Texte intégral

An Option Pricing Formula for the GARCH
Diffusion Model
Claudia Ravanelli
Submitted for the degree of Ph.D. in Economics
University of Lugano, Switzerland
Prof. G. Barone-Adesi, University of Lugano, advisor
Prof. M. Chesney , University of Zurich
Prof. P. Vanini, University of Lugano
March 2004

To my darling mom Silvana

Acknowledgments
It is a great pleasure to thank Professor Giovanni Barone-Adesi, my thesis advisor, for invaluable
suggestions and discussions and for his support during the last three years.
I am grateful to Professor Paolo Vanini for his useful comments and suggestions and for accepting
to be a member of the thesis committee.
I really wish to thank Professor Marc Chesney for accepting to be co-examiner of this work.
Thanks go to my friends Francesca Bellini, Ettore Croci, Silvia Cappa, Talia Cicognani, Clizia
Lonati and Silvia Zerbeloni for making everyday life enjoyable.
My gratitude to my mom Silvana and to my brother Renato for their continuous support during
these years.
Specially thanks go to Loriano for his love and support and for always enriching my life.
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Contents
Acknowledgments ii
Introduction 1
1 Stochastic Volatility Models 6
1.1 Motivations of Stochastic Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 TheGARCHDiffusionModel................................ 10
2 Option Pricing under the GARCH Diffusion Model 14
2.1 TheOptionPricingFormula................................. 14
2.2 MonteCarloSimulations................................... 20
2.3 Effects of Stochastic Volatility on Option Prices . . . . . . . . . . . . . . . . . . . . . . 24
2.4 ImpliedVolatilitySurfaces.................................. 25
3 Inference based on Nelson’s Theory 40
3.1 Simple Estimators for the GARCH Diffusion Model . . . . . . . . . . . . . . . . . . . . 41
3.2 EmpiricalApplication .................................... 42
4 Conclusions 51
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A Proof of Propositions and other Mathematical Issues 53
A.1 ProofofProposition2.1 ................................... 53
A.1.1 First conditional moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.1.2 Second conditional moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A.2 Conditional Moments of VTunder Log-Normal Variance . . . . . . . . . . . . . . . . . 58
A.3 The Market Price of Risk and the Novikov’s Condition . . . . . . . . . . . . . . . . . . 61
A.4 Nelson’s Moment Matching Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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