A Multivariate Stochastic Levy Correlation Model with Integrated Wishart Time Change and Its Application in Option Pricing
Madan, Dilip B
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We develop a new multivariate Levy correlation model which is formulated by evaluating Levy processes subordinate to the integral of a Wishart process. This new model captures not only stochastic mean, stochastic volatility, and stochastic skewness, but also stochastic correlation of cross-sectional asset returns while still being analytical tractable. It is a multivariate extension of the time changed Levy process introduced by Carr, Geman, Madan and Yor, which can capture the individual dynamics as well as the interdependencies among several assets. In this dissertation, two different methods are employed to simulate paths of the instantaneous rate of time change matrix A(t), followed by a Wishart process. The simulation paths successfully display desirable clustering and persistence features. In addition, we analyze the behavior of the joint log return distribution generated in this new model and show that the model provides a rich dependence structure. The option pricing problem involves computing the closed form of the characteristic functions, which are usually not easily obtained in the multivariate correlated case. In this thesis, we derive explicit forms of both marginal and joint conditional characteristic functions by applying the `Matrix Riccati Linearization' technique creatively. Our work is distinguished from existing multivariate stochastic volatility models, with the advantage that it can deal with stochastic skewness effects introduced by Carr and Wu. Finally, we derive pricing methods for multi asset options as well as single asset options by using both simulation and Fast Fourier transform methods. More important, this model can be well calibrated to the real market. We chose options on two major FX currencies to perform the calibration and remarkable consistency has been observed.