Numerical techniques for the Heston collocated volatility model

In the collocating volatility (CLV) model, the stochastic collocation technique is used as a convenient representation of the terminal distribution of the market option prices. A specific dynamic is added in the form of a stochastic driver process, which allows more control over the prices of forward starting options. This is reminiscent of the Markov functional models. (Grzelak uses a single-factor Ornstein–Uhlenbeck process as the driver for the CLV model, and Fries uses a single-factor Wiener process with time-dependent volatility in his equity Markov functional model. Van der Stoep et al consider a Heston stochastic volatility driver process and show that it offers more flexibility to capture the forward smile in the context of foreign exchange options.) In this paper, we discuss all aspects of derivative pricing under the Heston– CLV model: calibration with an efficient Fourier method; a Monte Carlo simulation with second-order convergence; and accurate partial differential equation pricing through implicit and explicit finite-difference methods.