The Estimation of Stochastic Models in Finance with Volatility and Jump Intensity

This thesis covers the parametric estimation of models with stochastic volatility, jumps, and stochastic jump intensity, by FFT. The first primary contribution is a parametric minimum relative entropy optimal Q-measure for affine stochastic volatility jump-diffusion (ASVJD). Other attempts in the literature have minimized the relative entropy of Q given P either by nonparametric methods, or by numerical PDEs. These methods are often difficult to implement. We construct the relative entropy of Q given P from the Lebesgue densities under P and Q, respectively, where these can be retrieved by FFT from the closed form log-price characteristic function of any ASVJD model. We proceed by first estimating the fixed parameters of the P-measure by the Approximate Maximum Likelihood (AML) method of Bates (2006), and prove that the integrability conditions required for Fourier inversion are satisfied. Then by using a structure preserving parametric model under the Q-measure, we minimize the relative entropy of Q given P with respect to the model parameters under Q. AML can be used to estimate P within the ASVJD class. Since, AML is much faster than MCMC, our main supporting contributions are to the theory of AML. The second main contribution of this thesis is a non-affine model for time changed jumps with stochastic jump intensity called the Leveraged Jump Intensity (LJI) model. The jump intensity in the LJI model is modeled by the CIR process. Leverage occurs in the LJI model, since the Brownian motion driving the CIR process also appears in the log-price with a negative coefficient. Models with a leverage effect of this type are usually affine, but model the intensity with an Ornstein-Uhlenbeck process. The conditional characteristic function of the LJI log-price given the intensity is known in closed form. Thus, we price LJI call options by conditional Monte Carlo, using the Carr and Madan (1999) FFT formula for conditional pricing