A General Stochastic Volatility Model on VIX Options

Abstract

Abstract In this dissertation, we study a general stochastic volatility model for the VIX options (Chicago Board Options Exchange) volatility index, which is a stochastic differential equation with 8 unknown parameters. It originated from a nested stochastic model based on several known models in the paper [7], stochastic volatility models and the Pricing of VIX Options. To estimate the parameters in these models from the real financial data a commonly used approach is the Generalized Method of Moments of Hansen (1982). We will study the model in more generality and we shall provide a completely different parameter estimation technique using the ergodic theory. Since our equation is more general and new and since our equation is singular in the sense it does not satisfy the global Lipschitz condition, we shall first study the existence, uniqueness and positivity of the solution of the SDE, in which Feller’s test will be used to calculate a criteria of all parameters such that the SDE has a unique and positive weak solution. The positivity property of the solution is crucial, since volatility is always positive. Then, we use the strong large law of numbers theorems given e.g. in [4] to give the region for the parameters to live in order that the model is ergodic. In important condition for the ergodicity is the positive recurrency. We give verifiable condition on the parameters so that process is positive recurrent. This results also provide ways to calculate the invariant distribution (limiting distribution). The next step is to provide a theoretical methodology of parameter estimation. Simulation process will be introduced with giving an example for each case. In the future study, I will work on testing the model using numerical schemes. Keywords: Stochastic volatility model, VIX options, Feller’s test, ergodicity, parameter estimation.