Dissertation Finance Kahl

8/12/2019 Dissertation Finance Kahl

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Modelling and Simulation

of Stochastic Volatil ity in Finance

Christian Kahl

DISSERTATION.COM

Boca Raton


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Modelling and Simulation of Stochastic Volatility in Finance

Copyright 2007 Christian Kahl

All rights reserved.

Dissertation.com

Boca Raton, Florida

USA 2008

ISBN: 1-58112-383-3

13-ISBN: 978-1-58112-383-8


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Modelling and simulation of stochasticvolatility in finance

Dissertation

zur Erlangung des akademischen Grades eines

Doktor der Naturwissenschaften (Dr. rer. nat.)

dem Fachbereich C - Mathematik und Naturwissenschaften -der Bergischen Universitat Wuppertal vorgelegt von

Dipl. Math. Christian Kahl

Promotionsausschu:

Gutachter/Prufer: Prof. Dr. Michael Gunther

Gutachter/Prufer: Dr. Peter JackelGutachter/Prufer: Prof. Dr. Ansgar JungelPrufer: Prof. Dr. Andreas Frommer

Dissertation eingereicht am: 2. Februar 2007Tag der Disputation: 20. April 2007


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Danksagung/Acknowledgement

Danksagung:

Die vorliegende Dissertation entstand aus einer Kooperation der Bergischen Universitat Wup-pertal und der Product Development Group von ABN Amro London. Daher mochte ich michzuerst und ganz besonders bei meinen Betreuern Prof. Dr. Michael Gunther und Dr. PeterJackel fur die intensive und erfolgreiche Zusammenarbeit bedanken.

Peter Jackels fachliche Kompetenz und sein unermudlicher Einsatz haben die vorliegen-den Forschungsergebnisse erst moglich gemacht. Sowohl die Doktorarbeit, als auch meinepersonliche Entwicklung haben stark von seinem Einfluss profitiert, wofur ich mich ganz herz-lich bedanken mochte. Der Dank fallt umso herzlicher aus, da ich wei, welche Muhen esgekostet hat trotz standigen Zeitmangels immer Zeit zu finden, sich meiner Arbeit zu widmen.

Mein Dank richtet sich auch an die Product Development Group von ABN Amro Londonunter der ehemaligen Leitung von Dr. Bernd van Linder und nun Dr. Andrew Greaves. Es falltmir schwer, einzelne Mitglieder dieses Teams besonders hervorzuheben, dennoch mochte ichbei den Personen bedanken, die mein Arbeitsumfeld mageblich gepragt haben. Dr. IoannisChryssochoos hat entscheidend dazu beigetragen, meine anfanglichen Probleme mit C++ undMicrosoft Visual Studio zu bewaltigen, und auch sonst immer Zeit gefunden, meine Fragen zubeantworten. Es war mir eine groe Freude, mit Dr. Andrew Betteridge und Dr. Mark Rytenmathematische Probleme zu diskutieren und ich mochte mich auch bei all denen bedanken,die ich an dieser Stelle unerwahnt gelassen habe.

Michael Gunther gebuhrt mein ganz personlicher Dank dafur, da er mir diese Dissertationim Rahmen der Kooperation mit ABN Amro London ermoglicht hat. Insbesondere mochteich mich dabei fur sein Vertrauen und die Moglichkeit, meine Forschungsschwerpunkte freizu gestalten, bedanken. Michael hat mir in Wuppertal ein optimales Forschungsumfeld zurVerfugung gestellt und auch auf personlicher Ebene war es mir eine Freude mit ihm zusam-menzuarbeiten.

Bei der Arbeitsgruppe Angewandte Mathematik/Numerische Analysis mochte ich michfur das angenehme Arbeitsumfeld bedanken. Ganz besonders danke ich hier Dr. Andreas Bar-

i


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tel und Cathrin van Emmerich fur intensive und hilfreiche Diskussionen. Auch Dr. RolandPulch hat sich immer die Zeit genommen meine nicht durchweg sinnvollen Fragen zu beant-worten. Nicht unerwahnt bleiben durfen die anderen derzeitigen und ehemaligen Mitgliederder Arbeitsgruppe Stephanie Knorr, Dr. Michael Striebel und Kai Tappe.

An dieser Stelle mochte ich auch meine beiden Co-Autoren Roger Lord und Prof. Dr. HenriSchurz erwahnen. Es war mir eine groe Freude mit ihnen zusammenzuarbeiten.

Dank richten mochte ich auch an meine Familie, die es geschafft hat, mich schon in jungenJahren fur Mathematik und Naturwissenschaften zu interessieren und zu begeistern. Danke,da Ihr mir das Studium der Mathematik ermoglicht habt!

Acknowledgement:

The present dissertation bases on a cooperation of the University of Wuppertal and the Prod-uct Developement Group of ABN Amro London. Hence foremost and special thanks goes tomy supervisors Prof. Dr. Michael Gunther and Dr. Peter Jackel for the intensive and successfulcollaboration.

Peter Jackels professional capacity and unfatiguing commitment are the foundations ofthese research results. Both this thesis as well as my personal progress have strongly benefitedfrom his impact. Thank you very much Peter.

I further like to address my thanks to the Product Developement Group of ABN AmroLondon under the former direction of Dr. Bernd von Linder and now Dr. Andrew Greaves.Though it is difficult to highlight some members of this team I would like to thank in particularthose, who significantly contributed to my working life in London. Dr. Ioannis Chryssochooshelped me to overcome my initial difficulties with C++ and Microsoft Visual Studio and wasalways available to answer my questions. Moreover it was a great pleasure to discuss mathe-matical problems with Dr. Andrew Betteridge and Dr. Mark Ryten and furthermore I wantto thank all those who I did not mentioned explicitely.

Michael Gunther has my sincere thanks for supervising this cooperation project with ABNAmro London. I want to thank him particularly for the confidence he put in me and the free-dom he gave me to choose my research focus. Michael provided me with an optimal researchenvironment in Wuppertal and it was a pleasure to work with him.

Thanks also goes to the working group Applied Mathematics/Numerical Analysis for theenjoyable working enviroment. I particularly thank Dr. Andreas Bartel and Cathrin van Em-merich for all the intensive and helpful discussions. Dr. Roland Pulch always found the time toanswer my questions. I do not want to leave unmentioned the other current and former mem-


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iii

bers of this working group. Thank you, Stephanie Knorr, Dr. Michael Striebel and Kai Tappe.

It is my honour to thank my co-authors Roger Lord and Prof. Dr. Henri Schurz. It hasbeen a pleasure to work together with you.

Last but not least I want to thank my family for all the support and to inspire theenthusiasm for mathematics and science in me. Thanks for giving me the opportunity tostudy mathematics!

Wuppertal, Februar 2007 Christian Kahl


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Table of Contents

Mathematical notation xi

1 Introduction 1

2 Stochastic volatility models 5

2.1 Transformed Ornstein-Uhlenbeck models . . . . . . . . . . . . . . . . . . . . . 62.2 Affine diffusion stochastic volatility models . . . . . . . . . . . . . . . . . . . . 13

3 Monte Carlo methods 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Quasi-Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Path construction methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Fractional Fourier transformation for spectral path construction . . . . . . . . 46

4 European option pricing for transformed Ornstein-Uhlenbeck models 514.1 Control variate conditional Monte Carlo . . . . . . . . . . . . . . . . . . . . . 514.2 Asymptotic approximation using Watanabes expansion . . . . . . . . . . . . . 60

5 Optimal Fourier inversion for affine diffusion models 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Integration domain change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3 Optimal choice of alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Numerical integration schemes for stochastic volatility models 1136.1 Numerical integration of mean-reverting CEV processes . . . . . . . . . . . . . 1156.2 Numerical integration of stochastic volatility models. . . . . . . . . . . . . . . 1276.3 Multidimensional stochastic volatility models. . . . . . . . . . . . . . . . . . . 146

7 Conclusion 161

A Balanced Milstein Methods for ordinary SDEs 163

B Proofs 179

Bibliography 189

Index 200

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List of Figures

1.1 Readers guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Linear, exponential and hyperbolic transformation functions . . . . . . . . . . 82.2 Densities linear, exponential and hyperbolic transformation . . . . . . . . . . . 92.3 Variance and correlation - linear, hyperbolic and exponential transformation . 14

2.4 Trajectory Heston characteristic function in the complex plane . . . . . . . . . 222.5 Hestons characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Complex discontinuities - counterexample. . . . . . . . . . . . . . . . . . . . . 24

3.1 Cumulative variability explained- Brownian motion . . . . . . . . . . . . . . . 363.2 Brownian bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Cumulative variability explained - Ornstein-Uhlenbeck bridge. . . . . . . . . . 403.4 Cumulative variability explained - Ornstein-Uhlenbeck process . . . . . . . . . 463.5 Higher eigenvectors - Ornstein-Uhlenbeck process . . . . . . . . . . . . . . . . 47

4.1 Control variate and convergence hyperbolic-OU at-the-money . . . . . . . . . 574.2 Control variate and convergence hyperbolic-OU out-of-the-money . . . . . . . 58

4.3 Error in implied volatility for the hyperbolic-OU model . . . . . . . . . . . . . 584.4 Watanabe expansion Black-Scholes model . . . . . . . . . . . . . . . . . . . . . 644.5 Asymptotic implied volatility hyperbolic Ornstein-Uhlenbeck (A) . . . . . . . 764.6 Asymptotic implied volatility hyperbolic Ornstein-Uhlenbeck (B). . . . . . . . 774.7 Asymptotic implied volatility for Jackels hyperbolic volatility model . . . . . 804.8 Asymptotic implied volatility for the hyperbolic-hyperbolic model . . . . . . . 84

5.1 Pricing error Heston - different adaptive integration accuracy over varying . 995.2 Integrand of the Carr-Madan representation for different values of . . . . . . 1005.3 Integrand (5.49) and its first derivative - Heston model . . . . . . . . . . . . 1065.4 Black implied volatilities of the Heston model . . . . . . . . . . . . . . . . . . 1105.5 Error in the implied volatility surface for the saddlepoint approximation. . . . 111

5.6 Error in the implied volatility surface for the optimal . . . . . . . . . . . . . 1115.7 Function evaluation points for the optimal . . . . . . . . . . . . . . . . . . . 112

6.1 Pathwise approximation for the CIR/Heston model . . . . . . . . . . . . . . . 1286.2 Strong convergence CIR/Heston model - different values of volatility. . . . . . 1296.3 Strong convergence CIR/Heston model - different number generators . . . . . 1306.4 Strong convergence Brennan-Schwartz model . . . . . . . . . . . . . . . . . . . 1316.5 Strong convergence exponential/hyperbolic Ornstein-Uhlenbeck: = 0 . . . . 1366.6 Strong convergence exponential/hyperbolic Ornstein-Uhlenbeck: = 2/5 . . . 1376.7 Strong convergence exponential/hyperbolic Ornstein-Uhlenbeck: = 4/5 . . . 137

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viii LIST OF FIGURES

6.8 Efficiency exponential/hyperbolic Ornstein-Uhlenbeck: = 0 . . . . . . . . . . 1416.9 Efficiency exponential/hyperbolic Ornstein-Uhlenbeck: = 2/5 . . . . . . . . 1416.10 Efficiency exponential/hyperbolic Ornstein-Uhlenbeck: = 4/5 . . . . . . . . 1426.11 Efficiency Brennan-Schwartz and Heston model: = 0 . . . . . . . . . . . . . 144

6.12 Efficiency Brennan-Schwartz and Heston model: = 2/5 . . . . . . . . . . . . 1456.13 Efficiency Brennan-Schwartz and Heston model: = 4/5 . . . . . . . . . . . . 1456.14 2-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.15 Completed two dimensional example . . . . . . . . . . . . . . . . . . . . . . . 1496.16 Multi-dimensional correlation graph . . . . . . . . . . . . . . . . . . . . . . . . 1516.17 Graph of the correlation structure during Gaussian-elimination . . . . . . . . . 1526.18 Multidimensional volatility surface . . . . . . . . . . . . . . . . . . . . . . . . 1576.19 Multidimensional volatility model convergence . . . . . . . . . . . . . . . . . . 1586.20 Multidimensional implied volatility surface - Euler vs. IJK . . . . . . . . . . . 1596.21 Correlation surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
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List of Tables

3.1 CPU-time covariance split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 Table of conditional expectations for multiple Wiener integrals . . . . . . . . . 684.2 At-the-money skewness for local and stochastic volatility models . . . . . . . . 83

5.1 Maximum jump-diffusion - number of Newton-Raphson steps . . . . . . . . 1035.2 Function evaluations and error for the optimal alpha and standard method . . 1095.3 Tiny option prices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1 Average speed-up IJK compared with log-Euler . . . . . . . . . . . . . . . . . 1406.2 Average speed-up BMM & IJK compared with Euler & log-Euler . . . . . . . 1436.3 Number of non-positive stochastic volatility paths in figure6.13 (B) . . . . . . 144

ix
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Mathematical notation

1{...} indicator function

:= defined as

approximately

AT transpose of the matrix A RnmxT y inner product, xT y= ni=1 xiyixT transpose of the vector x Rn

short hand notation for the correlation of Wiener processes,i.e.W Z stands for dW dZ= dt

(x) Dirac delta distribution

i,j Kronecker symbol

e base of the natural logarithm (Napiers constant)

E[X] expectation of X, i.e.

XdPX

f(z) Fourier transform of f

Im(z) imaginary part of the complex variablez

f expectation with respect to the distribution densityln(x) natural logarithm, i.e. the solution of x = ez

C complex numbers

N positive integers

P(A) probability of A

PX distribution induced by the random variableX

R real numbers

R+ positive real numbers

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xii Mathematical notation

Z integers

C(U, V) continuous functionsf :U V

Ck(U, V) functionsf :U

V, withk-th derivative in

C(U, V)

N(, 2) normal distribution with mean and standard deviationO(n) Landau or asymptotic notation tensor productz complex conjugate

(u, t) characteristic function,(u, t) = E

eiuXt

Re(z) real part of the complex variablez

X standard deviation ofX,X=

V[X]

distributed according toBlack(S,K,,T) Black formula for a European call option,

Black(S,K,,T) =S N

ln( SK )T

+ T

2

K N

ln( SK )T

T

2

Cor[X, Y] linear correlation betweenX andY

Cov[X, Y] linear covariance betweenXand Y

(x) standard normal distribution density,(x) = 1

2ex

2/2

V[X] variance of X, V[X] = E

(X E[X])2f(k)(x) short hand notation for thek-th derivative off, i.e.

kfxk

(x)

f1(x) inverse function, i.e. f1(f(x)) =x

hn Hermite polynomial of degree n

i1

Is,t(1) Wiener integral on the interval [s, t], i.e. tsdWuIs,t multiple Wiener integral on the interval [s, t] with multi-indices

N(x) cumulative normal distribution function, N(x) =x (z)dz

Wt Brownian motion

x y maximum ofxandyx y minimum ofx andy


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CHAPTERI

Introduction

The famous Black-Scholes model [BS73] was the starting point of a new financial industryand has been a very important pillar of all options trading since. One of its core assumptionsis that the volatility of the underlying asset is constant. It was realised early that one hasto specify a dynamic on the volatility itself to get closer to market behaviour. There aremainly two aspects making this fact apparent. Considering historical evolution of volatilityby analysing time series data one observes erratic behaviour over time. Secondly, backing outimplied volatility from daily traded plain vanilla options, the volatility changes with strike.The most common realisations of this phenomenon are the implied volatility smile or skew.The natural question arises how to extend the Black-Scholes model appropriately.

Any modelling approach aiming at this problem must fulfil some fundamental require-ments. The range of model implied volatilities must be flexible enough to reflect marketbehaviour. In order to be of practical relevance it must additionally allow the fast and ac-curate pricing of European options to accomplish calibration within reasonable computationtime. The approach of driving volatility itself via a stochastic process turned out to be asintuitive as successful. Depending on the particular choice of the stochastic process for volatil-ity, these models allow different dynamics of implied volatility and thereby mapping of marketbehaviour.

This thesis focus on the different problems arising within the context of stochastic volatil-ity from a modelling as well as from a numerical point of view. At the very heart of modelling

volatility is the choice of a sensible process. Hereby it is of fundamental importance to bearin mind analytical properties such as positivity of volatility. Moreover one usually observesa mean-reverting behaviour of volatility over time and might want to address the analyticaltractability, i.e. a closed form solution of the distribution density or its characteristic function.Once a model is chosen one needs to address the problem of calculating or approximating theprice of plain vanilla options in order to calibrate the model to market data.

The thesis is divided into six chapters and we illustrate their dependency structure infigure1.1. The second chapter provides an introduction about the stochastic volatility models

1
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2 1Introduction

discussed throughout this thesis. We mainly focus on two different modelling approaches.First we introduce stochastic volatility models where volatility is given by a transformedOrnstein-Uhlenbeck process. Here, we concentrate on analysing the analytical propertieswith respect to different choices of transformation functions. The second class of stochasticvolatility models we consider are affine (jump) diffusion models, having the appealing featurethat the characteristic function of the underlying asset is known in closed form. In a firststep, we discuss a stable numerical implementation of the characteristic function which is arather challenging problem involving an appropriate handling of multivalued functions suchas the complex logarithm. In chapter five, we use the closed form solution of the characteristicfunction for semi-analytical option pricing of plain vanilla options.

5.4) Numerical tests

6.1) Meanreverting CEV

6.2) Stochastic volatility

6.3) Multidimensional

Chapter 6

Numerical integration

stochastic volatility

Chapter 1

Introduction

Chapter 3

3.1) Introduction

methods

Monte Carlo

relation

Chapter 2

Stochastic volatility

models

2.2) Affine diffusion

2.1) OrnsteinUhlenbeck

3.3) Path construction

3.4) Fast fractional FT

Chapter 4

transformed Ornstein

Uhlenbeck models

4.2) Watanabes approach

conditional Monte Carlo

4.1) Control variate

European options for

3.2) QuasiMonte Carlo

Chapter 5

Fourier transform

option pricing

5.1) Introduction

5.2) Integration domain

5.3) Optimal alpha

Figure 1.1: Readers guidance.

The third chapter deals with Monte Carlo simulation methods. At the very heart of thischapter stands the optimal path construction of Wiener and Ornstein-Uhlenbeck processeswithin a quasi-Monte Carlo framework. In section3.3, we compare different path construction
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1Introduction 3

methods with regard to theirvariability explainedand their computational complexity. Usinga fast fractional Fourier transformation helps to reduce the computational effort for discreteand approximative spectral path construction methods of Wiener and Ornstein-Uhlenbeckprocesses as outlined in section3.4.

Next we discuss the computation and approximation of European option prices for trans-formed Ornstein-Uhlenbeck stochastic volatility models, where we mainly concentrate on twodifferent approaches. On the one hand, we use the efficient path construction of Ornstein-Uhlenbeck processes developed in chapter 3 to derive a fast and accurate control variateconditional Monte Carlo method. On the other hand, we develop asymptotic approximationsfor European option prices within transformed Ornstein-Uhlenbeck models using Watanabesapproach. Due to the generality of this method we extend the stochastic volatility model byadding a local volatility function which provides greater flexibility when calibrated to marketdata.

In chapter five, we study the semi-analytical option pricing for affine (jump) diffusion

models using Fourier inversion techniques. Fourier inversion allows to price European optionssolely based on the knowledge of the characteristic function of the underlying asset and thepayoff function by the aid of the Plancherel/Parseval equality and applying residual calculus.The numerical implementation of the inverse Fourier integral can be accomplished by trans-forming the unbounded integration domain to a finite interval using the limiting behaviour ofthe characteristic function. Furthermore we show that it is of fundamental importance for afast and robust implementation to choose an appropriate integration contour in the complexplane. Numerical tests provide further evidence for the effectiveness of this approach.

The sixth chapter deals with numerical integration schemes for stochastic volatility modelswhich are required to price path dependent options. Since the volatility process does notexplicitly depend on the underlying asset, the integration of a stochastic volatility model can

be divided into two separate steps. First we concentrate on the numerical integration of thestochastic volatility process itself. Based on the knowledge of the full volatility path, we focuson the simulation of the underlying asset in section 6.2. Using the strong approximationerror as a measure to compare different approximation schemes, we derive a fast and efficientapproximation method based on an interpolation of the drift and the decorrelated diffusion.Section6.3discusses multidimensional stochastic volatility models, where we particularly focuson how to complete a non-fully specified correlation matrix such that we obtain a symmetricpositive semi-definite matrix.

In chapter seven, we summarise the results developed and discussed throughout this the-sis and we provide an outlook about further research opportunities. Appendices A and B

supplement the previous work.
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CHAPTERII

Stochastic volatility models

The first publications on stochastic volatility models [Sco87, Wig87, HW88] were ahead oftheir time: the required computer power to use these models in a simulation framework wassimply not available, and analytical solutions could not be found. One of the first articles thatprovided semi-analytical solutions was published by Stein and Stein[SS91]. An unfortunatefeature of that model was that it did not give enough flexibility to represent observable marketprices, i.e. it did not provide enough degrees of freedom for calibration. To make matterseven worse the Stein and Stein model allows volatility to become negative which is a ratherundesirable feature. In 1993, Heston [Hes93] published the first model that allowed for areasonable amount of calibration freedom permitting semi-analytical solutions. Various otherstochastic volatility models have been published since, and computer speed has increasedsignificantly.

Throughout the following we consider stochastic volatility models to be given as a two-dimensional system of stochastic differential equations

dSt = Stdt+ vpt StdWt,(2.1)dvt = a(vt) dt+b(vt) dZt ,(2.2)

with scaling parameter R+. We assume the driving processes Wt and Zt to be correlatedBrownian motions satisfying dWt dZt = dt. The parameter p allows us to specify eitherinstantaneous variance for p = 1, or the instantaneous volatilityp = 1/2. In order to keep thestochastic volatility processv rather general, we will further specify the dynamics of the driftaand diffusionblater on.

The outline of this chapter is as follows. First we introduce so calledtransformed Ornstein-Uhlenbeck models where the stochastic volatility process is, according to its nomenclature,given by a deterministic transformation of a standard Ornstein-Uhlenbeck process. One ap-pealing feature of these model is that we can guarantee the positivity of the stochastic volatilityby choosing an appropriate transformation function.

In a second section 2.2, we draw our attention to affine diffusion models. These modelsare particularly attractive since the characteristic function of the transition density of the un-derlying asset is known in closed form, which allows for semi-analytical solutions of Europeanoption prices by the aid of Fourier inversion techniques. The numerical implementation of the

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6 2Stochastic volatility models

characteristic function is rather challenging since the characteristic function is of oscillatingnature and involves the evaluation of multivalued functions. In chapter5, we continue thediscussion with the implementation of the Fourier inversion, in order to calculate prices ofplain vanilla options.

2.1 Transformed Ornstein-Uhlenbeck models

Definition 2.1 An Ornstein-Uhlenbeck [UO30] process can be defined as the solution of thescalar stochastic differential equation

(2.3) dyt = ytdt+

2 dZt ,

with Brownian motionZt, initial valuey(0) =y0 and positive constants, R+.

The origin of this process goes back to Uhlenbeck and Ornsteins publication [UO30] inwhich they describe the velocity of a particle that moves in an environment with friction.Doob [Doo42] first treated this process purely mathematically and expressed it in terms ofa stochastic differential equation. In modern financial mathematics, the use of Ornstein-Uhlenbeck processes is almost commonplace. The attractive features of an Ornstein-Uhlenbeckprocess are that, whilst it provides a certain degree of flexibility over its auto-correlation struc-ture, it still allows for the full analytical treatment of a standard Gaussian process.

The parametrisation (2.3) permits a complete separation between the mean reversion speedand the variance of thelimitingorstationary distributionof the process [FPS99]. The solutionof (2.3) is

(2.4) yt = ety0+

t0

eu

2 dZu

,

with initial timet0 = 0. In other words, the stochastic process at time t is Gaussian with

(2.5) yt N

y0et, 2

1 e2t ,

and thus the stationary distribution (t ) is Gaussian with variance 2: a change inparameter requires no rescaling of if we wish to hold the long-term uncertainty in theprocess unchanged. It is straightforward to extend the above results to the case when (t)

and(t) are functions of time [Jac02]. Throughout the following we assume the process (2.3)to start iny(0) = 0. Since the variance of the driving Ornstein-Uhlenbeck process is the maincriterion that determines the uncertainty in volatility for the financial underlying process, allfurther considerations are primarily expressed in terms of

(2.6) (t) := 1 e2t .

The Ornstein-Uhlenbeck process is particularly attractive to be used as a stochastic volatil-ity driver since it is full analytical tractable. Despite this it is not suited for modelling the
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2.1Transformed Ornstein-Uhlenbeck models 7

stochastic volatility itself since the process is fluctuating around zero which is a rather un-desirable feature. The idea is to apply a monotonic transformation function which was firstproposed in [Sco87, equation (7)] using the exponential function. The model is intuitivelyvery appealing: for any future point in time, volatility has a lognormal distribution which is avery comfortable distribution for practitioners in financial mathematics. Alas, though, recentresearch [AP04] has cast a shadow on this models analytical features. It appears that, in itsfull continuous formulation, the log-normal volatility model can give rise to unlimited highermoments of the underlying financial asset. However, as has been discussed and demonstratedat great length for the very similar phenomenon of infinite futures returns when short ratesare driven by a lognormal process [HW93,SS94, SS97a,SS97b], this problem vanishes as soonas the continuous process model is replaced by its discretised approximation which is whylognormal volatility models remain numerically tractable in applications.

In order to avoid this problem altogether, we introduce an alternative to the exponentialtransformation function which is given by a simple hyperbolic form. In the following, we refer

to

flin(y) := y+ 1 , lin(y) := 0 flin(y)(2.7)

as the linear volatility transformation also known as Stein-Stein [SS91] or Schobel-Zhumodel[SZ99], to

fexp(y) : = ey , exp(y) := 0 fexp(y)(2.8)

as the exponential volatilitytransformation also known as Scotts model [Sco87], and to

fhyp(y) := y+

y2 + 1, hyp(y) := 0 fhyp(y)(2.9)

as the hyperbolic volatility transformation as introduced by Kahl and Jackel [KJ06]. Thedensities of these processes are given by

(2.10) (, 0, ) = (f1(/0), )

d/ dy

with

(2.11) f1lin (/0) = 0, (2.12) dlin/ dy = 0

(2.13) f1exp(/0) = l n (/0) , (2.14) dexp/ dy = exp

(2.15) f1hyp(/0) = (/0 0/)/ 2, (2.16) dhyp/ dy = 20

2hyp

20+2hyp
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and

(2.17) (y, ) := e

12(

y )

2

2

.

The hyperbolic transformation has been chosen to match the exponential form as closely aspossible near the origin, and only to differ significantly in the regions of lower probabilitygiven for|y/| >1.

Even though the linear transformation (2.7) does not guarantee the volatility to remainpositive (see figure 2.2), we included this model to our discussion since it allows for semi-analytical solutions for pricing plain-vanilla options by the aid of Fourier inversion.

In figure 2.1, we compare the functional form of the linear, exponential and hyperbolictransformation function.

-1

0

1

2

3

4

5

6

7

8

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

fexp

(y)

fhyp

(y)

flin(y)

Figure 2.1: Linear, exponential and hyperbolic transformation functions.

2.1.1 Linear vs. Exponential vs. Hyperbolic transformation

In figure2.2we compare the densities of the transformed Ornstein-Uhlenbeck process. Oneclearly sees that the linear transformation has a positive probability for negative values. Atfirst glance on linear scale one cannot recognise any reasonably difference between the hyper-bolic and exponential transformation. On a logarithmic scale, the differences become clear:

the exponential transformation has significantly fatter tails than the hyperbolic transforma-tion. Thus the probability of very low as well as very high values is noticeable lower for thehyperbolic transformation.

Returning to the analytical form of the density function (2.10), it is interesting to notethat, given that y = f1(/0) is Gaussian, the volatility distribution implied by the inversehyperbolic transformation (2.15) is nearly Gaussian for large values of instantaneous volatility 0 since we have

(2.18) hyp 20 y for hyp 0.


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0

0.5

1

1.5

2

2.5

3

3.5

4

0% 20% 40% 60% 80% 100%

exp

(,0,)

hyp

(,0,)

lin(,0,)

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0% 50% 100% 150% 200%

exp

(,0,)

hyp

(,0,)

lin(,0,)

Figure 2.2: Densities of instantaneous volatility using the linear, exponential and the hyper-bolic transformation of the driving Ornstein-Uhlenbeck process for 0 = 25% and = 1/2 ona linear (left) and logarithmic (right) scale. In order to illustrate the distinctly different tailsof the distribution for the hyperbolic and exponential transformation on a logarithmic scale,we only plotted the positive part of the density, even though the linear transformation has apositive probability to become negative which is already evident on a linear scale.

This feature is particularly desirable since it ensures that the tail of the volatility distribution

at the higher end is as thin as the Gaussian process itself, and thus no moment explosionsare to be feared for the underlying. Conversely, for small values of instantaneous volatility 1, the volatility distribution implied by the hyperbolic volatility model is nearly inverseGaussian because of

(2.19) hyp 0/ (2 y) for hyp 0.In a certain sense, the hyperbolic model can be seen as a blend of an inverse Gaussian modelat the lower end of the distribution, and a Gaussian density at the upper end, with theirrespectively thin tails. In contrast, exponential volatility is simply lognormally distributed,which in turn gives rise to distinctly fatter tails than the normal (at the high end) or inversenormal (at the low end) density. We concretise this behaviour in the following lemma.

Lemma 2.2 The transformation methods of the Ornstein-Uhlenbeck process have the follow-ing behaviour in the tails of the distribution

1. the linear transformationis Gaussian distributed at the upper and lower end,

2. the exponential transformationis lognormal distributed at the upper and lower end,

3. the hyperbolic transformation is Gaussian distributed at the upper end and inverse-Gaussian at the lower end.


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To obtain even more information about the analytical behaviour of the transformed Ornstein-Uhlenbeck process, we derive the driving stochastic differential equation using Itos lemma.

Lemma 2.3 The stochastic differential equation for the linear transformed Ornstein-Uhlenbeckprocess is given by

(2.20) d= (0 ) dt+0

2 dZ ,

thus by setting 0= and = SZ02

, we obtain Schobel-Zhus original SDE[SZ99,equation

(3)].

Lemma 2.4 The stochastic differential equation for the exponential transformed Ornstein-Uhlenbeck process is given by

(2.21) d=

2 ln(/0)

dt +

2 dZ .

This corresponds to the original SDE used by Scott [Sco87,equation (7)].

Lemma 2.5 The stochastic differential equation for the hyperbolic transformed Ornstein-Uhlenbeck process is given by

(2.22) d= 6 +420 (82 + 1)240 60

(2 +20)3 dt +

8

02 +20

dZ ,

as first mentioned by Kahl and Jackel [KJ06,equation (36)].

Particularly the diffusion terms of these equations are of great help for the construction ofefficient numerical integration schemes for the whole two-dimensional stochastic volatilitymodel which we will discuss in section 6.2. In order to obtain further insight we analyse themoments of the different transformation methods

2.1.2 Moments

Since the hyperbolic Ornstein-Uhlenbeck process is designed to match the exponential as close

as possible we can expect that the moments reflect this behaviour. Throughout the followingwe assumey N(0, 2).

Lemma 2.6 The moments of the exponential transformationare given by

(2.23) E[(fexp(y))m] = e

12m22 .

The moments of the linear and hyperbolic transformation both involves Kummers hypergeo-metric function 1F1(a,b,z) and the confluent hypergeometric function U(a,b,z).
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Definition 2.7 Kummers hypergeometric function has the following integral representa-tion [AS84,13.2.1]

(2.24) 1F1(a,b,z) = (b)(a)(b a)

10

eztta1(1 t)ba1 dt

and an integral representation of the confluent hypergeometric function is given by [AS84,13.2.5]

(2.25) U(,,) = 1

()

0

s1(1 + s)1es ds .

Lemma 2.8 The moments of the hyperbolic transformationare given by

(2.26) E[(fhyp(y))m] = 2

32n4

n1+n2

1F1(n2 , 1n, 122 ) + 2

32n4

n1n2

1F1(

n2

, 1+n, 122

),

where we have a simpler expression for the first two moments

(2.27) E[fhyp(y)] =

2 U(12

, 0, 122

), E

(fhyp(y))2 = 1 + 22 .

Proof: The first moment follows directly from the integral representation of the confluenthypergeometric function (2.25) by setting s = y2, := 1/2, := 2, and := 1/22 and the useof the relationship [AS84,13.1.29]

(2.28) U(, , ) = 1

U(1 +

, 2

, ).

To calculate the second moment we can use the symmetry and second moment of the Gaussiandistribution.

Lemma 2.9 The moments of the linear transformationare given by

(2.29) E[(flin(y))n] =

21+n2n1

22(1+n2 )1F1

1n2 , 3

2, 1

22

ifnis odd,21+

n2n

2 ( 1+n2 )1F1

n

2, 12, 1

22

ifn is even,

where the first moments can be calculated straightforwardly as

(2.30) E[flin(y) ] = 1, E

(flin(y))2 = 1 + 2 .

More revealing than the closed form for the moments of the respective transformation functionsis an analysis based on their Taylor expansions

fexp(y) = 1 + y+y2

2 + O y3 ,(2.31)

fhyp(y) = 1 + y+y2

2 + O y4 .(2.32)
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Thus, for both of these functions,

(2.33) (f(y))n = 1 +n y+

n

2 +n

2 y2 + O y

3

.Since y is normally distributed with mean 0 and variance 2 as defined in (2.6) and since allodd moments of the Gaussian distribution vanish, this means that for both the exponentialand the hyperbolic transformation we have

(2.34) E[(f(y))n] = 1 +

n

2

+

n

2

2 + O 4 .

The implication of (2.34) is thatall moments of the exponential and the hyperbolic transfor-mation function agree up to orderO (3).

2.1.3 CorrelationNext we analyse the correlations of the different transformation methods in order to providefurther insights about their interconnections. Whilst the moments of the hyperbolic andexponential transformation agree up to third order it turns out that distributions of the linearand hyperbolic transformation are even closer due to their similarity in the upper tail.

Proposition 2.10 The correlation of the linear and hyperbolic transformationis given by

(2.35) Cor [flin(y), fhyp(y)] =

1 + 22 22 U

12

, 0, 1222

,

withy N(0, 2). Furthermore we have the following limiting behaviour

(2.36) lim

Cor[flin(y), fhyp(y)] =

2( 1) 0.85.

Proof: We already calculated the expectation and the variance of the linear and hyperbolictransformation in (2.29) and (2.26). Putting together these results we obtain

Cov[flin(y), fhyp(y)] = Ex(x +

x2 + 1)

2 U

1

2, 0,

1

22(x, )= 2 ,(2.37)

due to the symmetry of the density (2.17) of the normal distribution. Combining (2.30),(2.27) and (2.37) verifies (2.35). Moreover, since

(2.38) limx

U

1

2, 0,

1

x

=

1

we obtain (2.36).


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2.2Affine diffusion stochastic volatility models 13

Proposition 2.11 The correlation of the linear and exponential transformation function isgiven by

(2.39) Cor [flin(y), fexp(y)] = exp(2) 1

,

withy N(0, 2). Therefore we obtain in the limit(2.40) lim

Cor[flin(y), fexp(y)] = 0.

Proof: The first moments of the exponential transformationare given in equation (2.23) oflemma2.6. Furthermore we obtain for the covariance of both transformation methods

Cov[flin(y), fexp(y)] = E

((1 + x) 1)

ex e

2

2

(x, )

= e2

22

R

xe(x)2

2 dx

= e2

2 2 .(2.41)

Combining (2.30), (2.23) and (2.41) leads to (2.39).

Remark 2.12 The correlation of the hyperbolic and exponential transformation function iscomputed numerically, since we could not derive an analytical solution.

In figure 2.3, we present the variances and correlations of the different transformation

methods. Comparing the variances in figure 2.3 (A) already indicates that the linear andhyperbolic transformationare more similar with increasing uncertainty of the normal distri-bution. This becomes even more evident considering the correlations in figure2.3(B), eventhough for small values of < 1 we have the highest linear dependency between the expo-nential and hyperbolic transformation. Bearing in mind that represents the variance of thedriving Ornstein-Uhlenbeck process (2.5) and (2.6) we expect that the high correlation of thelinear and hyperbolic transformation reflects in a high codependency of plain vanilla optionprices in the linear and hyperbolic Ornstein-Uhlenbeck model. Since the linear Ornstein-Uhlenbeck model admits closed form solutions for European options, we use this model as acontrol variate for the hyperbolicin a Monte Carlo simulation in section4.1.

2.2 Affine diffusion stochastic volatility models

In this section, we discuss another type of modelling stochastic volatility: affine diffusionmodels. These models have the appealing feature that the characteristic function of theunderlying assets marginal distribution is known in closed form. This result is not restrictedto the modelling of a two-dimensional system of stochastic differential equations but canbe generalised to the characterisation of Levy processes in n-dimensions. Duffie, Pan andSingleton [DPS00] first noticed that the computation of the characteristic function of an
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(A)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Hyperbolic

Exponential

Linear

(B)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Hyperbolic-Linear

Exponential-Linear

Hyperbolic-Exponential

Limit

Figure 2.3: (A) variance and (B) correlation of the different transformation methods (y-axis)over (x-axis) as given in equation (2.6).

affine diffusion is reduced to solving a system of ordinary differential equations of Riccatitype. This result has been generalised to linear-quadratic models by Gaspar [Gas04] andCheng and Scaillet [CS05]. In general one cannot expect to solve this system of Riccatiequation analytically such that solutions have to be computed numerically.

The focus of this section is not to derive another affine diffusion model but to analyse howto implement the standard affine diffusion stochastic volatility models such as Heston[Hes93]

and Schobel-Zhu [SZ99]. Despite the fact that the characteristic function is known in closedform, the implementation is a rather challenging problem since the characteristic function istypically oscillating and involves the evaluation of a complex logarithm which is a multivaluedfunction.

To concretise the characteristic features of an affine diffusion model, we follow the leadby Duffie, Pan and Singleton [DPS00] and Duffie, Filipovic and Schachermeyer [DFS03] andconsider the following general diffusion model

(2.42) dXt= (Xt) dt+ (Xt) dWt ,

with X = (X1, . . . , X n) being an n-dimensional stochastic process, C1(Rn,Rn), C2

(Rn

,Rn

n

) and Wa standard Brownian motion inRn

.The aim is to show that under certain conditions on the structure of the drift and thediffusion the characteristic function

(2.43) (u, x , t) := E

eiuXt

,

with u = (u1, . . . , un) is given as a solution of a system of Riccati equations. By virtue ofFeynman-Kacs theorem we know that equations (2.42) and (2.43) are equivalent to solving

(2.44) (u,x,t)

t + (x)

(u,x,t)

x +

1

2tr

(x) (x)T

2(u,x,t)

x2

= 0,
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where tr(A) is the trace of a matrix A and (u,x,t)x

=

x1

, x2

, . . . , xn

is the vector of the

partial derivatives. The boundary condition of (2.44) is given as

(2.45) (u,x, 0) = e

i

u

x

.

wherex denotes the initial value ofX. Next we specify the notation of an affine diffusion:

Definition 2.13 A diffusion process is called affine if the following conditions are satisfied

(2.46) (x) =K0+n

i=1

K(1,i) xi , for K0 Rn and K(1,i) Rn

and

(2.47)(x) T(x) =H0+ n

i=1

H(1,i) xi , for H0 Rnn andH(1,i) Rnn .

The key result of Duffie, Pan and Singleton [DPS00] is that the characteristic function ofan affine diffusion model is given as the solution of a system of Riccati equations, which weconcretise in the following proposition:

Proposition 2.14 Let Xbe an affine diffusion with drift and diffusion specified in defini-tion2.13, then the characteristic function is given as

(2.48) (u,x,t) = eA(t,u)+B(t,u)Tx ,

where the functions A C1 (R,C) and B C1 (R,Cn) satisfy the complex valued ordinarydifferential equations

(2.49) Bi(t)

t =KT(1,i) B(t) +

1

2B(t)T H(1,i) B(t),

for i = 1, . . . , nand

(2.50) A(t)

t =KT0B(t) +

1

2B(t)T H0 B(t),

subject to the boundary conditions B(0) =i uand A(0) = 0.Proof: Applying the suggested solution (2.48) to the partial differential equation (2.44) andcomparing the components with respect to the powers ofxverifies this result.

Note that in the case of stochastic volatility models we are, at least for plain vanilla options,only interested in the characteristic function with respect to the underlying financial driverusually the first element X1 of the vector valued stochastic process X (2.42). We show insection2.2.3that knowing the full multivariate characteristic function (2.43) is particularlyusefull for calculating the characteristic function conditionally on a future fixing date.


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2.2.1 Characteristic function of affine diffusionstochastic volatility models

The affine diffusion stochastic volatility model we consider here is characterised by the follow-

ing 2-dimensional system of stochastic differential equations

dS(t) = rS(t) dt+ v(t)pS(t) dWt ,(2.51)dv(t) = ( v(t))dt +v(t)1p dZt ,(2.52)

with correlated Brownian motions dWt dZt = dt. In order to categorise these models asaffine diffusions, we have to transform the underlying asset (2.51) to logarithmic coordinatesby virtue of Itos formula. Settingp = 0, we have the standard Black-Scholes [BS73] model,whilst for p = 1/2 we obtain the Heston [Hes93]stochastic volatility model. The third modelof affine structure is the Stein-Stein [SS91]and Schobel-Zhu [SZ99] model with p = 1. Eventhough this model is actually non-affine in its original coordinates (2.52), we can augment

the state vector by v2 such that it is affine in ln(S), v and v2 as recently became evidentfrom the study of quadratic models by Gaspar [Gas04] and Cheng and Scaillet [CS05]. Thelatter paper showed that any linear-quadratic jump-diffusion model is equivalent to an affine

jump-diffusion model with an extended state vector.Using the previous results about affine diffusion models, we now derive the univariate

characteristic functions of the underlying financial asset

(2.53) (u,x,t) := E

eiulnFt

,

whereFt= S0ert denotes the forward price of the underlying asset. We start with the easiest

example, the Black-Scholes model [BS73], where we even know the distribution density of theunderlying asset in closed form.

Lemma 2.15 The Black-Scholes model is affine in Xt = ln Ft,

(2.54) dXt = 12

2 dt + dWt,

withK0 = 122,K(1,1)= 0, as well asH0= 2 andH(1,1)= 0, leading to the following systemof ordinary differential equations

B(t)

t = 0,(2.55)

A(t)

t

=

1

2

2B(t) +1

2

2B(t)2 ,(2.56)

subject to the boundary conditions B(0) =iu and A(0) = 0. The solution can be derived as

B(t) = iu ,(2.57)

A(t) =

1

22u(u +i)

t ,(2.58)

such that the characteristic function is given by

(2.59) (u,x,t) = eiux122u(u+i)t .
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Next we show how to derive the characteristic function of the Heston model.

Lemma 2.16 The Heston model is affine in Xt = ln Ft andvt,

dXt = 1

2vtdt+ vtdWt ,(2.60)dvt = ( vt) dt+vtdZt,(2.61)

with the auxiliary variables given by

(2.62) K0 =

0

, K(1,1) = 0, K(1,2) =

12

,

as well as

(2.63) H0= 0, H(1,1) = 0, H(1,2) =

1 2

.

Thus we have to solve the following system of ordinary differential equationsB1(t)

t = 0,(2.64)

B2(t)

t = (u) (u)B2(t) + B2(t)2 ,(2.65)

A(t)

t = B2(t)(2.66)

subject to the boundary conditions B1(0) = iu, B2(0) = 0 and A(0) = 0. The auxiliaryvariables we introduced are

(2.67) (u) = 1

2u(i +u), (u) = ui and =1

22

.

The solution can be derived as

B1(t) = iu ,(2.68)

B2(t) = (u) + D(u)

2

eD(u)t 1

c(u)eD(u)t 1

,(2.69)

A(t) =

2

((u) + D(u)) t 2 ln

c(u)eD(u)t 1

c(u) 1

,(2.70)

where

(2.71) c(u) = (u) + D(u)(u) D(u) ,

and

(2.72) D(u) =

(u)2 4(u) .We further refer to the reciprocal ofc as

(2.73) G(u) =(u) D(u)(u) + D(u)

.


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Remark 2.17 The Riccati equation forB2is essential when analysing the problem of momentexplosion of the characteristic function as first observed by Andersen and Piterbarg [AP04].

Proof:[Lemma2.16]According to

(2.74) T =

v v v 2 v

,

we can directly compute H0, H(1,1) and H(1,2), and by the aid of proposition2.14set up thesystem of Riccati equations. The solution of the first differential equation (2.68) is obvious.Recasting the second Riccati equation (2.69) as

(2.75) B2(t)

t = c (B2 a)(B2 b),

immediately leads to the following solution

(2.76) B2(t) =abe(ab)ct 1ae(ab)ct b ,

where the roots of the Riccati equation are given by

(2.77) a=((u) + D(u))

2 , b=

((u) D(u))2

, c= ,

with

(2.78) D(u) = (u)2 4(u) .Hence we can rewrite the solution as

(2.79) B2(t) =(u) + D(u)

2eD(u)t 1

c(u)eD(u)t 1 .

The solution for A now follows immediately

t0

B2(s) ds = (u) + D(u)

2

t0

eD(u)s 1c(u)eD(u)s 1ds

= 2

((u) + D(u))t 2 lnc(u)eD(u)t

1

c(u) 1

,(2.80)

which completes the proof.

Remark 2.18 Evaluation of the characteristic function in the Heston model mainly involvestwo different kind of problems. The first is the fact that the characteristic function is typ-ically of oscillatory nature, a problem which we will discuss in greater length in chapter 5.The second problem is that the characteristic function requires the evaluation of a complexlogarithm in equation (2.80) which is a multivalued function. In the next section, we show


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that continuity of the characteristic function can be preserved by either applying the so calledrotation count algorithmas introduced by Kahl and Jackel [KJ05] or by rearranging termsin the representation of the characteristic function as suggested by Duffie, Pan and Single-ton [DPS00], Schoutens, Simons and Tistaert [SST04] and Gatheral [Gat06], such that takingthe principal branch of the complex logarithm guarantees continuity as outlined and discussedby Lord and Kahl[LK06,LK07a] and Albrecher, Mayer, Schoutens and Tistaert [AMST06].

Remark 2.19 It is numerically more stable to rewrite (2.69) as

(2.81) B2(t) =(u) D(u)

2

1 eD(u)t

1 G(u)eD(u)t

,

since we assume that Re(D(u)) > 0, such that the exponential terms can cause numerical

difficulties as limu|D(u)| = , which is shown in the proof of proposition5.10.

The characteristic function of the Schobel-Zhu model can be deduced in an analogousmanner. Despite the fact that the original approach of Schobel-Zhu used different argumentsto derive the characteristic function, it turns out that the affine diffusion approach allows tofind a representation which is similar to the characteristic function of the Heston model. Thussolving the problem of complex discontinuities in the Heston models also solves this problemfor the Schobel-Zhu model.

Lemma 2.20 The Schobel-Zhu model is affine in Xt = ln Ft,vt and zt = v2

t ,

dXt = 12

zt dt +

zt dWt ,(2.82)

dvt = ( vt) dt+ dZt ,(2.83)dzt =

2zt+ 2vt+2 dt + 2zt dZt .(2.84)We can compute the necessary auxiliary parameters to calculate the system of ordinary dif-ferential equations

(2.85) K0=

02

, K(1,1) = 0, K(1,2) = 0

2

, K(1,3) = 120

2

,

as well as(2.86)

H0=

0 0 00 2 0

0 0 0

, H(1,2) =

0 0 0 22

0 22 0

, H(1,3) =

1 0 20 0 0

2 0 42
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and H(1,1)= 0. Hence the system of ordinary differential equation is given by

B1(t)

t = 0,(2.87)

B2(t)t

= 2B3(t) +

B3(t) 12

(u)

B2(t),(2.88)B3(t)

t = (u) (u)B3(t) + B3(t)2 ,(2.89)

A(t)

t = B2(t) +

1

22B2(t)

2 +2B3(t),(2.90)

subject to the boundary conditions B(0) = (iu, 0, 0). The auxiliary variables we introducedare

(2.91) =

1

2u(i+u), (u) = 2(

ui) and = 22 .

Remark 2.21 The system of ordinary differential equations in the Schobel-Zhu model isclosely related to the Heston one. Following remarks by both Heston and Schobel-Zhu, weknow that when = 0, the Schobel-Zhu model collapses to a particular instance of the Hestonmodel as can be see from equation (2.84) - the variance then has a mean-reversion speed of2, a volatility of variance equal to 2 and a mean-reversion level of2/2.

Proposition 2.22 If we denote the characteristic function of the Heston model by Heston theSchobel-Zhu characteristic function becomes

(2.92) SZ(u, S0,

0,,,,,t) =Heston(u, S

0, 2

0, 2, 2/2, 2,,t)

eAz+B2v(0) ,

whereAz follows the ordinary differential equation

(2.93) Az(t)

t =B2(t) +

1

22B2(t)

2 .

Tedious though straightforward manipulations show that Az and B2 can be solved in closedform as:

B2(t) = (u) D(u)

D(u)2

1 e 12D(u)t

1

G(u)eD(u)t

,(2.94)

Az(t) = ((u) D(u))22

2D(u)32

(u)(D(u)t 4) + D(u)(D(u)t 2) + A(t)

,(2.95)

A(t) =4e

12D(u)t

D(u)22(u)2(u)+D(u) e

12D(u)t + 2(u)

1 G(u)eD(u)t .(2.96)

Proof: The solution for B2 can be directly verified by inserting (2.94) into the definingordinary differential equation (2.88), whilst the proof ofAz requires to integrateB2 overt dueto equation (2.93).


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Remark 2.23 By recognising a special case of the Heston model within the Schobel-Zhumodel it is immediately clear that the discontinuities can be avoided in the same way asin the Heston model. Moreover we can also link the asymptotics of both models using therelationship (2.92).

2.2.2 Continuous complex logarithm

An easy way to avoid the problem of complex discontinuities is to integrate the ordinarydifferential equations in (2.66) numerically, as this would automatically lead to the correctand continuous solution. A comparative advantage of the Heston and Schobel-Zhu modelis, that it has a closed-form characteristic function, something which distinctly reduces thecomputational effort. From a computational point of view, it is therefore certainly worthwhileto investigate how to avoid discontinuities in the closed-form solution. By taking a closer lookat the characteristic function, it is clear that two multivalued functions are present, both of

which could cause complex discontinuities. The first candidate is the square root used in D(u)as defined in equation (2.78). It turns out that the characteristic function is even in D, sothat we will from here on use the common convention that the real part of the square root isnonnegative. The complex logarithm used in equation (2.80) is the second candidate.

Definition 2.24 Letz= a + iba complex variable with a, b R. Throughout the following,we denote the real and imaginary part zr = Re(z) =a and zi = Im(z) =b. Furthermore letz= rei be the representation of a complex variable in polar coordinates. We use the notationzr = |z| =r for the radius and z = arg(z) = for the phase or argument.Definition 2.25 The logarithm of a complex variablez= a +ib= rei(t+2n) witht [, ),r

R+

and n Z

is defined as(2.97) ln z= ln(r) + i(t + 2n).

In this case the branch cut of the complex logarithm is (, 0], and the logarithm is discon-tinuous along it.

Since there are numerous ways to rewrite solution (2.80), we only consider these two inthe following:

Definition 2.26 The original formula of Heston, which we derived in proposition 2.16 isreferred to as formulation 1:

(2.98) A(t, u) =

2(((u) + D(u)) t 2ln(1(t, u))) , 1(t, u) =c(u)eD(u)t

1

c(u) 1 .Definition 2.27 We refer toformulation 2as the numerically more stable representation

(2.99) A(t, u) =

2(((u) D(u)) t 2ln(2(t, u))) , 2(t, u) = G(u)e

D(u)t 1G(u) 1 .

Remark 2.28 Kahl and Jackel developed the so called rotation count algorithmto preservecontinuity of the complex logarithm in (2.98) whilst (2.99) was used in Duffie, Pan and Sin-gleton [DPS00], Schoutens, Simons and Tistaert [SST04] and Gatheral [Gat05].
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To obtain a first impression of the problem of discontinuities of the complex logarithms, weshow the trajectory of 1(t, u) in the complex plane as u varies from 0 toin figure2.4.

(u) := 1(, u)lnln |1(,u)||1(,u)|

Re[(u)]

Im[(u)]

hue: ln10(u + 1) mod1

4 2 2 4 6 8

4

2

2

4

Figure 2.4: Part of the trajectory of the function 1(, u) (2.98) in the complex plane hasthe structural shape of a spiral which gives rise to repeated crossing of the negative real axis.Here: =

0.8, = 1, = 2, time to maturity = 10 and u

[106, 103].

Every crossing of the negative real axis corresponds to a discontinuity of the characteristicfunction as shown in figure2.5.

The problem of complex discontinuities was first encountered by Schobel and Zhu. Theymention: Therefore we implemented our formula carefully keeping track of the complex loga-rithm along the integration path. This leads to a smooth CF .... This is an involved technicalprocedure which would make the implementation of this model rather complicated.

Fortunately, there is a much simpler procedure to guarantee the continuity of 1(t, u):the rotation count algorithm, which we present in the following. To show the full capabilityof this algorithm we keep the notation as general as possible. Therefore let a, b

C(R,C)

continuous complex functions and c R such that(2.100) d(u) =a(u)eb(u) +c .

One can think of the functiondas either the numerator or denominator of the critical function1 in equation (2.98). So let us state the main principle to preserve continuity:

Proposition 2.29 Algorithm 1 on page 23 is able to preserve continuity if the functiona(u)eb(u) does not cross the real axis in [min(0, c), max(0, c)].


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(A)

-5

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5 4

(B)

-2.5e-32

-2e-32

-1.5e-32

-1e-32

-5e-33

0

5e-33

1e-32

1.5e-32

2e-32

2.5e-32

3e-32

0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 2.5: Complex discontinuities in the Heston model. Parameters taken from Duffie etal. [DPS00] implied from market data of S&P500 index options: = 6.21, = 0.61,= 0.7, = 0.019, v(0) = 0.010201, F = 1 and = 20. (A): The principal argument/phase of1 (2.98) with (red solid line) and without (green dashed line) correction. (B): Characteristicfunction with and without correction.

Require: a, b C, c R1: Calculate the phase interval ofaeb asn= int

12(a+bi+)

, equal to 0 if it is in [, )

2: setdr = |aeb +c|3: setd = arg aeb +c+ 2n where arg denotes the principal argument

Algorithm 1: Rotation count algorithm.

Proof: Using

(2.101) a(u)eb(u) =arebrei(a+bi) ,

we see that ifa+ bi / [, ) the principal argument ofaeb is not the correct one. Thus wehave to correct it as done in step 1 of algorithm 1.

Corollary 2.30 One can compute the complex logarithm of the function 1in equation (2.98)

by setting

d1(u, t) = c(u)eD(u)t 1,(2.102)

d2(u) = c(u) 1,(2.103)

such that

(2.104) ln (1(t, u)) = ln

|d1(u, t)||d2(u)|

+i (d1,(u, t) d2,(u)) .
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The remaining problem is now to check that the premise of the rotation count algorithm istrue. Thus we have to verify that neither the functionc(u)eD(u)t nor c(u) do cross the realaxis in the interval [0, 1]. In order to visualise the problem of adding a constant, we considerthe following complex function

(2.105) f(x) = (1 x)

1

2 x

i ,

wherex R. Figure2.6compares the graphs off(x) + 1,f(x),f(x) 1 andf(x) 2 in thecomplex plane, as well as their principal arguments:

(A)

-1.5

-1

-0.5

0

0.5

1

1.5

-3 -2 -1 0 1 2 3

Im(f(x

))

Re(f(x))

f(x)+1

f(x)

f(x)-1

f(x)-2

(B)

-3

-2

-1

0

1

2

3

-1 -0.5 0 0.5 1 1.5 2

Arg(f(x))

x

f(x)+1

f(x)

f(x)-1

f(x)-2

Figure 2.6: Figure (A) shows the real and imaginary part of the function (2.105). In figure(B) the argument of all functions is compared. The functionf(x) crosses the real axis in [0, 1]thus we recognise a jump in the difference of arg(f(x)) arg(f(x) 1) for x = 1/2.

Whilstf(x)+1 andf(x) do have a continuous principal argument, we obtain a jump in theargument off(x) 1 andf(x) 2. The lesson from this stylised example is clear: subtracting1 can cause a discontinuity in the phase. So, if the rotation count algorithm applied to theevaluation of 1(t, u) does not yield any discontinuities, it must be that subtracting 1 fromthe numerator (2.102) and denominator (2.103) indeed does not change the phase of bothc(u)and c(u)eD(u)t. From the example we can conclude that these discontinuities can only arisewhenc(u) orc(u)eD(u)t cross the imaginary axis, and their real parts are in the interval [0, 1].The following observation is essential that this cannot happen.

Lemma 2.31 If

(2.106) / , or Im(u) /() and / < 2/ ,we have

(2.107) |c(u)| =(u) + D(u)(u) D(u)

1,where the inequality is strict, except where Re(u) = 0 or = 2/.


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Proof: The proof can be found in the appendixB. We only need one additional lemma before we can prove the main theorem of this section:

Lemma 2.32 It is only possible that

(2.108) c(u)eD(u)t = 1, or c(u) = 1

ifD(u) = 0 which further requires Re(u) = 0. In this case neither 1 nor 2 will cross thenegative real line and should be evaluated as

(2.109) limD(u)0

1(t, u) = limD(u)0

2(t, u) = 1 + (u)t .

Proof: Equation (2.109) can be verified by applying lHopitals rule.

Theorem 2.33 The rotation count algorithm can be applied to the characteristic functionof the Heston model for the parameter configuration (2.106) given in lemma2.31 where weexclude the case = 2/.

Proof: Since we use the convention that the real part ofD(u) is positive, we obtain

(2.110)c(u)eD(u)t = |c(u)| eD(u)t |c(u)| 1,

where the first equality directly follows from the convention that the real part ofD is positiveandt >0. If the inequality is strict none of the critical terms can cross the real axis in [0 , 1].

Since we excluded the case = 2/, we only have to consider Re(u) = 0. Moreover we seefrom the definition ofc in (2.71) that|c(u)| = 1 implies D(u) = 0. Thus we have to evaluate1(u, t) as in equation (2.109).

Remark 2.34 Calibrating the Heston model to a market given implied volatility surface,the restriction (2.106) is almost always satisfied, i.e. for equity markets one typically finds1 <

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