Exact simulation of stochastic volatility models based on conditional Fourier-cosine method

Abstract

The traditional methodology used for the exact simulation of stochastic volatility models based on the Gil–Pelaez formula presents implementation problems that are observed by many researchers and practitioners. In particular, although conventionally considered exact, such a method presents a difficult control of the error. The bias of the Monte Carlo simulation estimator can only be computed numerically and is controlled by two parameters, typically determined by running time-consuming simulations under different tuning parameter configurations until an optimal setup is found. In this paper, we propose a new exact simulation scheme based on the Fourier-cosine method, which approximates a probability density given the characteristic function as follows: the density is truncated on a finite interval, and approximated by a classical Fourier-cosine series. The method allows full error control via an effective automatic identification of the tuning parameters given a user-supplied error tolerance. The new approach offers the following advantages: improved control of the error, simplified implementation, and reduction in computing time. The error is controlled by only one parameter instead of two. This parameter has a clear interpretation: it is the maximum tolerable bias. This facilitates the implementation, since the maximum bias becomes an input of the simulation algorithm, instead of an output, and can be set a priori, before running simulations. Our analysis shows that the proposed exact simulation scheme is computationally faster than the traditional one, and presents an improved speed-accuracy profile with respect to alternative state-of-the-art fast approximated sampling schemes.