On Optimization Techniques for Calibration of Stochastic Volatility Models

The aim of this paper is to study stochastic volatility models and their calibration to real market data. This task is formulated as the optimization problem and several optimization techniques are compared and used in order to minimize the difference between the observed market prices and the model prices. At first we demonstrate the complexity of the calibration process on the popular Heston model and we show how well the model can fit a particular set of market prices. This is ensured by using a deterministic grid which eliminates the initial guess sensitivity specific to this problem. The same level of errors can be reached by employing optimization techniques introduced in the paper, while also preserving time efficiency. We further apply the same calibration procedures to the recent fractional stochastic volatility model, which is a jump-diffusion model of market dynamics with approximative fractional volatility. The novelty of this paper is especially in showing how the proposed calibration procedures work for even more complex SV model, such as the introduced long-memory fractional model. Keywords—stochastic volatility models; Heston model; fractional SV model; option pricing; calibration; optimization