussiske modeller, men mere interessant finder vi at man skal væreganske papasselig nar man specificerer multidimensionale Levy processer.Hvis afhængigheden mellem de underliggende faktorer er restriktiv, kan ef-fekten ved at tilføje en ekstra faktor være mere eller mindre usynlig. Enmodel baseret pa uafhængige faktorer, ser derimod ud til at klare sig bedre.

Vores mest interessante resultat er, at den faktor, som pavirker hele rentekur-ven, udviser meget høj kurtosis. Vi mener den høje grad af kurtosis ernødvendig for at beskrive smil og ’skews’ som er tilstede i volatilitets over-fladen. Ydermere sa tyder vores resultater pa, at en anden faktor, der opførersig mere lig en Brownsk bevægelse, er nødvendig for at modellerne ikke ud-viser meget skarpe smil for korte løbetider.

Abstract

This thesis considers financial modelling using Levy processes. More specifi-cally it considers, as in Eberlein & Raible (1999), interest rate modelling byusing a Heath-Jarrow-Morton framework based on Levy processes.

We start by introducing Levy processes, their basic properties, stochasticintegral with respect to Levy processes and change of measure for Levy pro-cesses. We also cover pricing of derivatives in models, where no closed formexpression can be obtained. We use the Fourier inversion methods, such asdescribed in Heston (1993) and further enhanced by Carr & Madan (1999)to use the Fast Fourier Transform-algorithm.

The thesis also describes the Heath-Jarrow-Morton model based on Levy pro-cesses, for example the Heath-Jarrow-Morton drift condition (which can bereformulated to more general processes though the characteristic function)and Change-of-Numeraire. Furthermore we also describe how to price stan-dard LIBOR derivatives, namely caps and floors, using the Levy processbased Heath-Jarrow-Morton framework and the numerical methods men-tioned mentioned above.

Finally we also implement the model in C++, and calibrate the model to datafrom one specific trading day. More precisely our data consists of caps linkedto the 6 month EURIBOR rate.

Rather obviously we find that models based on Levy processes outperformGaussian models, but more interestingly we find that one has to be quite care-ful when specifying multivariate Levy processes. If the dependence betweenthe driving factors is restrictive, the effect from adding additional drivingfactor are hardly visible. A model based on independent factors on the otherhand seems to perform far better.

Our main finding, is that the factor that affects the entire yield curve, exhibita high degree of excess kurtosis. We believe that this high degree of excesskurtosis is need to capture the smiles and skews found in the volatility surface.Furthermore a factor that behaves more similar to Brownian motion is neededin order not to produce very sharp smiles for shorter maturities.

CONTENTS i

Contents

1 Introduction 1

2 Levy Processes 5

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Levy-Khintchine formula . . . . . . . . . . . . . . . . . . 6

2.3 The Levy decomposition . . . . . . . . . . . . . . . . . . . . . 9

2.4 Describing the jumps of Levy processes . . . . . . . . . . . . . 11

2.4.1 Finite activity . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.2 Finite variation . . . . . . . . . . . . . . . . . . . . . . 12

2.4.3 Infinite variation . . . . . . . . . . . . . . . . . . . . . 14

2.5 Stochastic Calculus for Levy processes . . . . . . . . . . . . . 16

2.6 Subordination . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Change of measure . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Non-homogeneous Levy processes . . . . . . . . . . . . . . . . 27

3 Pricing using the Fast Fourier Transform 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Option pricing using Fourier inversion . . . . . . . . . . . . . . 31

3.3 European Call options . . . . . . . . . . . . . . . . . . . . . . 33

3.4 European Put options . . . . . . . . . . . . . . . . . . . . . . 34

3.5 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . 35

3.6 Approximation of the Fourier integral . . . . . . . . . . . . . . 35

3.7 Outline of the pricing algorithm . . . . . . . . . . . . . . . . . 37

3.8 FFT pricing in the Black-Scholes model . . . . . . . . . . . . . 37

4 The Levy HJM model 41

4.1 Fixed income basics . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 The standard HJM framework . . . . . . . . . . . . . . . . . . 42

4.3 The general Levy HJM framework . . . . . . . . . . . . . . . . 43

4.4 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Pricing Caps and Floors . . . . . . . . . . . . . . . . . . . . . 50

ii CONTENTS

5 Implementation of the Levy HJM model 57

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Multivariate Variance Gamma . . . . . . . . . . . . . . . . . . 58

5.3 Volatility Structures . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . 67

5.6 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . 69

5.6.1 Gaussian Models . . . . . . . . . . . . . . . . . . . . . 70

5.6.2 Variance Gamma Models . . . . . . . . . . . . . . . . . 74

5.6.3 Implied risk neutral distributions . . . . . . . . . . . . 81

6 Conclusion & Future Research 87

A Proofs 95

A.1 Proof of theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . 95

A.2 Proof of lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.3 Proof of lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.4 Proof of lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . 97

A.5 Proof of theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . 97

A.6 Proof of lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.7 Proof of lemma 7 . . . . . . . . . . . . . . . . . . . . . . . . . 100

B The Characteristic Function 101

B.1 Definition & Basic Properties . . . . . . . . . . . . . . . . . . 101

B.2 Inversion Theorems . . . . . . . . . . . . . . . . . . . . . . . . 102

B.3 Characteristic Function & Moments . . . . . . . . . . . . . . . 103

C Convergence of integrals: Test rules 104

D Remarks on the C++ implementation 105

E MATLAB code for FFT pricing in the Black-Scholes model108

1 INTRODUCTION 1

1 Introduction

According to BIS (2007) estimates, the total Over-the-Counter (henceforthOTC) interest rate derivatives markets amounted to 346,937 billion USDby end June 2007, or about 67 % of the total notional outstanding in theentire global OTC derivatives market. In terms of year-on-year growth, thiscorresponds to an advance in market size of 32 % compared to June 2006.Interest rate swaps amounted for 78.4 % of the notional outstanding, forwardrate agreements 6.6 % and interest rate options 15.0 %. Although notionalis a measure of market size and not market value, the numbers indicate thatlarge amounts of money can be lost if one uses a misspecified model.

Stylized facts from fixed income markets show that market prices of interestrate derivatives, or rather the implied Black volatilities of these derivatives,are inconsistent with most short rate models (such as Vasicek (1978) or Cox,Ingersoll & Ross (1985)). Recent developments in interest rate modelling,known as the LIBOR market model (Miltersen, Sandmann & Sondermann(1997) and Brace, Gatarek & Musiela (1997)) have tried to model interestrate derivatives by modelling simple forward rates as opposed to instanta-neous short- or forward rates. This approach will give at-the-money modelprices that are consistent with market prices, but for in-the-money or out-of-the-money derivatives mispricing will still happen.

This mispricing is due to the fact that the underlying stochastic process ofthe market prices are different from the processes postulated by the differentmodels. Recently a number of papers have investigated stochastic volatilityin the setting of affine term structure models (Feldhutter (2006)), Heath-Jarrow-Morton (henceforth HJM) models (de Jong & Santa Clara (1999)and Schwartz & Trolle (2007)) and LIBOR market models (Andersen & An-dreasen (2000) and Andersen & Brotherton-Ratcliffe (2001)). These authorsconclude that the introduction of stochastic volatility greatly improves thefit to market prices.

Another interesting stylized fact about fixed income markets, is found byCollin-Dufresne & Goldstein (2002). These authors find, opposite to whata large class of models predicts, that interest rate derivatives can not beperfectly hedged by using zero coupon bonds. Collin-Dufresne & Goldstein(2002) name this feature Unspanned Stochastic volatility. The implicationof Unspanned Stochastic volatility is that interest rate derivatives no longerare redundant assets, and must be used in the hedging of other interest ratederivatives.

Although stochastic volatility can account for some of the dynamics in the

2 1 INTRODUCTION

short term interest rates and implied Black volatilities, Andersen, Benzoni, &Lund (2004), find that jumps are an integral part of describing the evolutionof the short term interest rate, and thereby relieves the stochastic volatilityprocess in generating extreme outliers. Jarrow, Li & Zhao (2007) estimatea LIBOR market model with stochastic volatility and jumps. These authorsfind, as well, that jumps should be a part of an interest rate model in orderto describe the behaviour of interest rates and volatility surfaces. However asargued in Schwartz & Trolle (2007), the model of Jarrow, Li & Zhao (2007)may be misspecified, as it does not allow for correlation between innovationsand volatility, and thus obtain rather unrealistic estimates for the jumps ofthe process.

This thesis will consider a class of fixed income models where the drivingprocess is not a Brownian motion (as in the stochastic volatility models), butrather a Levy process. Levy processes are a very general class of stochasticprocesses allowing for both skewness and excess kurtosis in the distribution;these features should be a valuable input in describing the observed impliedBlack volatilities. The use of Levy processes (or rather more general Semi-Martingales) was introduced in Bjork et al (1997) and a class of interest ratemodels based on the HJM framework was described in Eberlein & Raible(1999) and Raible (2000), where an extension allowing for non-homogeneousLevy processes can be found in Kluge (2005).

Modelling with Levy processes seem to be a reasonable compromise betweenmodel generality and tractability, as the processes are well studied and secu-rity prices can be computed using the same numerical methods as used in forinstance stochastic volatility models. Furthermore as shown in Bjork et al(1997) when modelling with processes exhibiting jumps, interest rate deriva-tives can not longer be perfectly hedged1; this implies that models based onLevy processes will introduce unspanned stochastic volatility.

The thesis is structured as follows: Section 2 will, in a rather applied man-ner2, describe Levy processes and the behaviour and stochastic properties ofthese processes. The section will also describe examples of Levy processes.Section 3 describes the pricing of derivatives when the underlying asset is

1More precisely Bjork et al (1997) find that for a 1-dimensional driving process, the riskneutral martingale measure is unique, but derivatives cannot not be perfectly hedged. Thisleads to approximate completeness, as defined in Bjork et al (1997). When the dimensionof the driving process is larger than one, then the models no longer exhibit approximatecompleteness, and hence the martingale measure is not unique.

2For instance we refrain from topics such as analytical continuity, contour integrationwhen deriving the characteristic function of different processes. Furthermore most resultsin section 2 are stated without proof.

1 INTRODUCTION 3

driven by a Levy process. Furthermore the section also describes how pric-ing of a large amount of derivatives can be done in an efficient manner usingthe Fast Fourier Transform. Section 4 will describe basic fixed income def-inition and the HJM framework based on both Brownian motion and Levyprocesses. The section will also discuss topics such as change-of-numeraireand the pricing of caps and floors. Section 5 describes the implementation ofthe Levy HJM model and will show calibration results based on interest capsfrom a single trading day. Finally section 6 concludes and proposes areas forfuture research.

4 1 INTRODUCTION

2 LEVY PROCESSES 5

2 Levy Processes

This section will introduce the basics of Levy processes. It will describe thethe stochastic properties of Levy processes and concepts specifically usefulto finance such as stochastic integrals, Ito’s formula and change of measure.Many of the theorems in this section will be stated without proof, hencefor a more detailed treatment we refer to Cont & Tankov (2004) and Prot-ter (2005). An applied approach to Levy processes can also be found inSchoutens (2003). Our exposition follows Ballotta (2007) to a large extent.

The outline of this section is to cover the basics of Levy processes, how Levyprocesses and their behaviour is described via the characteristic function.Furthermore stochastic calculus, subordination and change of measure will becovered. Finally a short description of non-homogeneous Levy processes (alsocalled additive processes) is given. A basic knowledge of non-homogeneousLevy processes is needed, as modelling interest rates with homogenous Levyprocesses, leads to (log-)bond prices that are driven by non-homogenous Levyprocesses. More on this in section 4.

In terms of notation, we use the word random or stochastic variable, for bothone- and multidimensional variables. It should be clear from the contextwhen one is considering the first or the latter. Hopefully this will not because of any confusion.

2.1 Definition

In this section assume as given a filtered probability space(Ω,F , Ftt≥0 , P

)

satisfying the usual conditions (see Protter(2005)), ie. where Ω is the sam-ple space, F is a σ-algebra, Ftt≥0 is a right continuous filtration, P is aprobability measure and finally that F0 contains all P -null sets of F .

A Levy process is then defined as

Definition 1. (Levy process) An adapted, cadlag3 process L ≡ Lt, t ≥ 0with L0 = 0 almost surely (a.s.) is a Levy process if

(i) L has increments independent of the past, ie. Lt − Ls is independentof Fs, 0 ≤ s < t < ∞;

3Cadlag is an abbreviation for continu a droit, limite a gauche, ie. french for right

continuous with left limits. Similarly caglad is an abbreviation for continu a gauche,

limite a droit, ie. french for left continuous with right limits.

6 2 LEVY PROCESSES

(ii) L has stationary increments, ie. Lt −Ls has the same distribution lawas Lt−s, 0 ≤ s < t < ∞;

(iii) L is continuous in probability, ie. ∀ǫ > 0, lims→0 P (|Lt+s−Lt| ≥ ǫ) = 0.

It is clear from the definition that Levy processes are a very general class ofstochastic processes, and can most easily be viewed as continuous time ran-dom walks. It is evident from the definition of Levy processes that Brownianmotion is also a Levy process, however other Levy processes include (com-pound) Poisson and Gamma processes. Furthermore Levy processes formformidable building blocks for both Markov processes and semimartingales.

Using the Levy characterization of Brownian motion (see Shreve (2004), The-orem 4.6.4 and Theorem 4.6.5), we get that the only (additive) stochasticprocess with continuous sample paths is Brownian motion, hence as Levyprocesses is more general than Brownian motion it must exhibit jumps. Fur-thermore, as it is defined as a cadlag process, we know that the left limitexists:

Lt− = lims→t−

Ls

giving us the following definition of jumps in a Levy process:

Definition 2. Let L be a Levy process, then the jump at t of L is ∆L =Lt − Lt−.

2.2 The Levy-Khintchine formula

As Levy processes are a very general class of stochastic processes, a closedform expression for the probability density can not in general be derived.Fortunately stochastic variables can be fully described by their characteristicfunction4. If we can derive the characteristic function of Levy processes wecan then describe the processes. This have been done in a very generaltheorem, namely the celebrated Levy-Khintchine formula. Before describingthe Levy-Khintchine formula, we will first describe the concept of infinitedivisibility :

4The characteristic function of a random variable X is defined as the Fourier transformof the probability density function:

φX(u) = E

[

eiu⊤x]

=

∫

Rd

eiu⊤xdP (x)

More on the properties of the characteristic function can be found in appendix B.

2 LEVY PROCESSES 7

Definition 3. (Infinite divisibility) A random variable X has an infinitelydivisible distribution if for any n, there exist a sequence of i.i.d. random

variable Y(n)1 , . . . , Y

(n)n such that X

D=∑n

i=1 Y(n)i .

Next we wish to show that a Levy process possesses an infinitely divisibledistribution. To to this consider the characteristic function of a Levy processL

φL(u, t) = E

[

eiu⊤Lt

]

Now consider an equidistant partition of the time [0, t] such that 0 = t0 <t1 < . . . < tn = t and ∆t = ti − ti−1 = t/n. Then by the assumption ofindependent and stationary increments we have

Lt =n∑

i=1

Lti − Lti−1

D= nLt/n

Therefore the characteristic function will be

φL(u, t) = E

[

eiu⊤Lt

]

= E

[

eiu⊤nLt/n

]

=(

E

[

eiu⊤Lt/n

])n

As infinite divisibility must hold for any n, we can let n = t and therebyobtain

φL(u, t) =(

E

[

eiu⊤L1

])t

= etϕ(u)

where ϕ(u) = log E

[

eiu⊤L1

]

.

The specific form of ϕ(u) was derived in the 1930’s by Paul Levy and A. Y.Khintchine, and is usually called the Levy-Khintchine formula (here statedwithout proof - a proof can for instance be found in Cont & Tankov (2004)):

Theorem 1. (Levy-Khintchine formula) Consider a triple (a, Σ, ν), wherea is a d × 1 vector, Σ is positive definite d × d matrix and ν is a positivemeasure on R

d0 ≡ (R \ 0)d (ie. the d dimensional space excluding the zero

vector) such that∫

Rd0

(1 ∧ |x|2)ν(dx) < ∞ (1)

Let L be a d dimensional Levy process; then

φL(u, t) = E

[

eiu⊤Lt

]

= etϕ(u)

8 2 LEVY PROCESSES

where

ϕ(u) = iu⊤a − 1

2u⊤Σu +

∫

Rd0

(

eiu⊤x − 1 − iu⊤x1|x|<1

)

ν(dx) (2)

The measure ν is called the Levy measure, the triple (a, Σ, ν) is called thecharacteristic triple (or characteristics) of the process L and the functionϕ(u) is called the Levy exponent (or Levy symbol).

Remark 1. Strictly speaking the Levy measure ν is defined on Rd excluding

some small ball of size ǫ > 0 around origo, and not just one point as statedabove. However the description above is standard in the literature.

Remark 2. For the reader familiar with the characteristic function of thecompound Poisson process, the term

∫

Rd0

iu⊤x1|x|<1ν(dx) may seem a bit

odd. This term ensures the the convergence of the integrals in the char-acteristic function by truncating the jumps that has absolute value smallerthan one, so that the expected value of these jumps are zero. The necessarycondition for the convergence of these integrals is given by equation (1), seeCont & Tankov (2004) for a more detailed treatment.

In order to familiarize the reader with the Levy-Khintchine formula we pro-vide two examples of well known processes to see how they fit into to theLevy-Khintchine formula.

Example 1. Consider a one dimensional Brownian motion. In this caseWt ∼ N (0, t). We can then find the characteristic function as

φW (u, t) = E[eiuWt

]=

1√2πt

∫ ∞

−∞

eiuxe−x2

2t dx

Then by completing the square and using that a density integrates to one,we arrive at

φW (u, t) = et(

−u2

2

)

Hence Brownian motion fits nicely into the Levy-Khintchine formula, andfurthermore the example tells us that the term 1

2u⊤Σu relates to a d dimen-

sional Brownian motion with covariance structure given by the matrix Σ.Finally the characteristic triple of the one dimensional Brownian motion isgiven as (0, 1, 0).

2 LEVY PROCESSES 9

Example 2. Consider a Poisson process N with intensity λ. We can thenderive the characteristic function as

φN(u, t) =E[eiuNt

]=

∞∑

x=0

eiux e−λt(λt)x

x!

=e−λt

∞∑

x=0

(λteiu)x

x!= eλt(eiu−1)

where we in the last equality have used the expansion form of the exponentialfunction. This form of the characteristic function implies that the Levyexponent is given as

ϕ(u) = λ(eiu − 1

)

Comparing this to the Levy-Khintchine formula we see that

ϕ(u) =

∫

R0

(eiux − 1)λδ(x − 1)dx

where δ is the Dirac delta function. Hence in this case there is only assigneda positive measure to jumps of size one. This is intuitive as the Poissonprocess is a counter process and only has jumps of size one. The last termin the Levy-Khintchine formula,

∫

R0iux1|x|<1ν(dx), vanishes as ν is only

positive whenever the indicator function is zero and vice versa. Finally thecharacteristic triple of the Poisson process is given as (0, 0, λδ(x − 1)dx)

The two examples has given us an idea on how the Levy-Khintchine formulatells us about the process in hand. The following subsection will explore thisinto further detail.

2.3 The Levy decomposition

As stated in theorem 1 the characteristic function of a Levy process is givenby

φL(u, t) = E

[

eiu⊤Lt

]

= etϕ(u)

where

ϕ(u) = iu⊤a − 1

2u⊤Σu +

∫

Rd0

(

eiu⊤x − 1 − iu⊤x1|x|<1

)

ν(dx)

10 2 LEVY PROCESSES

Next consider re-writing this as

ϕ(u) = iu⊤b − 1

2u⊤Σu +

∫

Rd0

(

eiu⊤x − 1)

ν(dx) (3)

where

b = a −∫

Rd0

x1|x|<1ν(dx)

Now, let us consider each part of the characteristic triple. First assume that

Σ = ν = 0, then we have that E

[

eiu⊤Lt

]

= eiu⊤b and hence the process can

be written as Lt = bt, ie. a deterministic motion along a straight line inthe d dimensional plane. The parameter b then determintes the drift of theprocess.

Next let Σ 6= 0, then the characteristic function will be E

[

eiu⊤Lt

]

= eiu⊤b− 1

2u⊤Σu,

implying that the process is a d dimensional arithmetic Brownian motion.Hence Σ is the diffusion coefficient/matrix of the process.

Finally, as we saw in the case of the Poisson process, the term involvingthe Levy measure, ν, governs the jumps of the process. Furthermore themultiplicative structure of the characteristic function tells us that the jumppart and Brownian motion part of the process are independent.

This leads us to following result:

Theorem 2. (Levy decomposition) A Levy process can be decomposedinto the sum of three parts

Lt = bt + Σ1/2Wt + Jt

where bt is deterministic, Σ1/2 is a matrix square root, ie. a matrix such that(Σ1/2

) (Σ1/2

)⊤= Σ, Wt is a d dimensional Brownian motion with W i

t⊥W jt

for i 6= j and Jt is the purely discontinuous part of Lt, which is independentof Wt. Finally the characteristic function of this decomposition of L is givenin equation (3).

Remark 3. In some cases it is practical to consider the continuous part ofthe Levy process. We denote this part as Lc

t = bt + Σ1/2Wt.

As shown in the theorem above, a Levy process can be decomposed intothree parts. As we assume that the reader is familiar with Brownian motion(otherwise Shreve (2004) provides an excellent treatment), we will focus onthe jump part of the process.

2 LEVY PROCESSES 11

2.4 Describing the jumps of Levy processes

In definition 2 we defined the jumps of a Levy process L : ∆Lt = Lt − Lt− .The Levy measure ν can tell us about the behaviour of the jumps of a Levyprocess. The Levy measure can also be defined as (see Cont & Tankov(2004)):

Definition 4. The Levy measure ν over a Borel set, A ∈ B(R

d), is the

expected number of jumps of the given size A in the time interval [0, 1], ie.

ν(A) = E [#t ∈ [0, 1] : ∆Lt 6= 0, ∆Lt ∈ A]

Hence the Levy measure tells us about the expected number of jumps of aspecific size per unit of time. Then on one hand the Levy measure tells usabout the intensity of the process. On the other hand it also tells us about thedistribution of the jumps. In two simple cases (the Poisson and compoundPoisson process) the two effects can be separated into two separate parts ofthe Levy measure, but in general the two effects can not be isolated.

Example 3. Consider the compound Poisson Nt =∑Nt

i=1 Xi where the Xisare i.i.d. and has density function f(x). We find the characteristic functionas

φN(u, t) = E

[

eiuNt

]

= E

[

E

[

eiu∑Nt

i=1Xi∣∣Nt = n

]]

= E

[

E[eiuX1

]Nt]

= E[φX(u)Nt

]

Next following the same steps as in example 2 we arrive at

φN(u, t) = eλt(φX(u)−1)

where φX(u) is the characteristic function of the jump sizes Xi. When re-writing the characteristic function in the same way as in the Levy exponentwe get

ϕ(u) = λ(φX(u) − 1) =

∫

R0

(eiux − 1

)λf(x)dx

which tells us that the Levy measure is given by ν(dx) = λf(x)dx. Asmentioned above, in case of the compound Poisson process, the Levy mea-sure factorizes into a part governing the intensity of the process, and a partgoverning the distribution of the jump sizes.

12 2 LEVY PROCESSES

In general we can characterize the behaviour of the jumps of the process Lvia Levy measure, by three different classifications

• Process with finite activity (FA process)

• Process with infinite activity and finite variation (FV process)

• Process with infinite variation

We will in the following sections describe the meaning of all three classifica-tions

2.4.1 Finite activity

A process is said to have finite activity, if on any finite time interval thenumber of jumps is finite.

By the definition of the Levy measure, it tells us about the number of jumpsin a specific finite time interval, namely the [0, 1] time interval, hence if aprocess has a finite number of jumps on this interval, it will also have a finitenumber of jumps on any other finite time interval. Using the definition of theLevy measure we can conclude that a Levy process will have finite activityif and only if

ν(R

d0

)=

∫

Rd0

ν(dx) < ∞

2.4.2 Finite variation

When defining finite activity, all we did was to count the number of jumps.However in general we can have a countably infinite number of jumps, whichpossibly could imply that the sample paths of the process could have un-bounded variation.

To examine when the process is of bounded variation we describe the jumppart of the process via its jump measure N(t, x). This measure counts thenumber of jumps at time t of size x that the process have had on the givensample path:

N(t, A) = #s ∈ [0, t], ω ∈ Ω : ∆Ls(ω) 6= 0, ∆Ls(ω) ∈ A, A ∈ B(R

d)

2 LEVY PROCESSES 13

Hence the jump part of the process can be written as

Jt(A) =

∫

A

xN(t, dx)

By the definition of the Levy measure we have

E [Jt(A)] = E

[∫

A

xN(t, dx)

]

=

∫

A

xE [N(t, dx)] = t

∫

A

xν(dx)

where we have used Fubinis theorem to change the order of integration.

Next recall the definition of the p-th variation process

Definition 5. Let X be a stochastic process. For p > 0, the p-th variationprocess is defined by

limsup ∆tk→0

∑

tk<t

∣∣Xtk+1

− Xtk

∣∣p

where 0 = t0 ≤ t1 ≤ . . . ≤ tn = t and ∆tk = tk+1 − tk. For p = 1 the processis called the Total Variation process and for p = 2 it is called the QuadraticVariation process, usually denoted [X,X]t

For the p-th variation of a jump process we have

VpJ(A)(t) = lim

sup ∆tk→0

∑

tk<t

∣∣Jtk+1

(A) − Jtk(A)∣∣p

=

∫

A

|x|pN(t, dx)

which follows from the definition of the jump measure. Furthermore theexpected value of the variation process can be found as

E

[

VpJ(A)(t)

]

= E

[∫

A

|x|pN(t, dx)

]

=

∫

A

|x|pE [N(t, dx)] = t

∫

A

|x|pν(dx)

where we again have used Fubinis theorem.

Now let us define two sets

Λ = x : |x| ≥ 1, Λ0 = x : |x| < 1

These two sets will be useful because, as seen from the Levy-Khintchineformula, the process behaves differently on the two sets. More specificallywe have the condition

∫

Rd0

(1 ∧ |x|2

)ν(dx) =

∫

Λ

ν(dx) +

∫

Λ0

|x|2ν(dx) < ∞

14 2 LEVY PROCESSES

From the definition of finite activity we see that on the set Λ the process isa finite activity process. Furthermore considering the last part of the aboveequation we see that this implies the jumps have finite quadratic variationon Λ0 (and naturally on Λ since there is only a finite number of jumps onthis set). Using the intuition above, we argue that a pure jump process willhave finite variation if and only if (for a more detailed treatment see Cont &Tankov (2004)):

∫

Rd0

(1 ∧ |x|) ν(dx) < ∞

2.4.3 Infinite variation

As shown in the discussion above, a Levy process will have infinite variationif

∫

Rd0

(1 ∧ |x|) ν(dx) = ∞

and furthermore that the condition on the Levy measure stated in the Levy-Khintchine formula makes sure that the jumps of a Levy process (and hencethe entire Levy process) always will have finite quadratic variation.

Example 4. (Gamma process) A stochastic process G ≡ Gt, t ≥ 0, is agamma process if the increments over non-overlapping time intervals [t, t+s]are independent and follow a gamma distribution with Γ(αs, λ), such thatGt has density

fG(x) =1

Γ(αt)λαtxαt−1e−λx

This implies that the characteristic function can be found as

E[eiuGt

]=

∫ ∞

0

1

Γ(αt)λαtxαt−1e−(λ−iu)xdx

=

[λ

λ − iu

]αt ∫ ∞

0

1

Γ(αt)(λ − iu)αtxαt−1e−(λ−iu)xdx

=

[λ

λ − iu

]αt

= exp

tα logλ

λ − iu

where the third equality follows from the fact that the integrand is the densityof a gamma distributed variable with parameters αt and λ − iu.

2 LEVY PROCESSES 15

Next to find the Levy measure we use Frullani’s integral, ie. that for afunction f(x) where f ′(x) is continuous and the following integral converges,then

∫ ∞

0

x−1 [f(ax) − f(bx)] dx = [f(0) − f(∞)] logb

a

Now let f(x) = αe−x and let a = λ − iu and b = λ, then we have that

α logλ

λ − iu=

∫ ∞

0

αx−1[e−(λ−iu)x − e−λx

]dx

=

∫ ∞

0

αx−1e−λx(eiux − 1

)dx

Which tells us, when comparing to the Levy-Khintchine formula that theLevy measure is given by ν(dx) = αx−1e−λx1x>0dx.

To see if the process is a finite activity process we must check the convergenceof the following integral

∫

R0

ν(dx) =

∫ ∞

0

αx−1e−λxdx < ∞

Applying the convergence rules given in appendix C we see that the integraldoes not converge for x → 0+

limx→0+

f(x)xa = limx→0+

αx−(1−a)e−λx

which is only finite if a ≥ 1, hence the integral does not converge and theGamma process is not a finite activity process.

If we need to check that the process has finite variation, then we need tocheck:

∫

R0

(1 ∧ |x|)ν(dx) =

∫

R0

(1 ∧ |x|)αx−1e−λxdx < ∞

However since any increment of the process is positive, then the total varia-tion of the process will be equal to value of the process 5, hence the gammaprocess is a finite variation process

5To see this, consider the total variation process

V 1G(t) = lim

sup ∆tk→0

∑

tk<t

|Gtk+1 − Gtk| = lim

sup ∆tk→0

∑

tk<t

(Gtk+1 − Gtk) = Gt

which follows from using that all increments are positive, that G0 = 0 and from recognisingthe telescopic sum.

16 2 LEVY PROCESSES

2.5 Stochastic Calculus for Levy processes

To be able to use Levy processes in financial modelling we need to have sometools that allows us to perform transformations of the Levy processes. As inthe case of Brownian motion, stochastic integrals and the Ito formula play acentral role.

To define the stochastic integral consider a time partition on the interval [0, t]such that

0 = t0 ≤ t1 ≤ . . . ≤ tn = t

We then define the stochastic integral as (see Protter (2005) pp. 58)

Definition 6. Let H be a bounded previsible6 process in Rdof the form H(t) =

Hj if tj < t ≤ tj+1, where Hj ∈ Ftj . Then the stochastic integral of H withrespect to the d-dimensional process X is given by

IX(H) =n−1∑

j=0

H⊤j (Xtj+1

− Xtj)

Often we express the stochastic integral as

IX(H) =

∫ t

0

H⊤t dXt = H · X

and for two previsible processes H and G (H in Rd and G in R) it holds that

if Y = IX(H) then

∫ t

0

GsdYs =

∫ t

0

GsH⊤s dXs

which follows from the definition of the stochastic integral (cf. Protter (2005),chap II, theorem 13).

So far we have posed no restrictions on the process X in order to make surethat the stochastic integral is well behaved. By well behaved we mean that asmall change in the process H should only cause a small change in stochasticintegral IX(H). As mentioned in Cont & Tankov (2004), pp. 253, not allprocesses satisfy this criteria. The processes that do satisfy this criteria arecalled Semimartingales. More formally we require that when H covergesuniformly, then IX(H) should converge in probability (see Cont & Tankov(2004), definition 8.2):

6A previsible process is a process X where Xt = Xt− a.s.

2 LEVY PROCESSES 17

Definition 7. (Semimartingale) An adapted cadlag process X is a semi-martingale, if the stochastic integral of simple predictable processes with re-spect to X verifies that when

sup(s,ω)∈[0,t]×Ω

|Hns − Hs| → 0 for n → ∞

for each t, then∫ t

0

Hns⊤dXs

P→∫ t

0

H⊤s dXs for n → ∞

At first sight, it might seem like a huge task to describe semimartingales.However it can be shown (see Protter (2005), chap III, Theorem 47) that asemimartingale can be decomposed into the sum of a local martingale and afinite variation cadlag process:

Xt = X0 + Mt + At

where Mt is a local martingale and At is a cadlag process with paths offinite variation. By the Levy decomposition we see that a Levy process is asemimartingale.

As was shown when we described the jump part of Levy processes, the samplepaths of the process may not always be of finite variation (which is never thecase, when considering a process containing a Brownian motion). Hence wewill need to consider the quadratic variation of the process to perform achange of varaibles. In definition 5 we defined the quadratic variation of astochastic process - in the case of semimartingales (and hence Levy processes)this can be defined as (Protter (2005), pp. 66)

Definition 8. Let X,Y be semimartingales. The quadratic variation processof X, denoted [X,X] ≡ [X,X]t, t ≥ 0, is defined by

[X,X]t = |Xt|2 − 2

∫ t

0

X⊤s−

dXs

The quadratic covariation of X and Y , denoted [X,Y ] ≡ [X,Y ]t, t ≥ 0, isdefined by

[X,Y ]t = X⊤t Yt −

∫ t

0

Y ⊤s−

dXs −∫ t

0

X⊤s−

dYs

Remark 4. Note the use of Xs− in the integrand. This is needed as Xs isnot a previsible process, whereas Xs− is. For a further elaboration of thistopic, see Protter (2005), pp. 65.

18 2 LEVY PROCESSES

Finally as we saw in the case of a Levy process, the process could be decom-posed into a continuous part and a discontinuous part, this in turn impliesthat the same thing, can be done with the quadratic variation process. Wecan decompose the quadratic variation into a continuous part and a discon-tinuous part:

Definition 9. For a semimartingale X, the process [X,X]c denotes the pathby path continuous part of [X,X]. We then have

[X,X]t = [X,X]ct +∑

0<s≤t

(∆Xs)2

If [X,X]ct = 0, then X is called a (quadratic) pure jump process.

Now by using the definition of the stochastic integral and quadratic variationthe following result should now be understood (Protter (2005), chap. II,theorem 33):

Theorem 3. (Ito formula) Let X be a d dimensional semimartingale andlet f : [0, t]×R

d 7→ R and f ∈ C1,2. Then f(X) is a semimartingale and thefollowing formula holds:

f(t,Xt) − f(0, X0) =

∫ t

0

∂f

∂s(s,Xs−)ds +

∫ t

0

d∑

j=1

∂f

∂Xj

(s,Xs−)dXj,s

+1

2

∫ t

0

d∑

j,k=1

∂f

∂Xj∂Xk

(s,Xs−)d[Xj, Xk]cs

+∑

0<s≤t

[

f(s,Xs) − f(s,Xs−) −d∑

j=1

∂f

∂Xj

(s,Xs−)∆Xj,s

]

As in the case for Ito calculus, this is the ”chain rule” for stochastic calculuswith semimartingales. The main part of the proof of course lies in a Taylorexpansion with appropiate limits, dictated by the total and quadratic vari-ation of the process. Furthermore, when letting X = L, where L is a Levyprocess we obtain

Theorem 4. (Ito formula) Let L be a d dimensional Levy process withcharacteristic triplet (a, Σ, ν) and let f : [0, t] × R

d 7→ R and f ∈ C1,2. Then

2 LEVY PROCESSES 19

f(L) is a semimartingale and the following formula holds 7:

f(t, Lt) − f(0, L0) =

∫ t

0

∂f

∂s(s, Ls−)ds +

∫ t

0

∂f

∂L⊤(s, Ls−)dLs

+1

2

∫ t

0

Tr

((Σ1/2

)⊤ ∂f

∂L⊤∂L(s, Ls−)

(Σ1/2

))

ds

+∑

0<s≤t

[

f(s, Ls) − f(s, Ls−) − ∂f

∂L⊤(s, Ls−)∆Ls

]

where Tr(A) is the trace of the matrix A and Σ1/2 is a matrix square root,

ie. a matrix such that(Σ1/2

) (Σ1/2

)⊤= Σ.

The difference in the two versions of the Ito formula, is of course the use of thediffusion matrix. This follows from the fact that the matrix Σ will determinethe continuous quadratic (co-)variation, since the Brownian motion part ofthe Levy process is the only source of continuous quadratic (co-)variation.

It is often more convenient to express the Ito formula in differential forminstead of integral form. The differential form is only a matter of notation,and does not make sense from a mathematical perspective. In differentialform we have

df(t, Lt) =∂f

∂t(t, Lt−)dt +

∂f

∂L⊤(t, Lt−)dLt

+1

2Tr

((Σ1/2

)⊤ ∂f

∂L⊤∂L(t, Lt−)

(Σ1/2

))

dt

+ f(t, Lt) − f(t, Lt−) − ∂f

∂L⊤(t, Lt−)∆Lt

Next recall that we can decompose a Levy process into a continuous partand a jump part, ie.

dLt = dLct + ∆Lt

which gives us the following differential notation

df(t, Lt) =∂f

∂t(t, Lt−)dt +

∂f

∂L⊤(t, Lt−)dLc

t + ∆f(t, Lt)

+1

2Tr

((Σ1/2

)⊤ ∂f

∂L⊤∂L(t, Lt−)

(Σ1/2

))

dt

We will now show an example on how to use the Ito formula to solve stochasticdifferential equations. This will be convenient as the extra terms in the Itoformula is a novelty compared to the Ito formula for Brownian motion.

7Note the change to matrix notation.

20 2 LEVY PROCESSES

Example 5. (Doleans-Dade exponential) Consider the stochastic inte-gral

Yt = 1 +

∫ t

0

Ys−dLs

where L is a one dimensional Levy process with characteristic triplet (a, σ2, ν)and Y is an unknown cadlag, adapted process. We now wish to solve for Yusing the Ito formula. Consider the stochastic integral in differential form

dYt = Yt−dLt

Next let Z = log Y then

dZt =1

Yt−

dY ct − 1

2

1

Y 2t−

d[Y, Y ]ct + ∆ log Yt

=dLct −

1

2σ2dt + ∆ log Yt

By the definition of the stochastic integral we have that

Yt =

Yt−(1 + ∆Lt) if ∆Lt 6= 0Yt− if ∆Lt = 0

This implies that

∆ log Yt = log Yt−(1 + ∆Lt) − log Yt− = log(1 + ∆Lt)

Which in integral form gives us Z

Zt =

∫ t

0

dLcs −

1

2σ2

∫ t

0

ds +∑

0<s≤t

log(1 + ∆Ls)

=Lct −

1

2σ2t +

∑

0<s≤t

log(1 + ∆Ls)

Where we find Y by exponentiating

Yt = exp

Lct −

1

2σ2t

∏

0<s≤t

(1 + ∆Ls)

We see that if the process has no jumps, then solution to Doleans-Dade expo-nential will (naturally) coincide with the solution to a Geometric Brownianmotion. Another nice feature of the Doleans-Dade exponential is that if Lis a martingale, then Y will also be a martingale (if the expectation is welldefined). This will not be the case with the standard exponential.

2 LEVY PROCESSES 21

2.6 Subordination

In this section we will consider the concept of subordination. Subordination isa way of constructing new Levy processes from existing ones. More preciselya subordinator is a one dimensional stochastic process that is non-decreasinga.s..

Let us formalise this a bit more. Let L be a Levy process with Levy exponentϕ(u) and let G be a non-decreasing Levy process with Levy exponent l(u),then we will call X a subordinated process if we can express X as

Xt = LGt

Hence we time change the Levy process L to run on a ”new clock” whose(stochastic) speed is dictated by the subordinator G. In order to justify theuse of a subordinator, think of how prices fluctuate in the market; as depictedby Brownian motion, prices fluctuate randomly, however trading does nothappen continuously - in fact traders do not know when the next trade isgoing to happen, ie. quotes are affected by the action of other investors. Inthis context we can view time, or rather the time of the next trade, to bestochastic. Hence the business time is random - this is what a subordinatormodels.

When the subordinator is a Levy process we have the following theorem

Theorem 5. Let L be a Levy process with Levy exponent ϕ(u), and let G bea Levy process and a subordinator with Levy exponent l(u). Then the processX ≡ Xt, t ≥ 0 defined for each ω ∈ Ω by X(t, ω) = L(G(t, ω), ω) is a Levyprocess with characteristic function given as

φX(u, t) = exp tl(−iϕ(u)

ie. the Levy exponent of X is given as a composition of the Levy exponentsof L and G.

We will now present an actual example of a subordinated process, namelythe Variance Gamma process introduced by Madan & Seneta (1990).

Example 6. (Variance Gamma process) The Variance Gamma process(henceforth VG) is a process which is obtained by subordinating a one dimen-sional arithmetic Brownian motion by a gamma process, ie. the VG processX is defined as

Xt = θGt + σWGt

22 2 LEVY PROCESSES

where W is a standard Brownian motion and G is a gamma process withparameters α > 0, λ > 0 which is independent of W . Then the VG processis said to be VG(θ, σ, α, λ).

It is shown in Madan & Seneta (1990) and Madan & Milne (1991) that it issufficient to consider a model where α = λ = 1/k where k > 0; this is calleda gamma process with unit mean rate, since the subordinator will have meanequal to one, when t = 1 (refer to Cont & Tankov (2004), chap. 4 for a moredetailed treatment of this issue). This can also be seen in figure 1 where asample path of two subordinators with unit mean rate is shown. Using thisspecification, X will be VG(θ, σ, 1/k, 1/k).

Next as we have shown above the Levy exponent of the arithmetic Brownianmotion is given by

ϕ(u) = iuθ − 1

2σ2u2

and the Levy exponent of the gamma process is given by

l(u) =1

klog

(1

1 − iku

)

Next using theorem 5 we obtain

φX(u, t) = exp

t

klog

(1

1 − iuθk + 12u2σ2k

)

=

[1

1 − iuθk + 12u2σ2k

]t/k

Using the properties of the cumulant generating function get that 8

E[Xt] =1

i

∂

∂ulog φX(u, t)

∣∣∣∣u=0

= θt

Var(Xt) =1

i2∂2

∂u2log φX(u, t)

∣∣∣∣u=0

=(σ2 + θ2k

)t

µ3(Xt) =1

i3∂3

∂u3log φX(u, t)

∣∣∣∣u=0

=(3σ2θk + 2θ3k2

)t

µ4(Xt) =1

i4∂3

∂u4log φX(u, t)

∣∣∣∣u=0

=(3σ4k + 12σ2θ2k2 + 6θ4k3

)t

here µ3 is a measure of skewness and µ4 is a measure of excess kurtosis. It isobvious that the VG process can exhibit both skewness and excess kurtosis,which should be an advantage compared to Brownian motion.

8for a detailed derivation on these moments we refer to Ballotta (2007)

2 LEVY PROCESSES 23

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Va

lue

of P

roce

ss

Gamma Process, k = 0.05Gamma Process, k = 0.01Deterministic time

0 0.2 0.4 0.6 0.8 190

95

100

105

110

115

120

125

130

135

140

Time

Va

lue

of e

xp(X

t)

VG Proces, θ = 0.05, σ = 0.2, k = 0.05

VG Proces, θ = 0.05, σ = 0.2, k = 0.01

Brownian Motion, θ = 0.05, σ = 0.2

Figure 1: Left: Sample paths of gamma processes with k = 0.05 and k = 0.01,compared to a deterministic motion. This is equivalent to the time changein a VG process with the two different gamma processes as subordinatorsand the case of no subordination. Right: Sample paths of expXt where Xis either a VG processes corresponding to the two gamma processes in theleft hand figure or an arithmetic Brownian motion, ie. no subordination. Inboth figures all processes are simulated using the same uniform numbers.

Figure 1 shows the sample path for the exponential of two different VG pro-cesses and an arithmetic Brownian Motion. It is evident that the VG pro-cesses have a more jump-like behaviour compared to the arithmetic Brownianmotion. Further more we see that the parameter k in the VG process governsthe activity of the process; for k approaching zero the process behaves similarto the arithmetic Brownian motion.

So far we have considered the construction of the process and its moments.But as shown above Levy processes can fit into three different categories,namely finite activity, finite variation and infinite variation. We now wish toshow which one of the three categories that the VG process belongs to.

To do this we need to consider the Levy measure of the process, this howeveris at the present stage not known. To find the Levy measure, consider writingthe characteristic function as

φX(u, t) =

[1

1 − iuθk + 12u2σ2k

]t/k

=

[λ+

λ+ − iu

]t/k [λ−

λ− + iu

]t/k

(4)

By solving for λ+ and λ− we find

λ+ =2

k(√

θ2 + 2σ2/k + θ) , λ− =

2

k(√

θ2 + 2σ2/k − θ)

24 2 LEVY PROCESSES

The trick above tells us that we can express the VG process as the differenceof two independent gamma processes

Xt = G1t − G2

t

where G1 is a gamma process with parameters 1/k and λ+ and G2 is agamma process with parameters 1/k and λ−. This implies that we can writethe characteristic function as

E[eiuXt

]= E

[

eiu(G1t−G2

t )]

= E

[

eiuG1t

]

E

[

ei(−u)G2t

]

Which is equivalent to equation (4). This tells us that the Levy measureagain will be the same as the difference of two gamma processes

ν(dx) =1

k|x|−1

(e−λ+x1x>0 + eλ−x1x<0

)

Now since the Levy measure can be expressed as difference of two gammaprocesses the VG process must, as the gamma process, be a finite variationprocess (with infinite activity). Further more, as stated in Cont & Tankov(2004), pp. 117, the small jumps (∆Xt → 0) of the VG process have relativelylow activity. A hint of this fact is given in figure 1.

2.7 Change of measure

In this section we will present change of measure for Levy processes, namelythe Girsanov theorem for Levy process.

First we recall absolute continuity and equivalence of probability measures

Definition 10. Given two measures P and P ∗ defined on the same σ-algebraF , we say that

(i) P is absolutely continuous with respect to P ∗, denoted P << P ∗, ifP (A) = 0 whenever P ∗(A) = 0, ∀A ∈ F .

(ii) if P << P ∗ and P ∗ << P , then we call P and P ∗ equivalent measures,denoted P ∼ P ∗.

In the context of finance, the equivalence of measures is important. Thisis due to the fact that equivalent measure has the same a.s. and null sets,hence by changing between measures, we do not alter the possible states in

2 LEVY PROCESSES 25

the economy, we only alter the probabilities assigned to each states. Thisensures the fairness of prices under changes of measures and numeraires.

To understand how change of measures works for Levy processes, we firstrecall how it works in the simpler setting of Brownian motion. Here wechange between two equivalent probability measures via a density process(Radon-Nikodym derivative)

γt =dP ∗

dP

∣∣∣∣Ft

= exp

∫ t

0

G⊤s dWs −

1

2

∫ t

0

|Gs|2ds

= 1 +

∫ t

0

GsγsdWs

where through the Girsanov theorem for Brownian motion we get that

W ∗t = Wt −

∫ t

0

Gsds

is a P ∗-Brownian motion. Hence according to this change of measure, we addand subtract drift (compensators9), and then shift the probability measureto make a P -martingale plus drift a P ∗-martingale.

Wt = Wt −∫ t

0

Gsds +

∫ t

0

Gsds = W ∗ +

∫ t

0

Gsds

Now consider doing the same for a Levy process, ie. we add and subtract driftGs for the Brownian motion as above, and add and subtract drift for the jumppart (the new compensator for the jump part) H(s, x)ν(dx). Afterwards wethen shift the measure to make both Brownian motion and the compensatedjump part P ∗ martingales.

Lt =at + Σ1/2Wt +

∫

Rd

x(N(t, dx) − tν(dx))

=at + Σ1/2

(

Wt ±∫ t

0

Gsds

)

+

∫

Rd

x(N(t, dx) − tν(dx) ± H(t, x)ν(dx))

=at + Σ1/2

∫ t

0

Gsds +

∫ t

0

∫

Rd

x(H(s, x) − 1)ν(dx)ds + Σ1/2W ∗t

+

∫ t

0

∫

Rd

x(N∗(ds, dx) − ν∗(ds, dx))

More formally we have the following theorem

9A compensator of the process X, is a finite variation process, X, such that X − X isa martingale.

26 2 LEVY PROCESSES

Theorem 6. (Girsanov) Assume P and P ∗ are two equivalent probabilitymeasures and let γ be the density process defined as

γt = 1 +

∫ t

0

γs−G⊤s dWs +

∫ t

0

∫

Rd

γs−(H(s, x) − 1)N(ds, dx) − ν(dx)ds))

where ν is the Levy measure, G is a d dimensional previsible process and H isa 1 dimensional previsible process such that E[γt] = 1. Suppose furthermorethat

E

[∫ t

0

|Gs|2ds

]

< ∞,

∫

Rd

(H(t, x) − 1)ν(dx) < ∞

Then under P ∗, the process

W ∗t = Wt −

∫ t

0

Gsds

is a standard Brownian motion, and the process

∫ t

0

∫

Rd

x(N∗(ds, dx) − ν∗(ds, dx)) =

∫

Rd

x(N(t, dx) − tν(dx))

−∫ t

0

∫

Rd

x(H(s, x) − 1)ν(dx)ds

is a (compensated) quadratic pure jump process with compensator

ν∗(ds, dx) = H(s, x)ν(dx)ds

Remark 5. By solving the expression for γ in the above theorem, we obtainthe following expression for the density process

γt = exp

∫ t

0

G⊤s dWs −

1

2

∫ t

0

|Gs|2ds −∫ t

0

∫

Rd

(H(s, x) − 1)ν(dx)ds

+

∫ t

0

∫

Rd

log H(s, x)N(ds, dx)

We see from the above theorem, that Girsanovs theorem for Brownian mo-tions is a special case of theorem 6, namely with ν(dx) = 0. Furthermorewe see that if G and H depends on time, then the process under P ∗ will notbe a Levy process, but rather a semimartingale or non-homogeneous Levyprocess as described in the following section.

2 LEVY PROCESSES 27

2.8 Non-homogeneous Levy processes

As mentioned above change of variables via the Ito formula or change ofMeasure can result in processes that are not Levy processes, but rather moregeneral semimartingales. A special case is non-homogeneous Levy processes(also known as additive processes).

The definition of the process is the same as a Levy process, but we relax theassumption of stationary increments:

Definition 11. (Non-homogeneous Levy process) An adapted, cadlagprocess X ≡ Xt, t ≥ 0 with X0 = 0 almost surely (a.s.) is a non-homoge-neous Levy process if

(i) X has increments independent of the past, ie. Xt − Xs is independentof Fs, 0 ≤ s < t < ∞;

(ii) X is continuous in probability, ie. ∀ǫ > 0, lims→0 P (|Xt+s −Xt| ≥ ǫ) =0.

Similar to the Levy-Khintchine formula for Levy processes there is a versionfor non-homogeneous Levy processes (see Cont & Tankov (2004), chap. 14)

Theorem 7. (Levy-Khintchine formula) Consider a collection of triples(a, Σ, ν) ≡ (at, Σt, νt), t ≥ 0, such that

1. For all t; at is a d× 1 vector, Σt is a positive definite d× d matrix andνt is positive measure on R

d0 ≡ (R \ 0)d with

∫

Rd0

(1∧|x|2)νt(dx) < ∞.

2. Positiveness: a0 = 0, Σ0 = 0, ν0 = 0 and for all s, t such that for s ≤ t;At −As is is a positive definite d× d matrix and νt(A) ≥ νs(A) for allmeasurable sets A ∈ B(Rd).

3. Continuity: If s → t then as → at, Σs → Σt and νs(A) → νt(A) for allmeasurable sets A ∈ B(Rd) such that A ⊂ x : |x| > ǫ for some ǫ > 0

Let X be a d dimensional non-homogeneous Levy process; then

φX(u, t) = E

[

eiu⊤Xt

]

= eϕ(u,t)

where

ϕ(u, t) = iu⊤at −1

2u⊤Σtu +

∫

Rd0

(

eiu⊤x − 1 − iu⊤x1|x|<1

)

νt(dx)

The triple (at, Σt, νt) is called the spot characteristics.

28 2 LEVY PROCESSES

Remark 6. The standard version of the Levy-Khintchine formula is natu-rally obtained as a special case of the above theorem with

at = bt, Σt = Γt, νt = µt

such that the characteristic triple of the homogeneous Levy process will be(b, Γ, µ).

One may wonder on how to construct non-homogeneous Levy processes; onepossibility is to define a non-homogeneous process X defined via the stochas-tic integral

Xt =

∫ t

0

f(s)⊤dLs

where L is a (homogeneous) Levy process and f(s) is a deterministic functionsuch that f : R 7→ R

d.

Since we are interested in describing the process and use it for derivativespricing we are interested in knowing the characteristic function

φX(u, t) = E [expiuXt] = E

[

exp

iu

∫ t

0

f(s)⊤dLs

]

Using the definition of the stochastic integral on a time partition 0 = t0 ≤t1 ≤ . . . ≤ tn = t with ∆tk = tk+1 − tk, we obtain

φX(u, t) = limsup∆tk→0

E

[

exp

iun−1∑

k=0

f(tk)⊤(Ltk+1

− Ltk)

]

Due to the independent increments and stationary increments of the Levyprocess we get

φX(u, t) = limsup∆tk→0

n−1∏

k=0

E[exp

iuf(tk)

⊤(Ltk+1− Ltk)

]

= limsup∆tk→0

n−1∏

k=0

E[exp

i(uf(tk)

⊤)L∆tk

]

= limsup∆tk→0

n−1∏

k=0

exp ϕ(uf(tk))∆tk

= limsup∆tk→0

exp

n−1∑

k=0

ϕ(uf(tk))∆tk

2 LEVY PROCESSES 29

Recognising the Riemann sum we get

φX(u, t) = exp

∫ t

0

ϕ(uf(s))ds

More formally we need to check that the integral actually converges, this hasbeen done in Kluge (2005) and gives us the following lemma

Lemma 1. Let L be a d dimensional Levy process with Levy exponent ϕ(u)and let f be a function such that f : R 7→ R

d, then the process X ≡ Xt, t ≥0 defined as

Xt =

∫ t

0

f(s)⊤dLs

is a non-homogeneous Levy process and has characteristic function

φX(u, t) = E

[

exp

iu

∫ t

0

f(s)⊤dLs

]

= exp

∫ t

0

ϕ(uf(s))ds

30 2 LEVY PROCESSES

3 PRICING USING THE FAST FOURIER TRANSFORM 31

3 Pricing using the Fast Fourier Transform

3.1 Introduction

One of the main purposes of using Levy processes in finance, is of courseto price derivatives. However in many cases where the underlying asset isdriven by a Levy process, no closed form solution for option prices exist.A few special cases are when the underlying asset is driven by a Brownianmotion (Black & Scholes (1973)) and a jump-diffusion where the jumps arenormally distributed (Merton (1976)) 10.

In other cases option prices have to be computed by numerical methods.This could for instance include special functions, such as modified Besselfunctions (Madan, Carr & Chang (1998)) or Fourier inversion method whereone calculates the delta and the in-the-money probability11 (Heston (1993)and Bates (1996)). However if one needs to calculate many option prices,for instance when performing model calibration, this may take a significantamount of time.

In this section we present a pricing method for pricing European options,that uses the computational power of the Fast Fourier Transform (henceforthFFT). The use of the FFT was introduced by Carr & Madan (1999) inthe case of European put and call options, where a more general approachcan be found in Raible (2000). For an introduction to Fourier methods infinance, both in discrete and continuous time, see Cerny (2004a, 2004b). Anintroduction to the FFT will be given in section 3.5, however a more detaileddescription of the computational methods behind the FFT, can be found inPress et al (2002).

3.2 Option pricing using Fourier inversion

In the following sections we assume that the given model parameters areobserved under the T -forward martingale measure QT . Assume furthermorethat we have a complete probability space

(Ω,F , Ftt≥0 , QT

). This implies

that, given no arbitrage opportunities, the asset price processes discountedwith the T zero coupon bond, are martingales.

10Strictly speaking, it is the jumps of the return process of the Merton (1976) modelthat are normally distributed, making the jumps in the underlying asset log-normallydistributed.

11The methods in this section are Fourier inversion methods as well, but are morerecent developments that require fewer calculations and seem to be more stable, see Carr& Madan (1999)

32 3 PRICING USING THE FAST FOURIER TRANSFORM

We place ourselves at time 0 and are pricing European options with expira-tion date T . We assume that the price of the underlying asset follows theprocess

XT = a exp(b⊤LT ) = exp(xT )

where xT = log XT = log a + b⊤LT , L is a d dimensional Levy process 12,a > 0, b ∈ R

d and a, b ∈ F0. Note as long as a, b ∈ F0, a and b can alsorepresent integrable processes. This will be used when we are consideringinterest rate derivatives.

Next consider a European option with contract function Φ(xT ) then the priceof the option can be found as V (0, X0) = p(0, T )ET [Φ(xT )], where p(0, T ) isthe time-0 price of a zero coupon bond maturing at time T . We can then findV (0, X0) by Fourier inversion methods, as stated in the following theorem:

Theorem 8. Consider an asset whose dynamics can be written as XT =a exp(b⊤LT ) and a European option with contract function Φ(log XT ). As-sume that there exist a β ∈ R such that x 7→ eβx|Φ(x)| is bounded and

integrable and ET[

e−βb⊤LT

]

< ∞.

Then the time-0 value of the option can be calculated as

V (0, x0) =p(0, T )e−β log a

2π

∫ ∞

−∞

e−iu log aΨ(β + iu)φL ((iβ − u)b, T ) du (5)

where φL is the characteristic function of the Levy process L and Ψ is theFourier transform of the contract function

Ψ(v) =

∫ ∞

−∞

Φ(x)evxdx

Proof. See appendix A.1

We see that the integral in (5) is a Fourier transform. This allows us tocalculate option prices using the FFT. More on the FFT algorithm is givenin section 3.5.

12In general the process need not be a Levy process, but any process where the charac-teristic function is known.


3.3 European Call options

In this section we will make theorem 8 a bit more concrete. We will considertwo plain vanilla options, namely European put and call options. Please notethat Raible (2000) provides the Fourier transform of the contract functionfor other pay off types as well - we however, confine ourselves to Europeanput and call options.

The European call option gives the holder of the option the right (but notthe obligation) to buy the underlying asset at a specified exercise price K.This implies that the terminal pay off is given by (XT − K)+, therefore thecontract function, Φ, is given by

Φ(xT ) = (exT − K)+

By applying the inverse Fourier transform to this function we can then pricethe option using theorem 8. However this will only be applicable for onesingle exercise price, moreover the FFT algorithm will give us prices for thissingle exercise price and several values of log a. In the case of European putand call options the following lemma will prove the solution to this problem:

Lemma 2. Let C(0, x0, K) denote the price of a European call option withexercise price K observed at time 0. Then we have the following relationship

C(0, x0, K) = KC(0, x0 − log K, 1)


The lemma tells us that instead of varying the exercise price, we can vary thestate variable and then multiply by the exercise of the option. The benefitis twofold - (a) we only need to derive the Fourier transform of the contractfunction for one exercise price, namely K = 1, and (b) varying the statevariable instead of the exercise price, enables us to use the FFT to obtainmultiple prices in an efficient way.

We now need to derive the Fourier transform of the contract function for theoption with exercise price 1. We can find it as:

Lemma 3. Let Φ(x) = (ex − 1)+ be the contract function of the Europeancall option with exercise price 1. Then let v ∈ C such that Re(v) = β < −1then

Ψ(v) =1

v(v + 1)



3.4 European Put options

In the above section we considered the pricing of European call options. Inthe case of European put options we could use the Put-Call-Parity to obtainthe prices. However for the sake of completeness, this section will considerthe pricing of European put options using the Fourier Inversion methodsdescribed above.

The European put option gives the owner the right (but not the obligation)to sell the underlying asset at a specified exercise price. Then the price ofthe option at maturity can then be written as (K −XT )+ which again givesus the contract function

Φ(xT ) = (K − exT )+

As for the call option we have the following relationship between state vari-able and exercise price

Lemma 4. Let P (0, x0, K) denote the price of a European put option withexercise price K observed at time 0. Then we have the following relationship

P (0, x0, K) = KP (0, x0 − log K, 1)

Proof. Analogous to the call option proof.

Again this lets us price put options, having different strike prices, by varyingthe state variable. So to price the the put option we only need to derive theFourier transform of the contract function for one specific strike price:

Lemma 5. Let Φ(x) = (1 − ex)+ be the contract function of the Europeanput option with exercise price 1. Then let v ∈ C such that Re(v) = β > 0then

Ψ(v) =1

v(v + 1)

Proof. See appendix A.4.

Remark 7. It is noticeable that the Fourier transform of the put and calloption contract function is the same, except the region where it is defined.


3.5 The Fast Fourier Transform

The Fast Fourier Transform (FFT) is an algorithm for calculation the discreteFourier transform of the sequence yjj=1,...,n, where the discrete Fouriertransform is a sequence zkk=1,...,n such that:

zk =n∑

j=1

e−i 2πn

(j−1)(k−1)yj, k = 1, . . . , n (6)

A simple implementation of the discrete Fourier transform will require O (n2)operations. However an efficient implementation of the discrete Fourier trans-form known as the Fast Fourier transform will only require O (cn log n) oper-ations. In general, if the number of terms in the sum, is not chosen carefully,the constant c can be rather large. For the FFT to be most efficient n hasto be a power of 2.

It should be quite obvious that if implemented such that the constant cis small, then the FFT can provide a significant enhancement in terms ofcomputational times. This will be useful when multiple option prices needto be calculated, for instance when calibrating models.

3.6 Approximation of the Fourier integral

We now turn to the actual implementation of the pricing method. Our pricingmethod consists of calculating

V (0, x0) =p(0, T )e−β log a

2π

∫ ∞

−∞

e−iu log ag(u)du

where

g(u) = Ψ(β + iu)φL ((iβ − u)b, T )

Now since g(u) consists of Fourier transforms of real valued functions, then ghas the property g(−u) = g(u), where g(u) is the complex conjugate of g(u)(see Raible (2000)).


This implies that

∫ ∞

−∞

e−iu log ag(u)du =

∫ 0

−∞

e−iu log ag(u)du +

∫ ∞

0

e−iu log ag(u)du

=

∫ ∞

0

eiu log ag(−u)du +

∫ ∞

0

e−iu log ag(u)du

=

∫ ∞

0

e−iu log ag(u)du +

∫ ∞

0

e−iu log ag(u)du

=2Re

(∫ ∞

0

e−iu log ag(u)du

)

since the complex conjugate eliminates the complex part of the integrals.

Next consider approximating the integral with a sum

V (0, x0) ≈p(0, T )e−β log a

πRe

(n∑

j=1

e−i(j−1)∆u log ag((j − 1)∆u)wj∆u

)

where wj is the jth weight in the integral approximation13. This approxi-mation can then be used if we only want to price the option for one specificexercise price (or equivalent value of the state variable log a).

Now let xk = log a and define

xk = −γ + (k − 1)∆x

which gives us

V (0, x0) ≈p(0, T )e−βxk

πRe

(n∑

j=1

e−i(j−1)∆u(−γ+(k−1)∆x)g((j − 1)∆u)wj∆u

)

=p(0, T )e−βxk

πRe

(n∑

j=1

e−i(j−1)(k−1)∆u∆xei(j−1)∆uγg((j − 1)∆u)wj∆u

)

and comparing this to equation (6) we see that if

∆u∆x =2π

nand yj = e−i(j−1)∆uγg((j − 1)∆u)wj∆u

we can calculate the integral for multiple values of xk using the FFT.

13For instance when using the trapezoid rule we have w1 = wn = 1/2 and wj = 1 forj = 2, . . . , n − 1.


3.7 Outline of the pricing algorithm

Using the results we derived above we can calculate multiple option pricesby using the FFT algorithm.

Let φL be the characteristic function of underlying Levy process and let Φbe the Fourier transform of the contract function. Then we can calculate theoption price by using the following steps:

• Choose a β such that x 7→ eβx|Φ(x)| is bounded and integrable and

ET[

e−βb⊤LT

]

< ∞.

• Choose a spacing in the integration variable ∆u and set the number ofintervals n. Furthermore let n be power of 2 to use the efficiency of theFFT algorithm.

• Calculate the state variable spacing ∆x = 2πn∆u

. Select a value of γ - if

one wants x1 = −γ and xn = γ then we can find γ as: γ = (n−1)πn∆u

• Calculate the sequence yjj=1,...,n defined as

yj = e−i(j−1)∆uγΨ(β + i(j − 1)∆u))φL ((iβ − (j − 1)∆u)b, T ) wj∆u

for j = 1, . . . , n.

• Perform the FFT on the sequence yjj=1,...,n and obtain the trans-formed sequence zkk=1,...,n.

• The price for log a = xk = −γ + (k − 1)∆x can then be found as

V (0, xk, 1) =p(0, T )e−βxk

πRe(zk)

• Find the strike prices using lemma 2 and 4, ie. Kk = elog a−xk whichgives us the prices

V (0, log a,Kk) = KkV (0, xk, 1)

3.8 FFT pricing in the Black-Scholes model

I this section we will, as an example, show the FFT method in the case ofthe Black-Scholes model. The choice of the Black-Scholes model is twofold;(a) the Black-Scholes model has an analytical solution for European put and


call options which allows assessment of the approximation error and (b) it isa good benchmark for debugging purposes. Appendix E shows the MATLAB

code of the implementation of the FFT pricing14.

In the Black-Scholes model it is assumed that under the risk neutral measureQ, the underlying asset follows a geometric Brownian motion 15

dSt = rStdt + σStdWt

which has the solution

ST = S0 exp

(

r − 1

2σ2

)

T + σWT

This results in the celebrated Black-Scholes option pricing formula for a Eu-ropean call option

C(0, S0, K) =S0N(d1) − e−rT KN(d2)

d1 =log S0/K +

(r + 1

2σ2)T

σ√

T, d2 = d1 − σ

√T

where S0 is the spot price of the underlying asset, K is the exercise price, ris the risk free interest rate, σ is the volatility of the underlying asset and Tis the time to maturity of the option. Finally N(d) is the cumulative normaldistribution evaluated at d.

In the context and notation of section 3.2, this implies that a = S0, b = 1and the characteristic function is given as

φB/S(u, T ) = exp

iu

(

r − 1

2σ2

)

T − 1

2u2σ2T

The remaining part of this section will consider the pricing of 101 call options(with different exercise prices) using the FFT method. We consider thefollowing parameters:

S0 σ r T100 0.20 0.05 1.00

14Although the actual implementation of the Levy HJM model is done in C++, we havechosen to implement this simpler example in MATLAB, so the code is more easily read.

15When interest rates are assumed to be deterministic, the dynamics under the T -forward measure is the same as under the standard risk neutral measure. See Bjork(2004), chap. 24 for further detail.


and will consider exercise prices ranging from 50 to 150 with a spacing of 1.

Since the FFT algorithm will give us equidistant prices in a log-exerciseprices, we will not have equidistant exercises prices. Hence to obtain equidis-tant exercise prices some kind of interpolation must be used. Here we chooseto interpolate by using a cubic spline. To avoid unnecessary (and inefficient)implementation of the cubic spline we use the in-built spline function inMATLAB, and to avoid unnecessary numerical calculations we only interpolatebetween exercise prices that are relevant, ie. only for the part of the pricefunction where the selected 101 exercise prices is in-between.

To assess the approximation error we use root mean squared error (henceforthRMSE):

RMSE =

√√√√

1

N

N∑

j=1

(CB/S(0, S0, Kj) − CFFT (0, S0, Kj)

)2

Where N is the number of strike prices that we are considering. We see thatRMSE gives us an average pricing error for the N exercise prices.

Following Carr & Madan (1999) we set ∆u = 0.25 and more or less arbitrarilywe set β = −10. We experimented with the β parameter and found forβ < −2 the pricing errors seemed stable. For β → −1 we experiencedincreasing and significant pricing errors.

∆u = 0.25, β = −10Points in integration (n) 128 256 512Comp. Time (Seconds) 0.004 0.006 0.007RMSE 1.91E-02 9.74E-04 5.52E-05Points in integration (n) 1,024 2,048 4,096Comp. Time (Seconds) 0.008 0.011 0.017RMSE 2.84E-06 1.51E-07 9.62E-09

Table 1: Computational times and RMSE for the FFT pricing algorithm forvarious choices of points in the numerical integration.

Table 1 shows the computational times and pricing errors for different numberof points in the numerical integration, n. We see that for all choices of n thecomputational time is very small; the largest computational time is 0.017seconds. Furthermore it is seen that the RMSE decreases very rapidly asone increases n, and is acceptable for all practical applications already forn ≥ 512. The rapid convergence is also shown in figure 2 which shows that


the order of convergence is 4, implying that when the number of points inthe integration is doubled, then the RMSE decreases by a factor 16.

102

103

104

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of points in integration

Roo

t Mea

n Sq

uare

d Er

ror

Figure 2: Convergence of the FFT pricing algorithm as the number of pointsin the integration increases. The figure is plotted on a log-log scale to showthe convergence of the algorithm. The slope of the line is −4.21, hence theorder of convergence is approximately 4.

4 THE LEVY HJM MODEL 41

4 The Levy HJM model

4.1 Fixed income basics

It this section we will the describe the basics of fixed income markets. Thiswill be the definition of different interest rates, such as continuously com-pounded rates and LIBOR rates. Most of this section follows Bjork (2004),chap. 20 16.

First we recall that a zero coupon bond with maturity T (also called T -bond),is a contract that pays the owner of the zero coupon bond 1 unit of currencyat time T . The zero coupon bond observed at time t that matures at timeT is denoted p(t, T ).

In the money market it is customary to consider simple compounding, ie. weare considering the simple forward rate (LIBOR foward rate ) L(t; S, T ). TheLIBOR forward rate agreement, is an agreement to borrow or lend betweentime S and T at a time t specified simple rate L(t; S, T ). By no-arbitrage wehave the following relationship between zero coupon bond prices and LIBORforward rates.

p(t, S)

p(t, T )= 1 + (T − S)L(t; S, T ) ⇔ L(t; S, T ) = − 1

T − S

p(t, T ) − p(t, S)

p(t, T )

The case of the simple spot rates (LIBOR spot rates), is denoted L(t, T ), ie.a simple compounded rate starting from today (time t) to some future pointin time T . This implies that the LIBOR spot rate is the same as a LIBORforward rate with S = t, ie. L(t, T ) = L(t; t, T ). Hence we have

L(t, T ) = − 1

T − t

p(t, T ) − 1

p(t, T )

In most models we are not considering simple rates (the LIBOR marketmodel being the exception), but instead it is more convenient to considercontinuously compounded rates.

When considering how to derive the the continuously compounded forwardrates from zero coupon bonds, we will use that it will not make a differencefrom investing in zero coupon bonds or an asset with continuously com-pounded rates

p(t, S)

p(t, T )= eR(t;S,T )(T−s) ⇔ R(t; S, T ) = − log p(t, T ) − log p(t, S)

T − S

16We do not cover swap rates in this section, as we are not pricing swaptions in thisthesis. The interested reader can refer to Bjork (2004), chap. 20 for more details.

42 4 THE LEVY HJM MODEL

Again we can find the continuously compounded spot rates, R(t, T ), by lettingS = t in the continuously compounded forward rates

R(t, T ) = − log p(t, T )

T − t

Finally in most continuous time interest rate models, it customary to useinstantaneous continuously compounded rates, ie. rates that prevail over aninstantaneous time interval [T, T + dt]. The instantaneous forward rates canbe obtained by letting (T −S) → 0 in the continuously compounded forwardrates

f(t, T ) = lim(T−S)→0

− log p(t, T ) − log p(t, S)

T − S= −∂ log p(t, T )

∂T

and similarly we define the instantaneous short rate as

r(t) = f(t, t)

Using the definitions above we have the following relationship for t ≤ s ≤ T

p(t, T ) = p(t, s) exp

−∫ T

s

f(t, u)du

and in particular

p(t, T ) = exp

−∫ T

t

f(t, u)du

4.2 The standard HJM framework

In this section we will briefly recap the standard Heath-Jarrow-Morton frame-work (henceforth HJM) as introduced in Heath, Jarrow & Morton (1992).We will do this to ensure the the reader is equipped to consider the moreadvanced HJM framework based on Levy process.

In the standard version of the HJM framework we assume as given a stochas-tic basis

(Ω,F , Ftt≥0 , Q

), where Q is the risk neutral martingale measure.

Then the instantaneous forward rates are described by the Ito process

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

∫ t

0

α(s, T )ds +

∫ t

0

σ(s, T )⊤dWs

where W is a d dimensional standard Brownian motion adapted to the fil-tration Ftt≥0, α(s, T ) is a process in R and σ(s, T ) is a process in R

d.


This specification of the forward rates implies that the stochastic differentialequation for the bond prices is given by (see Bjork (2004), prop. 20.5)

dp(t, T ) = p(t, T )

(

r(s) + A(t, T ) +1

2|S(t, T )|2

)

dt + p(t, T )S(t, T )⊤dWt

or equivalently the bond prices

p(t, T ) = p(0, T ) exp

∫ t

0

(r(s) + A(s, T )) ds +

∫ t

0

S(s, T )⊤dWs

where

A(s, T ) = −∫ T

s

α(s, u)du, S(s, T ) = −∫ T

s

σ(s, u)du

Heath, Jarrow & Morton (1992) then show that for absence of arbitrage wemust have the following condition on the drift function α(t, T ) (HJM driftcondition):

α(t, T ) = σ(t, T )⊤∫ T

t

σ(t, s)ds

The HJM drift condition implies that when specifying forward rate modelsunder the risk neutral martingale measure Q, all we need is to specify thevolatility structure. The volatility structure σ(t, T ) can both deterministicor stochastic.

When the volatility structure is deterministic we obtain a model where theforward rates are Gaussian, and hence the zero coupon bond prices are log-normal; this allows for analytically tractable models, but with serious empir-ical flaws.

If the volatility structure is stochastic the model is better able to capturethe empirical facts found in the fixed income market, such as the volatilitysurface for caps/floors and swaptions. An example of a stochastic volatilityHJM model can be found in Schwartz & Trolle (2007).

4.3 The general Levy HJM framework

In this section we will describe the Levy HJM framework as introduced inEberlein & Raible (1999). We will confine ourselves to models where thevolatility specification is deterministic. In this context it should be men-tioned that Raible (2000) shows that it is possible to include a volatility


specification that is driven by a stochastic process. Furthermore Carr et al(2003) show that when working with Levy processes stochastic volatility canbe introduced by a time change of the Levy process, where the subordinatorwill vary in intensity over time. Carr et al (2003) use an integrated squareroot process as the subordinator. Which of the two methods for generatingstochastic volatility that is the best is outside the scope of this thesis, andwill hence be left for future research.

When describing the Levy HJM framework, we assume as given a stochasticbasis

(Ω,F , Ftt≥0 , Q

), where Q is the risk neutral martingale measure. In

the Levy HJM framework we assume that the instantaneous forward ratesare driven by the process

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

∫ t

0

α(s, T )ds +

∫ t

0

σ(s, T )⊤dLs

where L is a d dimensional Levy process adapted to the filtration Ftt≥0,

α(s, T ) is a function in R and σ(s, T ) is a function in Rd. We furthermore

assume that σ(s, T ) is deterministic and bounded. Finally the Levy processhas Levy exponent given as

ϕ(u) = iu⊤a − 1

2u⊤Σu +

∫

Rd0

(

eiu⊤x − 1)

ν(dx)

This implies that the Levy process can be written as

Lt = at + Σ1/2Wt + Jt

where a is the drift, Σ is the diffusion matrix, and Wt is a Brownian motionwhich is independent from the jump process Jt.

We note that the only difference from the Gaussian HJM setting is thatthe driving source of uncertainty is now a Levy process instead of Brownianmotion. This in turn implies that the Gaussian HJM model is a special caseof the Levy HJM model.

As shown above the zero coupon bond prices can be expressed in terms offorward rates.

p(t, T ) = exp

−∫ T

t

f(t, s)ds

(7)

This leads to the explicit solution for bond prices in this model framework:


Theorem 9. The bond price p(t, T ) is given by

p(t, T ) = p(0, T ) exp

∫ t

0

(r(s) + A(s, T )) ds +

∫ t

0

S(s, T )⊤dLs

(8)

where

A(s, T ) = −∫ T

s

α(s, u)du, S(s, T ) = −∫ T

s

σ(s, u)du

where the integrals over σ(s, u) is to be understood elementwise, ie.

Si(s, T ) = −∫ T

s

σi(s, u)du


As we now know the expression for zero coupon bond prices, it will also beconvenient to be able have an expression for the bank account in this model.We recall that the bank account is given by

β(t) = exp

∫ t

0

r(s)ds

Letting T = t in equation (8) and solving for β(t) gives us

β(t) =1

p(0, t)exp

−∫ t

0

A(s, t)ds −∫ t

0

S(s, t)⊤dLs

Inserting this into the bond price we obtain the following equivalent expres-sion for the bond price

p(t, T ) =p(0, T )

p(0, t)exp

∫ t

0

A(s, T ) − A(s, t)ds +

∫ t

0

(S(s, T ) − S(s, t))⊤ dLs

Sometimes, for instance when simulating the bond prices, it can be convenientto know the stochastic differential equation for the bond prices. We have that:

Lemma 6. The dynamics of the bond price is given by

dp(t, T )

p(t−, T )=

(

r(t) + A(s, T ) + S(t, T )⊤a +1

2S(t, T )⊤ΣS(t, T )

)

dt (9)

+ S(t, T )⊤Σ1/2dWt +

∫

Rd0

(

eS(t,T )⊤x − 1)

N(dt, dx)



The main purpose of the rest of this section is to derive a condition onA(s, T ) that ensures no arbitrage. From the first fundamental theorem ofasset pricing, we know that all price processes discounted with the appropri-ate numeraire are martingales. When considering the standard risk neutralmeasure Q the numeraire will of course be the bank account.

Let Z(t) be the discounted zero coupon bond price process, hence we havethat

Z(t) =p(t, T )

β(t)= p(0, T ) exp

∫ t

0

A(s, T )ds +

∫ t

0

S(s, T )⊤dLs

By using the same methodology as given in the proof of lemma 6, we can getthe stochastic differential equation of this process

dZ(t)

Z(t−)=

(

A(t, T ) + S(t, T )⊤a +1

2S(t, T )⊤ΣS(t, T )

+

∫

Rd0

(

eS(t,T )⊤x − 1)

ν(dx)

)

dt

+ S(t, T )⊤Σ1/2dWt +

∫

Rd0

(

eS(t,T )⊤x − 1)

(N(dt, dx) − ν(dx)dt)

Note that compared to lemma 6, we have added and subtracted the compen-sator of the jump process, making the last line a martingale.

Hence for Z to be a (local) martingale we must eliminate the drift of theprocess. This will happen when

A(t, T ) = −(

S(t, T )⊤a +1

2S(t, T )⊤ΣS(t, T ) +

∫

Rd0

(

eS(t,T )⊤x − 1)

ν(dx)

)

for all t and T ≥ t. The term in the brackets is exactly the cumulantgenerating function of the Levy process evaluated at S(t, T ). Thus we musthave

A(t, T ) = −θ(S(t, T ))

To summarise we have the following theorem

Theorem 10. (HJM drift condition) Under the martingale measure Qthe processes A(t, T ) and S(t, T ) must satisfy the following relationship, forevery t and every T ≥ t:

A(t, T ) = −θ(S(t, T ))


where θ(u) is the cumulant generating function of the driving Levy process,ie. θ(u) = ϕ(−iu) = log E

Q[exp u⊤L1

].

As describing the HJM drift condition in terms of the cumulant generatingfunction is somewhat different from the usual HJM drift condition in theBrownian motion driven models, we here give an example that show theequivalence of the of the two drift conditions.

Example 7. (Gaussian HJM drift condition) Consider the GaussianHJM model where the forward rates are driven by a d dimensional Brownianmotion

df(t, T ) = α(t, T )dt + σ(t, T )⊤dWt

where W it⊥W j

t , for i 6= j, ie. the cumulant generating function of the Brow-nian motion is

θ(u) =1

2u⊤Iu

Then for a general deterministic volatility function σ(s, t) ∈ Rd we have that

A(t, T ) = −1

2S(t, T )⊤IS(t, T )

∫ T

t

α(t, u)du = −1

2

(∫ T

t

−σ(t, u)⊤du

)(∫ T

t

−σ(t, u)du

)

Differentiating wrt. T gives us

α(t, T ) = σ(t, T )⊤∫ T

t

σ(t, u)du

which is equivalent to the HJM drift condition in Bjork (2004), theorem 23.2.

4.4 Change of Numeraire

In this section we will present the change of numeraire technique, formallydescribed in Geman, El Karoui & Rochet (1995), in the setting of the LevyHJM model. First we recall the pricing of interest rate dependent deriva-tives, and the problems with using the bank account as numeraire. We willshow that by changing the numeraire, we will avoid the evaluation of a jointdistribution of two stochastic variables.


We recall that prices of derivatives discounted with the bank account haveto be martingales, or equivalently

Vt = β(t)EQ

[VT

β(T )

∣∣∣∣Ft

]

(10)

When pricing interest rate derivatives, the terminal pay-off and bank accountwill be correlated, making the above expectation particularly nasty. In thecase of the Levy HJM model, Eberlein & Raible (1999) manages to evaluatethe joint distribution of the bank account and a zero coupon bond. In generalit will be sensible to change the numeraire, to avoid the evaluation of a jointdensity. Thus we need to consider the Radon-Nikodym derivative

γt =dQT

dQ

∣∣∣∣Ft

The specific form of the Radon-Nikodym derivative can be found by knowingthat when using the p(t, T ) as a numeraire, we must have

V0 = p(0, T )ET

[VT

p(T, T )

]

= p(0, T )EQ

[VT

p(T, T )γT

]

Which gives us (together with equation (10)).

p(0, T )EQ

[VT

p(T, T )γT

]

= β(0)EQ

[VT

β(T )

]

or equivalently we can deduce

γt =β(0)

p(0, T )

p(t, T )

β(t)

By the abstract Bayes theorem (Bjork (2004), prop. B.41) we get that

Vt =p(t, T )ET

[VT

p(T, T )

∣∣∣∣Ft

]

= p(t, T )EQ

[VT

p(T, T )

γT

γt

∣∣∣∣Ft

]

=p(t, T )EQ

[VT

p(T, T )

β(0)

p(0, T )

p(T, T )

β(T )

∣∣∣∣Ft

](p(0, T )

β(0)

β(t)

p(t, T )

)

=β(t)EQ

[VT

β(T )

∣∣∣∣Ft

]

Which confirms that the Radon-Nikodym derivative is indeed valid.


So far we have not specified any dynamics of the bank account or the bondprices, so the above derivation must hold for any model. In particular in theLevy HJM framework we have that

γt =β(0)

p(0, T )

p(t, T )

β(t)=

p(t, T )

p(0, T )β(t)

= exp

∫ t

0

A(s, T )ds +

∫ t

0

S(s, T )⊤dLs

We see that when the HJM drift condition is satisfied, then γ will be amartingale.

To be able to perform pricing of derivatives under the T -forward measure,we must know the behaviour of the Levy process under this measure. Thuswe need to find the characteristic function (or triplet) under this measure.Using the Levy decomposition and the HJM drift condition, we can writethe Radon-Nikodym derivative as

γt = exp

∫ t

0

S(s, T )⊤dLs −∫ t

0

θ(S(s, T ))ds

= exp

−1

2

∫ t

0

S(s, T )⊤ΣS(s, T )ds −∫ t

0

∫

Rd0

(

eS(s,T )⊤x − 1)

ν(dx)ds

+

∫ t

0

S(s, T )⊤Σ1/2dWs +

∫ t

0

∫

Rd0

S(s, T )⊤xN(ds, dx)

Comparing to Girsanovs theorem, we see that in the notation of theorem 6we have

Gt =(Σ1/2

)⊤S(t, T ), H(t, x) = exp

S(t, T )⊤x

Since the volatility function is time dependent, the Levy process will be anon-homogeneous Levy process under the forward measure. In particular byusing Girsanovs theorem we can find the spot characteristics (bT

t , ΓTt , µT

t ) as

bTt = at + Σ

∫ t

0

S(s, T )ds +

∫ t

0

∫

Rd0

x(

eS(s,T )⊤x − 1)

ν(dx)ds

ΓTt = Σt

µTt (dx) =

∫ t

0

eS(s,T )⊤xν(dx)ds

Hence by changing the measure, we add drift to the process, and as in thecase of the model driven solely by Browninan motion, we do not change


the diffusion of the process. We however perform, what is know as, antime-inhomogeneous exponential tilting of the Levy measure. In its time-homogeneous form, this kind of measure is also know as the Esscher measure.

This change of measure implies that jump intensity and distribution of jumpswill change under the forward risk neutral measure QT , compared to the stan-dard risk neutral measure Q. This could significantly alter the probabilitiesassigned to each sample path of the jump process, and must therefore beinterpreted at as forward risk premium.

More specifically we see that the expected value of the jump process underthe QT forward measure can be expressed as

ET [Jt(A)] =

∫ t

0

∫

A

xeS(s,T )⊤xν(dx)ds, A ∈ B(R

d)

Thus it resembles the expectation under the standard risk neutral measureQ, but the expected value of jumps with size belonging to A, will be re-weighted by a factor eS(s,T )⊤x for all x ∈ A. This is exactly the factor thatcan be interpreted as a forward risk premium.

4.5 Pricing Caps and Floors

This section will cover the pricing of caps and floors in the Levy HJM model.We start by recalling the structure of caps and floors, thereby describingcaplets and floorlets. Finally we will show that the pricing can be done byusing the Fourier inversions methods given in section 3.

An interest rate caplet is a contract working on a principal amount N withstrike rate (or cap rate) K, expiry date T , and settlement date T +δ indexedon the LIBOR rate L(T, T + δ), is defined by the time T + δ pay off:

XT+δ = Nδ (L(T, T + δ) − K)+

The term δ is called the tenor of the caplet, and evidently the pay off attime T + δ is known at time T . A caplet is equivalent to a European calloption on the underlying LIBOR and thereby provides protection againstrising LIBOR rates. For unit notional we denote the price of a caplet withexpiry T , tenor δ and cap rate K, evaluated at time t as

CPLt(T, δ,K)

Similarly we can define an interest rate floorlet with the same characteristicsas the caplet, except that the pay off at time T + δ is given by

XT+δ = Nδ (K − L(T, T + δ))+


t T0 T1 T2. . . Tn−1 Tn

δ︷︸︸︷

δ︷︸︸︷

δ︷︸︸︷

TodayStart Date,

(reset)

Settlement/reset

Settlement/reset

Settlement/resetMaturity,

(Settlement)

Figure 3: Structure of a forward starting Cap/Floor.

This makes the floorlet a European put option on the LIBOR rate andthereby provides protection against falling LIBOR rates. For unit notionalwe denote the price of a floorlet with expiry T , tenor δ and cap rate K,evaluated at time t as

FLLt(T, δ,K)

An interest rate cap working on a principal amount N with cap rate K,length n, start date T0, maturity date Tn, reset dates Ti, i = 0 . . . , n− 1, Ti −Ti−1 = δ and settlement dates Ti, i = 1, . . . , n − 1 indexed on the LIBORrates L(Ti, Ti+1), i = 0, . . . , n − 1 is defined by a sequence of pay offs at thesettlement dates given by:

XTi= Nδ (L(Ti−1, Ti) − K)+

A time line of the reset and settlements of the cap is given in figure 3.

A cap is seen to be a portfolio of almost identical caplets that differ only intheir specification of the reset dates. Thus, the contract provides protectionagainst rising LIBOR rates over a number of future dates. For unit notional,start date T0, maturity Tn, tenor δ, cap rate K and a number of paymentdates n, we denote the time t price of the cap as

CAPt(T0, δ,K, n) =n−1∑

i=0

CPLt(Ti, δ,K)

Analogously to the cap we can define an interest rate floor with the samecharacteristics as a the cap, except that the pay off at each settlement datewill be defined as

XTi= Nδ (K − L(Ti−1, Ti))

+

A floor is seen to be a portfolio of almost identical floorlets that differ only intheir specification of the reset dates. Thus, the contract provides protection


against falling LIBOR rates over a number of future dates. Again for unitnotional, start date T0, maturity Tn, tenor δ, floor rate K and a number ofpayment dates n, we denote the time t price of the floor as

FLRt(T0, δ,K, n) =n−1∑

i=0

FLLt(Ti, δ,K)

As we can price caps and floors, by pricing single caplets or floorlets, we willfocus on the pricing of these interest derivatives.

Using the definition above, we aim to price a caplet with unit notional. Sincethe pay off settled at time T + δ is FT -measurable then the time T + δ payoff, valued at time T must be equivalent to the discounted value of this payoff:

p(T, T + δ)XT+δ = p(T, T + δ)δ (L(T, T + δ) − K)+

= p(T, T + δ) ((1 + δL(T, T + δ)) − (1 + δK))+

Using the definition of the LIBOR rate we obtain

p(T, T + δ)XT+δ =p(T, T + δ)

(1

p(T, T + δ)− (1 + δK)

)+

= (1 − (1 + δK) p(T, T + δ))+

= (1 + δK)

(1

1 + δK− p(T, T + δ)

)+

Hence pricing a caplet with cap rate K, settled at time T + δ, is the sameas pricing 1 + δK European put options with maturity T on a zero couponbond maturing at T + δ, where the option has exercise price 1/(1 + δK):

CPLt(T, δ,K) = (1 + δK)PZCBt

(

T, T + δ,1

1 + δK

)

where PZCBt(U, T,K) is the price of a European put option with exerciseprice K and maturity U on a zero coupon bond maturing at time T .

Similarly we can find that the price of a floorlet with floor rate K, settled attime T +δ, is the same as pricing 1+δK European call options with maturityT on a zero coupon bond maturing at T + δ, where the option has exerciseprice 1/(1 + δK):

FLLt(T, δ,K) = (1 + δK)CZCBt

(

T, T + δ,1

1 + δK

)


where CZCBt(T, U,K) is the price of a European call option with exerciseprice K and maturity T on a zero coupon bond maturing at time U .

Since the pricing of caps and floors are equivalent to pricing options on zerocoupon bonds, we now focus on pricing options on zero coupon bond in theLevy HJM framework. We will first consider the pricing of call options, andthen the pricing of put options will follow analogously.

In the following we are considering a European call option with exerciseprice K and maturity T on a zero coupon bond maturing at time U . We findourselves under the T -forward measure, hence we have

CZCB0 (T, U,K) = p(0, T )ET[(p(T, U) − K)+]

To price the options using the methodology described in section 3 we mustknow the characteristic function of the zero coupon bond price. To derive thisunder the T -forward measure, we will consider the following representationof the zero coupon bond price process

p(T, U) =p(0, U)

p(0, T )exp

∫ T

0

θ(S(s, T )) − θ(S(s, U))ds + XT

where

XT =

∫ T

0

(S(s, U) − S(s, T ))⊤ dLs

Hence in the notation of section 3.2, this implies that

a =p(0, U)

p(0, T )exp

∫ T

0

θ(S(s, T )) − θ(S(s, U))ds

and b = 1

Finally we need to have an expression for the characteristic function of XT

in order to be able to use the FFT pricing methods. This is given by

Lemma 7. Under the T -forward measure, the characteristic function of

XT =

∫ T

0

(S(s, U) − S(s, T ))⊤ dLs

is given by

φTX(u) = exp

∫ T

0

θ (iuS(s, U) + (1 − iu)S(s, T )) − θ (S(s, T )) ds

where θ(u) is the Q-cumulant generating function θ(u) = log EQ [expuL1]


Proof. See appendix A.7.

Next using the results derived above and in section 3 we get that

Theorem 11. Assume that there exist a β < −1 such that φTX (iβ) < ∞.

Then in the Levy HJM model, the price of a European call option with exer-cise price K and maturity T on a zero coupon bond with maturity U is givenby

CZCB0 (T, U,K) = KV (0, log a − log K)

where

V (0, log a) =p(0, T )e−β log a

2π

∫ ∞

−∞

e−iu log aΨ(β + iu)φTX (iβ − u) du

and φTX is the characteristic function given in lemma 7 and Ψ is the Fourier

transform of the contract function

Ψ(v) =1

v(v + 1)

and

a =p(0, U)

p(0, T )exp

∫ T

0

θ(S(s, T )) − θ(S(s, U))ds

Proof. The statement follows immediately from applying theorem 8, lemma2 and 3 together with the results from this section.

We see that using the above theorem together with the algorithm given insection 3.7, we can recover prices for call options on zero coupon bonds formultiple exercise prices, just as done in the example given in section 3.8. Aswe have theorem 11 we have the equivalent for put options on zero couponbonds:

Theorem 12. Assume that there exist a β > 0 such that φTX (iβ) < ∞. Then

in the Levy HJM model, the price of a European put option with exercise priceK and maturity T on a zero coupon bond with maturity U is given by

PZCB0 (T, U,K) = KV (0, log a − log K)

where

V (0, log a) =p(0, T )e−β log a

2π

∫ ∞

−∞

e−iu log aΨ(β + iu)φTX (iβ − u) du


and φTX is the characteristic function given in lemma 7 and Ψ is the Fourier

transform of the contract function

Ψ(v) =1

v(v + 1)

and

a =p(0, U)

p(0, T )exp

∫ T

0

θ(S(s, T )) − θ(S(s, U))ds


5 IMPLEMENTATION OF THE LEVY HJM MODEL 57

5 Implementation of the Levy HJM model

5.1 Introduction

In this section we will discuss the implementation of the Levy HJM model.We will consider pricing using a Gaussian HJM model, a model driven by amultivariate Variance Gamma model, as described in Luciano & Schoutens(2006) and a models where the factors are driven by independent VarianceGamma processes. Furthermore we will experiment with different volatilityspecifications, for instance allowing for humped volatility structures. Thecalibration will be done using caplet prices, more specifically using the datafound in Kluge (2005). To our knowledge, the use of multivariate processesand humped volatility structures in the implementation of models based onthe Levy HJM framework, are novelties to the literature, cf. discussion below.

We are aware of two implementations of the Levy HJM model, namely Raible(2000) and Kluge (2005) 17. In this context, Raible (2000) calibrates a onedimensional Levy HJM model to observed returns on zero coupon bonds,hence Raible is estimating the parameters under the real world measure, P .

Kluge (2005) on the other hand calibrates a one dimensional Levy HJM modelto observed caplet prices, and hence estimates the parameters under the riskneutral measure, Q. Furthermore Kluge (2005) considers two types of drivingprocesses; homogeneous and non-homogeneous Levy processes (both basedon the Normal Inverse Gaussian model, see Barndorff-Nielsen (1998)). Thenon-homogeneous Levy process is modelled as a composite of standard Levyprocesses, with the difference that the parameters vary on each time interval,ie. the model will have different parameters when t ∈ [0, 1[ compared tot ∈ [1, 5[. Finally in terms of volatility specifications, then Kluge (2005) onlyconsiders the Vasicek specification of the volatility structure, ie. σ(t, T ) =σe−κ(T−t).

In our opinion the use of one dimensional driving processes is not in anyway optimal. The use of a one dimensional process will, as in any singlefactor model, lead to interest rates that are perfectly correlated. To see thisconsider the correlation between two log-bond prices in a one factor LevyHJM model over a small time interval dt:

ρT,U =Cov(S(t, T )dLt, S(t, U)dLt)

√

Var(S(t, T )dLt)√

Var(S(t, U)dLt)= 1

17It should be mentioned that Eberlein & Kluge (2007) describe the calibration of in-terest models based on Levy processes. In the empirical part of their paper, they howeveronly consider a Levy LIBOR market model.

58 5 IMPLEMENTATION OF THE LEVY HJM MODEL

which follows using the rules for (co-)variance for one dimensional randomvariables and that S(t, T ) is deterministic.

When pricing derivatives that depend on the correlation between interestrates (this could be something as relatively simple as a swaption), it couldprove a serious problem to price and hedge in a one factor model. Kluge(2005) actually prices swaptions in the one factor model. This pricing onlyconsists of at-the-money swaptions, hence it is hard to assess the implicationsof using a one factor model.

5.2 Multivariate Variance Gamma

The multivariate Variance Gamma model, as introduced in Luciano & Schou-tens (2006), is a multi dimensional Levy process which is obtained fromsubordinating a multivariate arithmetic Brownian motion with a commongamma subordinator. When describing the dependence between the elementsof the process, we will refrain from discussing Copula functions. Althoughthis might be the natural setting, a full description of Copulae will be a thesison its own, thus we will try to explain the dependence using scatter plots.

We start by considering the arithmetic Brownian motion:

Yt = µt + Σ1/2Wt

where µ⊤ = (µ1, . . . , µd), Σ1/2 = diag(σ1, . . . , σd) and Wt is a d dimensionalstandard Brownian motion. This implies that the Levy exponent of Y isgiven by

ϕY (u) = iu⊤µ − 1

2u⊤Σu

where Σ = diag(σ21, . . . , σ

2d).

Next step in the construction is to introduce a common gamma subordinator,and as in the case of the univariate VG process, we use a unit mean rategamma process, ie. a gamma process, G, with parameters 1/k and 1/k:

Xt = µGt + Σ1/2WGt

Using theorem 5 together with the Levy exponent of the gamma process, weobtain the Levy exponent

ϕX(u) =1

klog

(1

1 − iu⊤µk + 12u⊤Σuk

)

= −1

klog

(

1 − iu⊤µk +1

2u⊤Σuk

)


or equivalently the cumulant generating function:

θX(u) = ϕX(−iu) = −1

klog

(

1 − u⊤µk − 1

2u⊤Σuk

)

From the specification of the process, we see that the number of parame-ters only increases linearly, when increasing the dimension of the process.This would for instance not be the case with a correlated Brownian motion.This makes the multivariate VG process tractable for modelling purposes.Furthermore it is obvious that the single elements of the process, are onedimensional VG processes, with the same properties and moments as givenin example 6:

E[Xj,t] =µjt

Var(Xj,t) =(σ2

j + µ2jk)t

µ3(Xj,t) =(3σ2

j µjk + 2µ3jk

2)t

µ4(Xj,t) =(3σ4

j k + 12σ2j µ

2jk

2 + 6µ4jk

3)t

As seen from the above equations, especially the variance has a nice interpre-tation. It can be described as an idiosyncratic component, σ2

j , and a commonexogenous component µ2

jk, which arises from the common time change.

So far we have only been concerned with the behaviour of the single ele-ments of the process. When describing the dependence structure betweenGaussian (or more precisely elliptic) random variables, it is fully given bythe correlation between the variables. This is however not the case with theVG process. To see this consider the correlation between the two elementsof the multivariate VG process

ρj,l =E [(Xj,t − E[Xj,t])(Xl,t − E[Xl,t])]

√

E [(Xj,t − E[Xj,t])2]√

E [(Xl,t − E[lk,t])2]

By conditioning on G we can obtain the following expression for the correla-tion

ρj,l =µjµlk

√

σ2j + µ2

jk√

σ2l + µ2

l k

To see that correlation does not work as a measure of dependence, considera symmetric VG process, ie. µj = µl = 0. Thus correlation will be zero.The two elements of the VG process, will still be dependent through thesubordinator.


−6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

1st Gaussian process

2nd

Gau

ssia

n pr

oces

s

−6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

1st Variance Gamma process

2nd

Var

ianc

e G

amm

a pr

oces

s

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

1st Gaussian process

2nd

Gau

ssia

n pr

oces

s

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

1st Variance Gamma process

2nd

Var

ianc

e G

amm

a pr

oces

s

Figure 4: Top: Gaussian process and VG process with same mean and covari-ance structure where the correlation is high. The parameters are µ1 = −0.35,σ1 = 0.2, µ2 = −0.30, σ2 = 0.25 and k = 2.5. The correlation is: ρ = 0.83.Bottom: Gaussian process and VG process with same mean and covariancestructure where the correlation is low. The parameters are µ1 = −0.10,σ1 = 0.2, µ2 = −0.15, σ2 = 0.25 and k = 1. The correlation is: ρ = 0.23.The Gaussian process and the Gaussian part of the VG process is based onthe same random numbers.

As mentioned above, to thoroughly describe the dependence we need to useCopulae. We however, will only use scatter plots to gain some intuitionon the behaviour the processes. For a full description via Copulae, refer toLuciano & Schoutens (2006).

Figure 4 shows scatter plot of a two dimensional VG process and a twodimensional Gaussian process, where the mean and covariance structure ofthe VG process and the Gaussian process is the same.

We see when correlation is high, the Gaussian process forms a dense andstretched scatter plot. The VG process on the other hand show a largerdegree of clustering in the upper tail, but as well extreme events in the lowertail of both marginals. When correlation is low, then the Gaussian process


show a relatively dense and evenly scattered plot. For the VG process this issimilar, but again with a higher tendency that the values will be in the lowertail.

In general, it is shown in Luciano & Schoutens (2006) that increasing k willincrease the dependence in the tails of the distribution.

5.3 Volatility Structures

When deriving the Levy HJM model, we did not make any assumptions onthe volatility structure, except that it should be deterministic and bounded.In order to implement the model, one will have to be more specific on theform of the volatility structure. A number of volatility structures are possible,some derived from short rate models, such as the Ho & Lee (1986) or Vasicek(1978) models. Others are specified directly for the purpose of forward ratemodelling. In this section we will describe possible forms of the volatilityspecification and discuss the advantages and disadvantages of the specificforms.

In this thesis we will consider a volatility specification taking the followingform 18:

σ(t, T ) = σp(n, T − t)e−κ(T−t)

where p(n, T − t) is a polynomial such that

p(n, T − t) = 1 + β1(T − t) + β2(T − t)2 + . . . + βn(T − t)n

This specification is quite flexible and nests well known models in the lit-erature, such as the Ho & Lee (1986) or Vasicek (1978) models. To keepour model specification parsimonious enough to handle in practice, we limitourselves to a model where p(n, T − t) is a first degree polynomial. This issimilar to what has been done in Schwartz & Trolle (2007).

The Ho-Lee model is obtained when κ = 0 and n = 0, giving us the volatilityspecification

σ(t, T ) = σ

which implies the bond volatility

S(t, T ) = −∫ T

t

σ(t, u)ds = −σ(T − t)

18When presenting the calibration results, we omit the constant σ as it can be factorizedinto the Levy process.


0 5 10 15 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Maturity, Years

σ(t,T

)

κ = 0.1κ = 0.2κ = 0.3

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Maturity, Years

|S(t

,T)|

κ = 0.1

κ = 0.2

κ = 0.3

Figure 5: Left: The forward rate volatility structure with a Vasicek volatilityspecification for various κ. Right: Absolute bond volatility for various κ. Forall plots σ = 0.05.

Naturally this specification implies flat forward rate volatilities and (abso-lute) bond volatilities that will increase linearly with increasing time to ma-turity. This very simple structure is very unlikely to be able to fit marketdata. However its simplicity will be a good benchmark for implementationand debugging purposes, especially due to the fact that a rather simple ex-pression for options on zero coupon bonds exists (when the driving processis Brownian motion).

Letting n = 0 and κ > 0 we obtain the Vasicek model

σ(t, T ) = σe−κ(T−t)

which implies the bond volatility

S(t, T ) = −σ

(1 − e−κ(T−t)

κ

)

As shown in figure 5 the forward rate volatility will decrease exponentially,where the speed will be dictated by the size of κ. The bond volatility willincrease as time to maturity increases, this will however not be linearly, butrather concavely. The long term level will be given by σ/κ. Again as in thecase of the Ho-Lee model, the Vasicek model, is a good benchmark, as it isa well studied model, again with a closed form formula for options on zerocoupon bonds when the model is driven by Brownian motion.

An exponentially decreasing volatility structure may not always be consis-tent with empirical observations. A humped volatility structure maybe moreappropriate. The hump can be interpreted as rates corresponding to short


0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

Maturity, Years

σ(t,T

)

κ = 0.1κ = 0.2κ = 0.3

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Maturity, Years

|S(t

,T)|

κ = 0.1

κ = 0.2

κ = 0.3

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

Maturity, Years

σ(t,T

)

β = 0.1β = 0.3β = 0.5

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Maturity, Years

|S(t

,T)|

β = 0.1

β = 0.3

β = 0.5

Figure 6: Top: Mercurio-Moraleda forward rate and bond volatility struc-ture for various choices of κ. The remaining parameters are: σ = 0.045and β = 0.2. Bottom: Mercurio-Moraleda forward rate and bond volatilitystructure for various choices of β. The remaining parameters are: σ = 0.045and κ = 0.3.

maturities, are less uncertain than rates corresponding to medium maturities.The volatility will then again flatten for rates corresponding long maturities,as these rates are mostly determined by macro economic variables, such asinflation.

The humped volatility shape can be obtained by letting p(n, T − t) be a firstdegree polynomial. This has been done in Mercurio & Moraleda (2000) andSchwartz & Trolle (2007). In this case the forward volatility is given by

σ(t, T ) = σ (1 + β(T − t)) e−κ(T−t)

This implies the bond volatility

S(t, T ) = −σ

(1 − e−κ(T−t)

κ

)

− σβ

(1 − e−κ(T−t) (1 + κ(T − t))

κ2

)

As seen from figure 6 this volatility specification can produce forward ratevolatilities that are similar to the ones from the Vasicek model, but more


important, it can create humped forward rate volatility structures. In termsof bond volatilities, the hump is not very evident, thus this specification stillproduces concave bond volatility structures. From the bond volatility, we seethat the long term level of the bond volatility is given by σ

κ

(1 + β

κ

).

Although a closed form formula for options on zero coupon bonds do existwhen the model is driven by Brownian motion (it can for instance be foundin Brigo & Mercurio (2006), pp 192.), it is rather lengthy and is hence notan ideal candidate for debugging purposes. On the other hand, as shownin Mercurio & Moraleda (2000), this volatility structure is preferable whencalibrating models, again making this specification preferable for actual im-plementations.

5.4 Data Description

As mentioned above we will use the caplet data presented in Kluge (2005).

This data consists of market data from February 19th, 2002. More specifi-cally the data consists of a Euro denominated default free yield curve, withmaturities ranging from one half to twenty years. The other part of the datais of course caplet prices. The caplets are linked to the 6-month EURIBORand have cap rates ranging from 2.5 % to 10 %.

The zero coupon bond prices are given in table 2 and the corresponding yieldcurve is given in figure 7. The shape of the yield curve is upward slopingwithout any humps. Thus it is very common shape of a yield curve.

In terms of caplet prices, our data is not measured in nominal values, butrather in implied Black volatilities. This is due to the fact, that it is marketconvention is to quote caplets by their implied Black volatility.

To get market prices, we then need to calculate prices from Black volatilitiesby using Blacks formula (where σIBV is the implied Black volatility and theremaining inputs are as given in section 4):

CPLt(T, δ,K) =p(t, T + δ)δ [L(t; T, T + δ)N(d1) − KN(d2)]

d1 =log L(t;T,T+δ)

K+ 1

2σ2

IBVT

σIBV

√T

, d2 = d1 − σIBV

√T

Finally, as mentioned in Kluge (2005), the volatilities are actually not capletvolatilities, but rather flat volatilities19 for a cap consisting of two caplets.

19A flat volatility, is the single volatility, for all caplets, that makes the cap equal to itsmarket value.


The caplets are such that, for instance the 2 year cap, will consist of a capletthat matures in one year (and settles in one and a half year) and a caplet thatmatures in one and a half year (and thus settles in two years). This impliesthat the one year cap only consists of one caplet; the caplet that matures inone half year.

T 0.5 1.0 1.5 2.0 2.5 3.0 3.5p(0, T ) 0.9834 0.9647 0.9436 0.9229 0.9007 0.8790 0.8568T 4.0 4.5 5.0 5.5 6.0 6.5 7.0p(0, T ) 0.8352 0.8134 0.7921 0.7707 0.7499 0.7292 0.7091T 7.5 8.0 8.5 9.0 9.5 10.0 10.5p(0, T ) 0.6893 0.6701 0.6515 0.6334 0.6160 0.5990 0.5822T 11.0 11.5 12.0 12.5 13.0 13.5 14.0p(0, T ) 0.5658 0.5498 0.5342 0.5190 0.5042 0.4899 0.4760T 14.5 15.0 15.5 16.0 16.5 17.0 17.5p(0, T ) 0.4627 0.4497 0.4370 0.4246 0.4127 0.4011 0.3897T 18.0 18.5 19.0 19.5 20.0p(0, T ) 0.3787 0.3678 0.3573 0.3472 0.3375

Table 2: Euro denominated Zero Coupon Bond prices on February 19th,2002.

0 5 10 15 202

2.5

3

3.5

4

4.5

5

5.5

6

Maturity, Years

Con

tinuo

usly

Com

poun

ded

Inte

rest

Rat

e, %

Figure 7: Euro denominated yield curve of February 19th, 2002.

The actual data is given in table 3 and the corresponding volatility surface isgiven in figure 8. The shape of the surface is quite common. At short matu-rities we have a shape similar to volatility smile, where for longer maturitiesit more like a volatility skew.


Mat., Cap RateYears 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.01 27.2 23.6 20.1 19.9 20.6 21.0 21.8 21.7 22.8 22.9 22.0 24.32 27.4 24.3 21.3 18.6 18.3 18.1 18.6 19.2 20.0 21.7 23.5 26.43 26.9 22.9 20.0 18.7 16.1 15.6 15.5 15.7 17.0 18.9 21.3 23.64 26.3 22.0 19.4 17.4 15.6 14.7 14.6 14.3 14.7 16.4 17.0 18.45 25.4 21.4 19.8 16.8 15.6 14.5 13.9 13.4 13.5 12.8 14.8 15.86 25.2 21.7 19.6 17.5 15.9 14.2 13.2 13.3 13.1 13.8 14.4 15.47 23.6 20.9 18.4 16.2 15.2 14.1 13.2 12.2 12.1 12.2 13.1 13.88 23.5 20.4 18.5 16.3 14.8 13.7 13.1 12.3 12.3 13.5 13.5 13.69 22.9 21.0 17.5 16.6 15.1 13.3 12.1 12.2 12.2 12.9 12.7 13.910 22.2 19.0 17.7 15.7 14.1 13.0 12.2 11.8 11.8 12.5 13.4 13.8

Table 3: EURIBOR caplet implied Black volatilities on February 19th, 2002.Numbers in bold indicate the two cap rates that are closest to At-The-Money.

2

4

6

8

10

34

56

78

910

10

12

14

16

18

20

22

24

26

28

30

Maturity, Years

Cap Rate, %

Impl

ied

Blac

k Vo

latili

ty

Figure 8: Implied volatility surface for the 6 month EURIBOR caplets onFebruary 19th, 2002.


5.5 Computational Aspects

In this section we will discuss the numerical methods used to calibrate theLevy HJM model to market data. This will include issues as numericalintegration, application of Fourier inversion methods and the use of objectsin the applied C++ code.

As indicated above, the implementation of the model has been done in C++.This has been done both to gain computational speed and to use objectorientated programming. The use of object orientated C++, greatly facilitatesthe implementation of different models, as all one need to change are theobjects that define the model (such as cumulant generating functions andvolatility structures).

As mentioned in the previous section we are calibrating the model to observedcap prices, and hence we need to implement the numerical pricing of option onzero coupon bonds. We recall that we can price these option using Fourierinversion methods. More precisely we need to evaluate the characteristicfunction

φTX(u) = exp

∫ T

0

θ (iuS(s, U) + (1 − iu)S(s, T )) − θ (S(s, T )) ds

For most cases the the above characteristic function will not be known inclosed form, thus we will need to calculate it numerically. We have chosento use Gauss-Legendre integration, and have implemented a general n-pointGauss-Legendre integrator object. We found that for a reasonable set ofparameters, then the numerical integral in the characteristic function, willbe stable already for 5 points. In the actual calibration, we have used 10points in the integration. This more conservative choice has been chosen, sothat for most parameter values the resulting characteristic function will bestable.

The relatively small number of points in the integration methods could bedue to the fact that the time from option maturity of the maturity of theunderlying zero coupon bond, only differ by the tenor of the caplet, ie. onlyby 6 months. For other applications, maybe pricing swaptions using theStochastic Duration approach of Munk (1999), the number of points in theabove integration may have to be revised.

The pricing of the caplets was done using the results given in theorem 12. Dueto the fact that the characteristic function involves numerical integration, itis computationally expensive, and one should avoid too many evaluations.This is not in favour of the FFT method outlined in section 3.7, as it will


require either 2,048 or 4,096 evaluations of the characteristic function to besufficiently precise. We have found that by evaluating the integral in theorem12 by using Gauss-Legendre integration with 100 points, the resulting priceis just as precise. Since we can save the values of the characteristic functionand then re-use them for different strike rates, we only need to evaluatethe characteristic function 100 times for one volatility smile/skew. This willprove to be a huge benefit in terms of computational time.

To show how big the benefit is, we price the entire volatility surface (228caplets) with the two methods, using parameters choices that will producesimilar precision in prices (both method uses 10 points in the evaluation ofthe characteristic function). With the Gauss-Legendre integration, using atruncation point of 2,000 and 100 points in the numerical integral, the eval-uation takes approximately 0.8 seconds20. Using the FFT with 2,048 pointsand a integration step size of 0.75 in the FFT procedure, this evaluation takes14.1 seconds. Thus the benefit from using the Gauss-Legendre integrationwhen evaluating caplets is huge.

When running a calibration that require multiple evaluations of the pricingroutine, the difference in computational times, will make the model possi-ble to calibrate within a reasonable time frame and hence making it usefulin practical applications as well. Furthermore, for the Gaussian models atruncation point of 2,000 is adequate, but as the characteristic function ofthe Variance Gamma process decay slowly (see Cont & Tankov (2004), chap.11), we need a higher truncation point. We have found that a truncationpoint of 10,000 and 200 points in the Gaussian quadrature produce stableresults.

As derivative prices vary significantly across moneyness and maturity, thena simple (Root Mean) Squared Error measure, will tend to put very differentweight to the different derivatives. To overcome this, we have, inspired by theapproach in Schwartz & Trolle (2007), chosen to minimise the the followingquantity

∑

j∈C

(

CAPMarketj − CAPModel

j

Vj

)2

where C is the set of all cap prices, CAP•j is the cap price in the market

or the model cap price and finally Vj is the market Vega of the cap basedon Blacks formula. Other types of measures could be squared pricing errordivided by the At-The-Money price (Kluge (2005)) or squared percentagepricing error (Jarrow, Li & Zhao (2007)).

20All calculations are performed on a Pentium M, 1.7 GHz with 768 MB RAM.


The optimisation performed to get the model parameters, is done using aderivative free optimisation routine, namely the Nelder-Mead downhill sim-plex. Our implementation is an object orientated version of the downhillsimplex method found in Press et al (2002), with the addition of an extraconvergence criteria21.

5.6 Calibration Results

This section will present the results from the calibration of the HJM modelsbased on Brownian motion (Gaussian models) and on the Variance Gammaprocess. We will also discuss our findings and present results such as thevolatility structure implied by the different models, and thus also the Q-distribution of different interest rates. In terms of wording, we use theword factor to describe the elements of the considered Levy process; e.g.a 1-dimensional Variance Gamma model is a 1-factor model, whereas a 2-dimensional Variance Gamma model is a 2-factor. In this way, we are con-sistent with the existing literature on interest rate modelling.

For both Gaussian and Variance Gamma models, we have found that the firstfactor is easily identifiable, the second factor however is harder to determine22. In many cases the minimal value of the objective function was obtainedwith unrealistic values of the factor correlation.

For the Gaussian models we have decided to handle this problem by imposingzero correlation between the factors. In the Variance Gamma models thecorrelation is an integrated part of the model, and is not easily set to zero.Hence we chosen to implement two Variance Gamma models. One modelis as described in section 5.2; the other is obtained by letting the elementsof the multidimensional process be independent Variance Gamma processes.One possible advantage is that the two factors can have different intensity,measured by the k parameter in the gamma process.

We believe that these identification problems are due to the fact, that thesecond factor (in the notation of Litterman & Scheinkman (1991), the slopeof the interest rate curve) mostly governs the correlation between different

21More precisely we have added the following criteria: If θLow is the set of parametersyielding the lowest value of the objective function and θHigh is the set of parametersyielding the highest value of the objective function, then if |θLow − θHigh| < ε, for somesmall ε > 0, then the method has converged.

22Due to the fact that we only have data for one specific date, we have chosen not toreport standard errors or another measure of sensitivity. We have found by computing theinverted Hessian of the objective function, that parameters relating to the second factorhave big sensitivities.


interest rates. Since caplets are options on single rates (or zero couponbonds) the second factor will be hard to identify only using caplet data fromone specific date. As our cap data consists of cap, where the caplets arenon-overlapping, we are left with the same problem. A full calibration toobserved derivative prices, of a multifactor Levy HJM model, should theninclude correlation sensitive derivatives such as swaptions, preferably acrossmoneyness.

Finally when performing the calibration, we saw that when excluding thecaps with one half year until maturity, the fit (measured by implied Blackvolatilities) to the observed prices, greatly improved (except for this excludedmaturity). Hence we have chosen to exclude this single maturity when cali-brating the models.

5.6.1 Gaussian Models

The estimated parameters in the 1- and 2-factor Gaussian models are pre-sented in table 4. As mentioned above we calibrate three different volatilitystructures namely the Ho-Lee, the Vasicek, and the Mercurio & Moraleda(henceforth MM) volatility structures. As a 2-factor Ho-Lee model is notidentifiable, we have excluded it from the estimation of the 2-factor Gaus-sian models.

1-Factor models 2-Factor modelsHo-Lee Vasicek MM Vasicek MM

σ1 0.007996 0.008709 0.008663 0.008594 0.008637σ2 - - - 0.004349 0.003556κ1 - 0.029613 0.064822 0.026784 0.000000β1 - - 0.040182 - -0.025780κ2 - - - 1.631713 2.831595β2 - - - - 0.644462Obj. Func. 0.017883 0.017126 0.017125 0.017124 0.017066

Table 4: Estimated parameters in the Levy HJM model based on Brownianmotion.

We see that the 1-factor Ho-Lee models has a slightly lower volatility thanthe Vasicek and MM models. This is naturally due the fact that the twolatter volatility structures allow for higher volatilities for short maturitiescompared to the Ho-Lee specification. However measured by the objectivefunction (see table 4) the improvement is only marginal. As seen from figure


2 3 4 5 6 7 8 9 1012

14

16

18

20

22

24

26

28

30

32

Cap Rate, %

Impl

ied

Vola

tility

, %

Market Implied Volatilities1−Factor Brownian Motion / Ho−Lee1−Factor Brownian Motion / Vasicek1−Factor Brownian Motion / MM

2 3 4 5 6 7 8 9 1010

12

14

16

18

20

22

24

26

Cap Rate, %

Impl

ied

Vola

tility

, %

Market Implied Volatility1−Factor Brownian Motion / Ho−Lee1−Factor Brownian Motion / Vasicek1−Factor Brownian Motion / MM

2 3 4 5 6 7 8 9 1010

12

14

16

18

20

22

24

Cap Rate, %

Impl

ied

Vola

tility

, %

Market Implied Volatility1−Factor Brownian Motion / Ho−Lee1−Factor Brownian Motion / Vasicek1−Factor Brownian Motion / MM

Figure 9: Implied volatilities generated by 1-factor Gaussian models com-pared to market implied volatilities. Top: 2-Year cap. Middle: 5-Year cap.Bottom: 10-Year cap.


2 3 4 5 6 7 8 9 1012

14

16

18

20

22

24

26

28

30

32

Cap Rate, %

Impl

ied

Vola

tility

, %

Market Implied Volatility2−Factor Brownian Motion / Vasicek2−Factor Brownian Motion / MM

2 3 4 5 6 7 8 9 1010

12

14

16

18

20

22

24

26

Cap Rate, %

Impl

ied

Vola

tility

, %


2 3 4 5 6 7 8 9 1010

12

14

16

18

20

22

24

Cap Rate, %

Impl

ied

Vola

tility

, %


Figure 10: Implied volatilities generated by 2-factor Gaussian models com-pared to market implied volatilities. Top: 2-Year cap. Middle: 5-Year cap.Bottom: 10-Year cap.


1 2 3 4 5 6 7 8 9 100

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Maturity

Fac

tor

load

ing

Factor 1Factor 2

1 2 3 4 5 6 7 8 9 100

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Maturity

Fac

tor

load

ing

Factor 1Factor 2

Figure 11: Left: Estimated factor loadings in the 2 factor Gaussian modelwith Vasicek volatility structure. Right: Estimated factor loadings in the 2factor Gaussian model with MM volatility structure.

9, the models based on Vasicek and MM volatility structures produce moreor less indistinguishable implied volatilities at the 2-, 5- and 10-year maturi-ties. Furthermore we see that the implied volatilities implied by the Ho-Leespecification seem more to be an average over maturities, compared to theVasicek and MM specification.

Furthermore it is evident that the Gaussian models cannot adequately de-scribe the observed smile. All the Gaussian models seem only to be able toproduce a skew, that does not have the correct slope. For the maturitiesshowing an implied volatility smile, the Gaussian models only seem to cap-ture one side of the smile, whereas the other part is neglected! Hence forthese maturities there are significant deviations from the normality assump-tion imposed by these models.

When introducing a second factor in the Gaussian models, we see from figure10 that the fit to implied volatilities only improve marginally. This is alsoconfirmed when looking at the objective function, that only improves slightly.

Furthermore figure 11 presents the factor loadings23 implied by the two Gaus-sian models. As found in Litterman & Scheinkman (1991) the first factordescribes a general impact to the yield curve, where the second factor has

23We calculate the factor loading by taking the effect of a change in the underlyingprocess on the yield curve, ie. by using the bond price expression (equation (8)) and therelationship between bond prices and zero coupon rates, we obtain:

Factor Loadingi(T ) = −Si(0, T )√

Var(Li,1)

T


a smaller impact on the yield curve, and it predominantly affects the shortmaturities. This can then be interpreted as slope effect.

Finally we see that for this specific date, there is hardly any difference be-tween a models with a Vasicek volatility specification, compared to a modelwith a MM volatility specification.

5.6.2 Variance Gamma Models

1-Factor Models:

We will now consider the models driven by Variance Gamma processes. Table5 present the estimated parameters in the 1-factor models.

1-Factor modelsHo-Lee Vasicek MM

k 3.581395 3.187198 3.465242µ1 0.000563 0.000598 0.000519σ1 0.008084 0.008478 0.007504κ1 - 0.016841 0.168142β1 - - 0.291164

Obj. Func. 0.011665 0.011285 0.011021

Table 5: Estimated parameters in the Levy HJM model based on 1-dimensional Variance Gamma processes.

We see from the parameters given in table 5, that for all three volatility speci-fications, the underlying Variance Gamma processes exhibit positive skewness(since µ1 is positive) and for all three volatility specifications the VarianceGamma processes yield a similar degree of excess kurtosis. Furthermore theparameters in table 5 yield variances (and thus standard deviations), thatis similar to what is found in the Gaussian models. Especially the positiveskew, has the nice interpretation that instantaneous forward rates will alsobe positively skewed, hence putting higher probability towards higher ratherforward rates.

From the value of the objective function of the three single factor VarianceGamma models, we see a significant improvement compared to the Gaussianmodels. When comparing the different volatility specifications, we see thatwhen improving the flexibility of the volatility specification, the objectivefunction only improves slightly.

The latter statement is also evident figure 12, that shows the implied volatil-ities arising from the single factor Variance Gamma models. We see from


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, %

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, %


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, %


Figure 12: Implied volatilities generated by 1-factor Variance Gamma modelscompared to market implied volatilities. Top: 2-Year cap. Middle: 5-Yearcap. Bottom: 10-Year cap.


this figure, that the MM volatility specification is slightly better when thematurity of the cap increases. For the 2-year maturity the Vasicek and MMspecifications yield more or less indistinguishable implied volatilities. Allthree models yield shapes of the implied volatility smile that are similar.This includes a very sharp smile for the 2-year maturity which is not consis-tent with the actual market data. For the middle maturity all three modelsseem to provide a decent fit to the implied volatility surface. Finally for the10-year maturity the all three models produce a volatility skew similar tothe ones found in the Gaussian models. Thus short and long maturity thefit seem inadequate, where for the middle maturities the fit is better.

The shape of the volatility surface, that we have found for the VarianceGamma process, is similar to what is found in Kluge (2005) for a time-homogeneous Normal Inverse Gaussian process. As mentioned in the intro-duction to the present section, Kluge (2005) improves the fit to the volatilitysurface by allowing for time-inhomogeneous Levy processes. It is known inthe equity derivatives literature, that models based on Levy processes willperform well for single maturities, but a time-inhomogeneous time change(stochastic volatility), such as in Carr et al (2003), will provide a good fit tothe entire volatility surface (see Cont & Tankov (2004)). We believe that thetime varying parameters of the model found in Kluge (2005), can be inter-preted the same way as a time-inhomogeneous time change. Hence allowingtime-inhomogeneity via stochastic volatility as described in Carr et al (2003)is a more intuitive way modelling the volatility surface.

As mentioned earlier, stochastic volatility in the Levy HJM framework isoutside the scope of this thesis, and to stay inside the framework presentedin section 4, we try to improve the fit to implied volatility surface by addingan additional dimension to the driving Levy process. It should be notedthat given data from only one specific date, it will probably hard to tellthe difference between adding stochastic volatility to a single factor model,compared to modelling with a multidimensional process. To assess whichmodelling approach that is preferable will require time series of data, ratherthan just data from one single date.

2-Factor models with dependent factors:

Table 6 presents the estimated parameters in the 2-factor models with de-pendent Variance Gamma factors.

For the Variance Gamma models with dependent factors, the intensity ofthe subordinating gamma process is similar to what is found for the 1-factormodels.

It is evident that the estimated parameters in the model with a Ho-Lee


2-Factor models (dependent VG)Ho-Lee Vasicek MM

k 3.581394 2.977842 3.289918µ1 0.017756 0.001586 0.000106σ1 0.007109 0.000000 0.000274µ2 -0.017193 -0.000026 0.000901σ2 0.003849 0.008519 0.007345κ1 - 0.712828 0.113100β1 - - -3.353658κ2 - 0.021488 0.212951β2 - - 0.378900

Obj. Func. 0.011665 0.011049 0.010775

Table 6: Estimated parameters in the Levy HJM model based on dependent2-dimensional Variance Gamma processes.

volatility specification are rather unrealistic. The two factors has a corre-lation of -0.97, which implies that the two factors could more or less befactorised into one. This is also evident from the standard deviation of thetwo factors, which is of the size 0.033, or far greater than what is found inthe Gaussian and one-factor Variance Gamma models. Of course the jointstandard deviation of the two factors is similar to what is found for the 1-factor models. As seen from figure 13 that the fit to implied volatilities doesnot improve compared to the 1-factor model.

When considering the two factor model based on dependent Variance Gammaprocesses and the Vasicek volatility specification, we see that one of thefactors (the second factor) has parameters similar to the 1-factor Vasicekmodel. This is also evident from figure 14, where this factor corresponds tothe a change in the level of the interest curve. The other factor on the otherhand mostly affects the short term maturities and can hence be interpretedas a slope factor. Although the two factors has this nice interpretation,we see from figure 14 that the improvement from adding the second factor ismarginal, if even visible at all. Finally the correlation between the two factorsis -0.005, which implies low dependence between the two factors. Howeveras mentioned earlier, estimation based on data from one day only and/or noswaption data, will probably not be able to fully estimate the dependence.

When considering the model with the MM volatility structure and dependentVariance Gamma processes, we again see that one of the factors (the secondfactors), has parameters similar to the one factor MM model. As seen fromfigure 14 this factor can again be interpreted as a level factor. The other


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, %


Figure 13: Implied volatilities generated by 2-factor Variance Gamma modelscompared to market implied volatilities. Top: 2-Year cap. Middle: 5-Yearcap. Bottom: 10-Year cap.


0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Maturity

Fac

tor

load

ing

Factor 1Factor 2

0 2 4 6 8 10−4

−2

0

2

4

6

8

10

x 10−3

Maturity

Fac

tor

Load

ing

Factor 1Factor 2

Figure 14: Left: Estimated factor loadings in the 2 factor Variance Gamma(common subordinator) model with Vasicek volatility structure. Right: Es-timated factor loadings in the 2 factor Variance Gamma (common subordi-nator) model with MM volatility structure.

factor can, as in the Vasicek model, describe a slope effect. Although thetwo factors has a nice interpretation, it is evident from figure 13, that theeffect from adding this second factor is hardly visible.

2-Factor models with independent factors:

Table 6 presents the estimated parameters in the 2-factor models with in-dependent Variance Gamma factors. The main purpose of estimating thesemodels is to assess the effect of letting the two Variance Gamma processeshaving different activity (the k parameter in the gamma processes).

An interesting observation, is that the two factors have very different activity.One factor has a very high activity, and thus a yield a very high degree ofexcess kurtosis. The other factor on the other hand has a very low activity,and then yield a very low degree of excess kurtosis.

Interestingly, as seen from figure 15, it is the factor with a high degree ofkurtosis, that can interpreted as the level factor, and hence has an effect onthe entire yield curve. Hence it seems to fully describe the volatility surface,one needs a factor with a high degree of kurtosis to fully capture the steepnessof the skew/smile of the volatility surface.

It is also visible from figure 16 (and the value of the objective function) thatallowing for different activity in the two Variance Gamma processes, greatlyimproves the fit to the volatility surface. Especially for the short maturitieswe do not see a sharp smile any more, and especially the model with the MMvolatility specification performs quite well in describing the volatility smile.

It can be seen from figure 16, that even though factor one can be interpreted


2-Factor models (independent VG)Ho-Lee Vasicek MM

k1 11.701450 5.517398 0.301341µ1 0.000251 0.000402 0.002286σ1 0.006546 0.007660 0.010046k2 0.075481 0.000725 29.627217µ2 0.000388 0.127268 0.000039σ2 0.004795 0.014176 0.001214κ1 - 0.000000 0.359492β1 - - -0.000532κ2 - 1.827485 0.218963β2 - - 3.829624

Obj. Func. 0.011039 0.009063 0.005676

Table 7: Estimated parameters in the Levy HJM model based on independent2-dimensional Variance Gamma processes.

as a level factor, then in the MM model, for shorter maturities the effect issmaller than for the middle and longer maturities. This lower factor sensi-tivity and the fact that the other factor behaves more similar to Brownianmotion, is what helps in not producing the very sharp smile given in theother Variance Gamma models.

When considering the effect from having different volatility structures, withthe Variance Gamma models based on independent factors, we see an effectfrom adding more flexibility to the volatility specification. As mentionedabove, the MM volatility structure produces the best fit to volatility struc-tures. We believe that this better fit, arises from the fact that the highkurtosis factor, more precisely can be assigned to the different maturities,more specifically, as mentioned above, that for short maturities the modeldoes not need to produce the sharp smile.

When considering the Vasicek volatility specification, we see that the firstfactor is very flat. This implies that the high kurtosis factor affects all matu-rities the same way. It is evident from figure 16 that this flat factor loading,produces a slightly sharper smile for the short maturities, compared to theMM volatility specifications. The kurtosis of level factor is also smaller thanin the MM volatility specification. This can be attributed to the lack offlexibility in the Vasicek structure compared to the MM structure; a lowerdegree of kurtosis is needed since it affects the entire yield curve, and notmostly the mid- and long term rates as in the MM structure.


0 2 4 6 8 100

0.005

0.01

0.015

Maturity

Fac

tor

Load

ing

Factor 1Factor 2

0 2 4 6 8 100

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Maturity

Fac

tor

Load

ing

Factor 1Factor 2

Figure 15: Left: Estimated factor loadings in the 2 factor Variance Gamma(independent Variance Gamma processes) model with Vasicek volatilitystructure. Right: Estimated factor loadings in the 2 factor Variance Gamma(independent Variance Gamma processes) model with MM volatility struc-ture.

Finally an interesting fact is that the model with the Ho-Lee volatility speci-fication, performs quite well too. With six parameters it clearly outperformsthe 2-factor Variance Gamma models with dependent factors and MM volatil-ity specification. This is a very clear indication that it is better to have moreflexibility in the driving processes, than in the volatility specification. Againthis could possible change when considering a time series of data. In the con-text of Gaussian HJM models, Mercurio & Moraleda (2000) find on the basisof time series data, that the MM volatility structures is indeed preferable.

When comparing our results to the results in Kluge (2005), we find that ourresults are slightly worse than what is produced by a 1-dimensional time-inhomogeneous Levy process. For the 2- and 5-year cap our results arecomparable, however for the 10 year cap, the model found in Kluge (2005),outperforms the best model found in this thesis. So it seems for a staticfit to the volatility surface a time-inhomogeneous model seems to performbetter than a multi factor model. If this still holds when pricing and hedgingcorrelation dependent derivatives, will be left for future research

5.6.3 Implied risk neutral distributions

As a final comment on the calibrated models, we will now briefly describe therisk neutral distributions implied by the different models. This will also leadto a comparison between the different driving processes. We will only focuson the 2-factor models with a MM volatility structure. Since the one factor


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Impl

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, %


Figure 16: Implied volatilities generated by 2-factor Variance Gamma modelswith independent factor, compared to market implied volatilities. Top: 2-Year cap. Middle: 5-Year cap. Bottom: 10-Year cap.


Gaussian and Variance Gamma models produces similar implied volatilities,we will only comment on the 2-factor models.

We here consider the probability density of the 6M LIBOR (in this case the6M EURIBOR), at different time points. We will consider the density of the6M LIBOR 2, 5 and 10 years ahead in time.

To derive the density, we use the definition of the spot LIBOR rate and thezero coupon bond expression in the Levy HJM, we obtain:

L(t, T ) =1

T − t

(p(0, t)

p(0, T )exp

−∫ t

0

θ(S(s, t)) − θ(S(s, T ))ds − Xt

− 1

)

where

Xt =

∫ t

0

(S(s, T ) − S(s, t))⊤ dLs

Now, since we know the characteristic function of Xt, we can use the inversiontheorem, to derive the corresponding probabilities for each value of Xt, andthus for the LIBOR rate. We have found that it is more numerically stable toretrieve the distribution function, rather than the density. We then retrievethe density by numerically differentiating the distribution function. Theresulting densities are given in figure 17.

One thing is immediately evident from figure 17; for the 5- and 10-year LI-BOR rates, there is definitely a positive probability of negative interest rate.This is of course also the case for the 2-year rate, however the probabilityof negative rates is definitely smaller. It is visible that for the models toproduce the implied volatilities given in the market, the models should showquite a degree of excess kurtosis, and hence giving rise to negative interestrates.

It is also clearly visible, that the sharp smile produced by the VarianceGamma models (1-Factor and 2-Factor with dependent factors), is a con-sequence of the fact that the distribution looks like a double exponentialdistribution. This is of course not a market fact, but rather a consequenceof using the Variance Gamma process for modelling. Again with two inde-pendent Variance Gamma processes, the distribution no longer resembles adouble exponential distribution, and has a more realistic shape.

Finally both Variance Gamma models produce a slight skew in the distri-bution, but more important the Variance Gamma models produce excesskurtosis. These fat tails will have implications on hedging and risk man-agement. Hence the models based on Variance Gamma processes should be


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−5 0 5 10 150

5

10

15

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2−Factor Variance Gamma (Independent Factors) / MM2−Factor Variance Gamma / MM2−Factor Brownian Motion / MM

−5 0 5 10 150

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15

20

25

30

6M EURIBOR, %

Prob

abilit

y

2−Factor Variance Gamma (Independent Factors) / MM2−Factor Variance Gamma / MM2−Factor Brownian Motion / MM

Figure 17: Risk neutral distributions implied by the different models. Allmodels are based on the MM volatility structure. Top: Probability densityof 6M EURIBOR at t = 2. Middle: Probability density of 6M EURIBOR att = 5. Bottom: Probability density of 6M EURIBOR at t = 10.


more realistic than Gaussian models. If HJM models based on Levy processesprovide a more realistic description of risk factors and dependence betweeninterest rates, than models based on affine jump diffusions (see Duffie, Pan &Singleton (2000)), will be interesting to explore in future research projects.


6 CONCLUSION & FUTURE RESEARCH 87

6 Conclusion & Future Research

In this thesis we have considered an alternative to usual financial modelling,where innovations are driven by Brownian motions. We have consideredfinancial modelling using Levy processes; more specifically we have describedinterest rate modelling in the Heath-Jarrow-Morton framework, where oneuses Levy processes instead on Brownian motion as the driving process. Oneadvantage of using Levy processes in interest rate modelling, is that this classof models will more or less automatically introduce Unspanned StochasticVolatility as described in Collin-Dufresne & Goldstein (2002).

We began by considering Levy processes, stochastic calculus using Levy pro-cesses and change of measure for Levy processes. The next section describedthe pricing of plain vanilla call and put options using Fourier inversion. Morespecifically we showed how one efficiently can price derivatives using the FastFourier Transform.

The main part of this thesis considered the Heath-Jarrow-Morton frameworkbased on Levy processes. We described the derivation of the modelling frame-work, herein a version of the Heath-Jarrow-Morton drift condition based thecharacteristic function of the underlying Levy process. We also related theLevy Heath-Jarrow-Morton framework the standard Gaussian Heath-Jarrow-Morton framework. Finally we showed the Change-of-Numeraire techniquein the present modelling framework, and hence we also considered the pricingof standard LIBOR derivatives, namely caps and floors.

The last section covered an implementation of the Levy Heath-Jarrow-Mortonmodel. We considered models both based on Brownian motion and VarianceGamma processes. We furthermore allowed for three different volatility struc-tures, namely the Ho & Lee (1986), Vasicek (1978) and Mercurio & Moraleda(2000) volatility structures. Model calibration was done using market datafrom one trading day, more specifically we used 6 month EURIBOR linkedcaps. The data was from February 19th, 2002 and can be found in Kluge(2005).

We have implemented the above mentioned framework in C++, which allowedus to use object orientated programming, which facilitated the use of dif-ferent volatility structures and driving processes. We found that the FastFourier Transform method, was slower than a standard numerical integra-tion method. This was due to the fact the characteristic function, used inthe Levy Heath-Jarrow-Morton model, contains a numerical integral, andit is thus computationally expensive. Therefore numerical integration usingGaussian quadrature was more efficient.

88 6 CONCLUSION & FUTURE RESEARCH

Based on the estimated models, we found (not surprisingly) that modelsbased on Variance Gamma processes outperforms the Gaussian models. Wealso found that when adding a second factor to the interest rate dynamics,model specification is crucial. We found that a model with a common gammasubordinator restricts the flexibility of the model, and the benefit from addingthis second factor is hardly visible. On the other hand we found a significantimprovement, when the second factor was based on an additional VarianceGamma process, which is independent from the first factor. Furthermorethe two factors had very different degree of excess kurtosis; one factor had avery high degree of excess kurtosis, where the other behaved more similar toBrownian motion.

Interestingly we found (in the notation of Litterman & Scheinkman (1991))that the level factor, showed a high degree of excess kurtosis. The inter-pretation is, that to produce the smiles and skews observed in the volatilitysurface, a factor with a high degree of kurtosis is needed. A simple Gaussianfactor can not produce smiles that are steep enough. The second factor, thatbehaves more like a Brownian motion, is needed at the shorter maturities toensure that the short term smiles not become too sharp. This is also evidentfrom the factor loadings in the model that performs the best; the factor withthe high degree of kurtosis does not affect the short term rates as much asthe long term rates.

Finally we considered the risk neutral densities generated by the differentmodels. We saw the models based on Levy processes generated a high degreeof excess kurtosis. The model based on the two independent Variance Gammaprocesses produced the most realistic densities; for example the VarianceGamma model with dependent factors produced implied risk neutral densitiesthat was comparable to a double exponential density. Finally we saw thatthe models did produce positive probabilities of negative interest rates, whichis indeed a critical point.

During the course of this thesis we identified several points for future research.Firstly it would be interesting to calibrate the model based on a time seriesof data. This will also show if the level factor, still show such a high degreeof excess kurtosis, when based on time series data. As part of this analysis,we could include realistic modelling of the dependence between the modelfactor is an issue. As mentioned above, a model based on a common gammasubordinator is insufficient. The independence between the factors may alsobe inadequate for time series data. Fortunately Levy processes allow forvery flexible dependence modelling, namely though Levy copula, see Cont& Tankov (2004), chap 5. This could include different dependence in thepositive and negative tails of the factor distributions. This could indeed be

6 CONCLUSION & FUTURE RESEARCH 89

interesting in terms of interest rate risk management.

It could also be interesting to implement a stochastic volatility Levy Heath-Jarrow-Morton model. As mentioned in the earlier sections, we believe thatthis could be a more theoretically consistent way of modelling the volatilitysurface, than the time in-homogeneous Levy processes found in Kluge (2005).

90 6 CONCLUSION & FUTURE RESEARCH

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APPENDIX 95

A Proofs

A.1 Proof of theorem 8

Proof. Let the pay off at maturity be given by

V (T, xT ) = Φ(xT )

Then we can write the initial value of the T claim as

V (0, x0) = p(0, T )ET [Φ(xT )]

Next consider finding coefficients such that

Φ(x) =

∫ β+i∞

β−i∞

Ψ(v)e−vxdv

As this in an inverse Fourier transform, we can find Ψ by applying a Fouriertransform. Let β be such that eβx|Φ(x)| is bounded and integrable, then

Ψ(v) =1

2π

∫ ∞

−∞

Φ(x)evxdx =1

2πΨ(v)

Now again look at the initial value of the T claim

V (0, x0) = p(0, T )ET [Φ(xT )] =p(0, T )

2πE

T

[∫ β+i∞

β−i∞

Ψ(v)e−vxT dv

]

=p(0, T )

2π

∫ β+i∞

β−i∞

e−v log aΨ(v)ET[

e−vb⊤LT

]

dv

=p(0, T )

2π

∫ β+i∞

β−i∞

e−v log aΨ(v)φL(ivb, T )dv

where we in the last equality have posed the restriction that β has to be such

that ET[

e−βb⊤LT

]

= φ(iβb) < ∞.

Then perform the substitution v = β + iu

V (0, x0) =p(0, T )

2π

∫ ∞

−∞

e−(β+iu) log aΨ(β + iu)φL (i(β + iu)b, T ) du

=p(0, T )e−β log a

2π

∫ ∞

−∞

e−iu log aΨ(β + iu)φL ((iβ − u)b, T ) du

which completes the proof.

96 APPENDIX

A.2 Proof of lemma 2

Proof. The price of the call option can be found as

C(0, x0, K) =p(0, T )ET [Φ(xT )]

=p(0, T )ET[(

exT − elog K)+]

=p(0, T )ET[

elog K(exT−log K − 1

)+]

=KC(0, x0 − log K, 1)

which completes the proof


Proof. The contract function for the European call option is given by

Φ(x) = (ex − 1)+

Next let v ∈ C with Re(v) = β, then by inversion we have that

Ψ(v) =

∫ ∞

−∞

evx (ex − 1)+ dx

In order for this to be bounded and integrable we must have that

limx→∞

|eβx (ex − 1)+ | < ∞

which implies that β < −1. Now calculating the integral gives us

Ψ(v) =

∫ ∞

0

evx (ex − 1) dx =

∫ ∞

0

e(v+1)x − evxdx

=

[e(v+1)x

v + 1

]∞

0

−[evx

v

]∞

0

=1

v− 1

v + 1

=v + 1 − v

v(v + 1)=

1

v(v + 1)

Which completes the proof.

APPENDIX 97


Proof. The contract function for the European put option is given by

Φ(x) = (1 − ex)+

Next let v ∈ C with Re(v) = β, then by inversion we have that

Ψ(v) =

∫ ∞

−∞

evx (1 − ex)+ dx

In order for this to be bounded and integrable we must have that

limx→−∞

|eβx (1 − ex)+ | < ∞

which implies that β > 0. Now calculating the integral gives us

Ψ(v) =

∫ 0

−∞

evx (1 − ex) dx =1

v(v + 1)


A.5 Proof of theorem 9

Proof. This proof follows Ozkan (2002) and Bjork, Di Masi, Kabanov &Runggaldier (1997). From equation (7) we can deduce that

log p(t, T ) = −∫ T

t

f(t, s)ds

Inserting the definition of the forward rates gives us

log p(t, T ) = −∫ T

t

[

f(0, u) +

∫ t

0

α(s, u)ds +

∫ t

0

σ(s, u)⊤dLs

]

du

= −∫ T

t

f(0, u)du −∫ T

t

∫ t

0

α(s, u)dsdu −∫ T

t

∫ t

0

σ(s, u)⊤dLsdu

98 APPENDIX

Next using Fubinis theorem for stochastic integrals (see Protter (2005), chap.IV, theorem 64), we change the order of integration

log p(t, T ) = −∫ T

t

f(0, u)du −∫ t

0

∫ T

t

α(s, u)duds −∫ t

0

∫ T

t

σ(s, u)⊤dudLs

= −(∫ T

0

f(0, u)du −∫ t

0

f(0, u)du

)

−∫ t

0

(∫ T

s

α(s, u)du −∫ t

s

α(s, u)du

)

ds

−∫ t

0

(∫ T

s

σ(s, u)du −∫ t

s

σ(s, u)du

)

dLs

= −∫ T

0

f(0, u)du −∫ t

0

∫ T

s

α(s, u)duds −∫ t

0

∫ T

s

σ(s, u)⊤dudLs

+

∫ t

0

f(0, u)du +

∫ t

0

∫ t

s

α(s, u)duds +

∫ t

0

∫ t

s

σ(s, u)⊤dudLs

Defining

A(s, T ) = −∫ T

s

α(s, u)du, S(s, T ) = −∫ T

s

σ(s, u)du

we obtain

log p(t, T ) = log p(0, T ) +

∫ t

0

A(s, T )ds +

∫ t

0

S(s, T )⊤dLs

+

∫ t

0

f(0, u)du +

∫ t

0

∫ t

s

α(s, u)duds +

∫ t

0

∫ t

s

σ(s, u)⊤dudLs

Next we know that

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

∫ t

0

α(s, t)ds +

∫ t

0

σ(s, t)⊤dLs

and equivalently∫ t

0

r(u)du =

∫ t

0

f(0, u)du +

∫ t

0

∫ u

0

α(s, u)dsdu +

∫ t

0

∫ u

0

σ(s, t)⊤dLsdu

Using Fubinis theorem we obtain∫ t

0

r(u)du =

∫ t

0

f(0, u)du +

∫ t

0

∫ t

s

α(s, u)duds +

∫ t

0

∫ t

s

σ(s, t)⊤dudLs

This is exactly the second line of the log bond price equation, hence

log p(t, T ) = log p(0, T ) +

∫ t

0

(r(s) + A(s, T )) ds +

∫ t

0

S(s, T )⊤dLs

Exponentiating log p(t, T ) completes the proof.

APPENDIX 99


Proof. Recall the log-bond price

log p(t, T ) = log p(0, T ) +

∫ t

0

(r(s) + A(s, T )) ds +

∫ t

0

S(s, T )⊤dLs

implying the dynamics

d log p(t, T ) = (r(t) + A(t, T )) dt + S(t, T )⊤dLt

=(r(t) + A(t, T ) + S(t, T )⊤a

)dt + S(t, T )⊤Σ1/2dWt

+

∫

Rd0

S(t, T )⊤xN(dt, dx)

where the last line follows from the Levy decomposition. Next finding thedynamics of p(t, T ) = explog p(t, T ) by the Ito formula

dp(t, T ) =p(t−, T )

(

r(t) + A(t, T ) + S(t, T )⊤a +1

2S(t, T )⊤ΣS(t, T )

)

dt

+ p(t−, T )S(t, T )⊤Σ1/2dWt + ∆p(t, T )

Next we need to consider the jumps ∆p(t, T ); we have that

log p(t, T ) = log p(t−, T ) + S(t, T )⊤∆Lt if ∆Lt 6= 0

log p(t, T ) = log p(t−, T ) if ∆Lt = 0

This gives us

∆p(t, T ) = p(t−, T )eS(t,T )⊤∆Lt − p(t−, T ) = p(t−, T )(

eS(t,T )⊤∆Lt − 1)

or by using the jump measure of the process we can write this as

∆p(t, T ) = p(t−, T )

∫

Rd0

(

eS(t,T )⊤x − 1)

N(dt, dx)

Inserting the above equation into the expression for dp(t, T ) completes theproof.

100 APPENDIX


Proof. The characteristic function can be found as

φTX(u) =E

T [exp iuXT] = EQ [exp iuXT γT ]

=EQ

[

exp

iu

∫ T

0

(S(s, U) − S(s, T ))⊤ dLs

×

exp

∫ T

0

S(s, T )⊤dLs −∫ T

0

θ(S(s, T ))ds

]

=EQ

[

exp

∫ T

0

(iuS(s, U) + (1 − iu)S(s, T ))⊤ dLs

]

× exp

−∫ T

0

θ(S(s, T ))ds

Next using lemma 1 we get that

φTX(u) = exp

∫ T

0

θ (iuS(s, U) + (1 − iu)S(s, T )) ds

× exp

−∫ T

0

θ(S(s, T ))ds


APPENDIX 101

B The Characteristic Function

B.1 Definition & Basic Properties

The characteristic function of a variable X is defined as the Fourier transformof the density of the random variable, more precisely we have

Definition 12. The characteristic function of a d dimensional random vari-able X is the function φX : R

d 7→ C defined by

φX(u) =

∫

Rd

eiu⊤xdP (x) =

∫

Rd

eiu⊤xfX(x)dx = E

[

eiu⊤X]

where i =√−1, ie. the imaginary unit.

Furthermore the characteristic function of a random variable is unique.

In the following we only consider 1 dimensional random variables; most ofthe results carry over to multiple dimensions, see Cont & Tankov (2004),chap. 2.

The characteristic function has several nice properties; first of all for a con-tinuous density function, it always exists and is finite in L1. This followsfrom

φX(u) = E[eiuX

]= E [cos(uX) + i sin(uX)]

and hence we have that

| cos(uX) + i sin(uX)| =√

cos2(uX) + sin2(uX) = 1

Then we have in L1 by Jensens inequality:

|E[eiuX

]| ≤ E

[|eiuX |

]= 1

For two independent random variables X and Y it holds that

φX+Y (u) = E[eiu(X+Y )

]= E

[eiuXeiuY

]= E

[eiuX

]E[eiuY

]= φX(u)φY (u)

and for a, b ∈ R such that Y = a + bX we have that

φY (u) = E[eiuY

]= E

[eiu(a+bX)

]= eiua

E[eiubX

]= eiuaφX(ub)

102 APPENDIX

B.2 Inversion Theorems

If we know the characteristic function we can evaluate the density at anygiven point. This section different inversion theorem that can be used toevaluate the density or distribution function. We omit proof, but proofs(although slightly heuristic) can be found in Ballotta (2007).

To evaluate the density, we have the following theorem:

Theorem 13. If X is continuous with characteristic function

φX(u) = E[eiuX

]

then the density function of X is given by

fX(x) =1

2π

∫

R

e−iuxφX(u)du

We can also find the distribution function as

Lemma 8. If X is continuous with characteristic function

φX(u) = E[eiuX

]

then the distribution function of X is given by

FX(x) = P (X ≤ x) =1

2+

1

2π

∫ ∞

0

φX(−u)eiux − φX(u)e−iux

iudu

Alternatively we have that

Lemma 9. If X is continuous with characteristic function

φX(u) = E[eiuX

]

then the distribution function of X is given by

FX(x) = P (X ≤ x) =1

2− 1

π

∫ ∞

0

Re

(φX(u)e−iux

iu

)

du

APPENDIX 103

B.3 Characteristic Function & Moments

Once we know the characteristic function of a random variable, we can derivethe central and non-central moments by the two following theorems. Notethat the proofs are omitted; they are fairly simple and relies on the expansionform of the exponential function.

Theorem 14. If for a random variable X, E [|X|n] < ∞ and the character-istic function has continuous derivatives at u = 0, then the ∀k < n the kthnon-central moment can be derived as

µk = E[Xk]

=1

ik∂φX(u)

∂u

∣∣∣∣u=0

For the central moments we have that

Theorem 15. If for a random variable X, E [|X|n] < ∞ and the character-istic function has continuous derivatives at u = 0, then the ∀k < n the kthcentral moment can be derived as

κk = E[(X − µ1)

k]

=1

ik∂ log φX(u)

∂u

∣∣∣∣u=0

=1

ik∂ϕX(u)

∂u

∣∣∣∣u=0

where ϕX(u) = log φX(u).

104 APPENDIX

C Convergence of integrals: Test rules

1. Improper integrals of type 1:

∫ ∞

a

f(x)dx,

∫ b

−∞

f(x)dx

The test size is

T = limx→±∞

f(x)xα

Then if

α > 1, T ∈ R the integral exists and is finiteα ≤ 1, T ∈ R \ 0 the integral does not existα ≤ 1, T = 0 no answer from the test

2. Improper integrals of type 2:

∫ b

a

f(x)dx

The test sizes are

T = limx→a+

f(x)(x − a)α, T = limx→b−

f(x)(b − x)α

Then if

α < 1, T ∈ R the integral exists and is finiteα ≥ 1, T ∈ R \ 0 the integral does not existα ≥ 1, T = 0 no answer from the test

APPENDIX 105

D Remarks on the C++ implementation

This section will briefly describe the C++ code used in the implementationof the Levy HJM model. It will describe implemented structures and showhow pricing of caplets can be performed. The full C++ code can be foundon the CD supplied with the thesis. The code has been written in Visual

C++ 2008 Express Edition, which is a free edition of Visual C++ that canbe obtained from http://www.microsoft.com/express/vc/Default.aspx.Although we have not tested it, the code should work with other compilers.

We have implemented a wide range of different objects, ranging from vectorand matrix classes. The only exceptions are a FFT object24 and a timerobject25, which is obtained from the internet.

The different files (that we have coded) and their main purpose is given below

File Purpose Linesvector.h Class to handle vector operations. The

class can handle both real and com-plex numbers, and has memory man-agement.

1010

matrix.h Class to handle matrix operations. Theclass can handle both real and com-plex numbers, and has memory man-agement.

1208

probability.h Different classes and functions relatingto probability theory. This could be in-verters of characteristic functions andthe Gaussian CDF.

445

ContingentClaims.h Functions and classes for evaluatingstandard Log-normal options and forgetting implied volatilities.

1239

NumericalMethods.h Functions and classes for numericalmethods, such as root-finding, theNelder-Mead downhill simplex opti-miser, interpolation and integration.

1155

MGFs.h MGFs and cumulants for the differentmodels implemented in the thesis.

756

FourierPricing.h Classes to price options by Fourier in-version as given in section 3.

791

24Obtained from http://www.jjj.de/fft25Obtained from http://oldmill.uchicago.edu/~wilder/Code/timer/

106 APPENDIX

File Purpose LinesMarketData.h Classes to store market data such as

zero coupon bonds and caplets.1402

LevyHJM.h Classes and functions that performsthe pricing in the Levy HJM frame-work. This is includes classes for thecharacteristic function, options on zerocoupon bonds, caps and a calibratorobject.

2240

The last part of this appendix show how pricing of a caplet can be done usingthe classes in the above files:

int main()

// get discount factors

DiscountFactors DFs("YC.txt");

// create caplet with mat = 2, tenor = 0.5 and cap rate = 5 %

capletData caplet( 2.0 , 0.5 , 0.05 );

// create cumulant for Brownian motion

CumulantBrownianMotion<complex<double> > Cumulant;

// create covariance matrix

Matrix<double> CovMat(1);

// assign sigma = 0.008;

CovMat(0,0) = 0.008 * 0.008;

// assign covmat to cumulant object

Cumulant.setCovarMatrix( CovMat );

// time has to be set to 1, cf. thesis

Cumulant.setTime( 1.0 );

// create Vasicek bond volatility

// first a vector with kappa value

Vector<double> Kappa(1);

Kappa( 0 ) = 0.25;

// create bond volatility ( dimension = 1, kappa value)

BondVolVasicek bondVol( 1 , Kappa );

// create pricer

LevyHJMcapletSinglePrice< complex<double> ,

CumulantBrownianMotion<complex<double> > ,

BondVolVasicek ,

FourierVanillaCallPut<complex<double> >> Pricer;

// setup pricer (Levy HJM char func, discount factors, Fourier pricer)

// pass cumulant and bond vol. Set the number of points in the

APPENDIX 107

// numerical integral of the char. func

Pricer.setCharFunc( &Cumulant, &bondVol , 10 );

// set market discount factors

Pricer.setDiscountFactors( &DFs );

// set Fourier pricer. beta = 20, 100 points in GL integration

// and a truncation value of 2000

Pricer.setPricer( 20.0 , 100 , 2000.0 );

// price caplet

Pricer( caplet );

// print caplet price

cout << caplet.getModelPrice() << endl;

return 0;

108 APPENDIX

E MATLAB code for FFT pricing in the Black-

Scholes model

% FFT pricing of Black-Scholes Call option

tic;

% imagenary unit

ii = 1i;

% Black-Scholes parameters

S0 = 100; % spot price

r = 0.05; % risk free rate

q = 0.00; % dividend yield

vol = 0.2; % volatility

mat = 1; % option maturity

% FFT parameters

beta = -10; % integration strip

n = 2^10; % number of points in integration

du = 0.25; % Step size in integration

u = [0:1:n-1]*du; % values of u

% state variable

dx = (2*pi)/(n*du); % state variables spacing to use FFT

gamma = (n-1)/n * pi / du; % set start variable

x = -gamma + [0:1:n-1]*dx; % values of state variable

% integration method variable

w = ones(1,n); % Create vector of ones

w(1) = 0.5; % change end points

w(n) = 0.5; % to use trapezoid rule

% calculate the fourier transform of the contract function

FourierContract = FourierVanillaCall(beta + ii*u);

% calculate the char. function

CharFunc = CF_BS(ii*beta -u ,r,q,vol,mat);

% calculate the sequence y

y = exp(ii.*u.*gamma).*FourierContract.*CharFunc.*w.*du;

% perform the FFT

z = fft(y);

% calculate the value of a ZCB

ZCB = exp(-r*mat);

% then calculate the price for exercise 1

temp = (ZCB.*exp(-beta.*x))./pi.*real(z);

% get exercise prices

strikes = exp(log(S0)-x);

% calculate final prices

APPENDIX 109

prices = strikes.*temp;

% find the relevant strike prices and option prices

minStrike = 50;

maxStrike = 150;

upperVal = round((gamma - log(minStrike/S0))/dx + 2);

lowerVal = round((gamma - log(maxStrike/S0))/dx);

% get the relevant data

relevantStrikes = strikes(lowerVal:upperVal);

relevantPrices = prices(lowerVal:upperVal);

% interpolate using a cubic spline

FinalStrikes = [50:1:150];

FinalPrices = spline(relevantStrikes,relevantPrices,FinalStrikes);

% get the computational time

compTime = toc;

% black Scholes

d1 = (log(S0./FinalStrikes)+(r+0.5*vol^2)*mat)./(vol*sqrt(mat));

d2 = d1 - vol*sqrt(mat);

BS = S0.*0.5.*erfc(-d1./sqrt(2)) - ...

ZCB.*FinalStrikes.*0.5.*erfc(-d2./sqrt(2));

% calculate RMSE

RMSE = sqrt((1/length(FinalStrikes))*sum((BS-FinalPrices).^2))

% plot the prices

plot(FinalStrikes,BS,FinalStrikes,FinalPrices);

% char. function for Brownian motion

function f = CF_BS(u,r,q,sigma,mat)

% char CF for Black-Scholes

f = exp( - 0.5.*u.*u.*sigma.*sigma.*mat );

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