Distributional Properties of Continuous Time Processes: From CIR to Bates

Abstract

We focus on returns defined as log-price differentials and generated by a diffusion process which incorporates stochastic volatility and jumps in prices. The jumps are properly compensated for this model. The stochastic volatility follows the well-known CIR process. We present general conditional and unconditional (co-)moment formulas for the solution of this process. By identifying these moments with those of a non-central chi-squared distribution, we derive distributional properties in a way that significantly differs from the historic approaches. Next, we derive the conditional and unconditional characteristic functions of log-returns which allows us to generate conditional and unconditional moments. We provide closed form expressions for the first four unconditional moments of log-returns.