A comparative study on time-efficient methods to price compound options in the Heston model

The primary purpose of this paper is to provide an in-depth analysis of a number of structurally different methods to numerically evaluate European compound option prices under Heston's stochastic volatility dynamics. Therefore, we first outline several approaches that can be used to price these type of options in the Heston model: a modified sparse grid method, a fractional fast Fourier transform technique, a (semi-)analytical valuation formula using Green's function of logarithmic spot and volatility and a Monte Carlo simulation. Then we compare the methods on a theoretical basis and report on their numerical properties with respect to computational times and accuracy. One key element of our analysis is that the analyzed methods are extended to incorporate piecewise time-dependent model parameters, which allows for a more realistic compound option pricing. The results in the numerical analysis section are important for practitioners in the financial industry to identify under which model prerequisites (for instance, Heston model where Feller condition is fulfilled or not, Heston model with piecewise time-dependent parameters or with stochastic interest rates) it is preferable to use and which of the available numerical methods.