Parameter Estimation And VIX Derivatives Pricing Under A Class Of Stochastic Volatility Jump Diffusion Models

Stochastic modeling of asset prices is one of the core issues of financial research.As people deepen their understanding of the stylized facts of asset returns,more and more stochastic models have been proposed to include these empirical characteristics of returns.The research in this thesis is based on a novel stochstic volatility jump diffusion model with a flexible jump structure.Under this model,we first derive the closed-form linear relationship between the VIX index and the instantaneous variance,and then study the parameter estimation problem of the stochastic volatility jump diffusion model and the pricing problem of VIX index derivatives.The first chapter introduces the research background and significance of this thesis,main references,and research innovations and deficiencies.Chapter 2 proposes a novel stochastic volatility jump diffusion model and com-pares it with some popular stochastic volatility jump diffusion models.In addition,we give a closed-form linear relationship between the VIX index and the instantaneous variance.This relationship serves as a bridge.On the one hand,when studying the pa-rameter estimation problem of the stochastic volatility jump diffusion model,we can use the observable VIX index data replaces the instantaneous variance for maximum like-lihood estimation.On the other hand,when studying the pricing problem of VIX index derivatives,we can indirectly derive the probability density function of VIX through the dynamic process of instantaneous variance to obtain the pricing formula.Chapter 3 studies the parameter estimation problem of the stochastic volatility jump diffusion model.Using the closed-form linear relationship between the VIX in-dex and the instantaneous variance,we give the maximum likelihood estimation of the transformed data of the SVCIJ model.In particular,this estimation method is still appli-cable to its special case models,such as SV,SVJ,SVCJ and SVIJ model.Using the data of the S&P 500 and VIX index for nearly 30 years,we found that the SVCIJ model can better capture the jump activities.In addition,the estimation results based on the sliding window method show that the 2007-2009 financial crisis shifted the pattern of market jumps.Prior to this,frequent small jumps were the mainstay,and after that,large jumps with lower frequency dominated the market’s jump activities.We also found that the enhancement of leverage effect during the crisis is mainly dominated by the jump part,and the leverage effect of the continuous part is almost unchanged.Chapter 4 studies the pricing problem of VIX index derivatives.We first give the analytical expression of the instantaneous variance characteristic function under the SVCIJ model,and then use the closed-form linear relationship between the VIX index and the instantaneous variance to obtain the characteristic function of VIX and use the Fourier inverse transformation to obtain its transition probability density function.By designing two auxiliary Laplace transformations,and using Fubini theorem to exchange the order of integration,we have obtained the pricing formula of VIX index futures and options in analytical form.The results of Monte Carlo simulation experiments show that our pricing formula is accurate.Chapter 5 summarizes the thesis and discusses the deficiencies of the research and further prospects for work.