An efficient Model is developed for option pricing under stochastic volatility and jump diffusion processes.
The Bates model is represented by the bivariate system of stochastic differential equations $$\hspace{2cm}\frac{dS_t}{S_t}=(\mu-\lambda\,\bar{J})dt+\sqrt{v_t}dW_1(t)+JdN(t)$$ $$dv_t=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2(t)$$ where $$\mathbb{E^Q}[dW_1(t)dW_2(t)]=\rho dt$$ and $N_t$ is a compound Poisson process with intensity $\lambda$ and independent jumps $J$ with $$\ln{(1+J)}\tilde{~} N\left(\ln(1+\beta)-\frac{1}{2}\alpha^2,\alpha^2\right)$$ The parameters $\beta$ and ${\alpha}$ determine the distribution of the jumps and the Poisson process is assumed to be independent of the Wiener processes.
