Wavelet Method for Pricing Options with Stochastic Volatility

We use the Heston stochastic volatility model for calculating the theoretical price of an option. While the Black-Scholes model assumes that the volatility of the asset is constant or a deterministic function, the Heston model assumes that the volatility is a random process. The explicit solution for the Heston model is unavailable for many types of options and therefore numerical methods have been proposed for pricing options under Heston model, e.g. Monte Carlo method, the finite difference method, the finite element method or the discontinuous Galerkin method. The Heston model is represented by a parabolic equation. For its efficient numerical solution, we use the theta scheme for the time discretization and we propose an adaptive wavelet method for the discretization of the equation on the given time level. We construct a piecewise linear wavelet basis and use it in the scheme. The advantage of wavelets is their compression property. It means that the representation of the solution in a wavelet basis requires a small number of coefficients and the computation of the solution can be performed with the small number of parameters. Numerical example is presented for the European put option.