Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model

Abstract

We deal with several efficient discretization methods for the simulation of the Heston stochastic volatility model. The resulting schemes can be used to calculate all kind of options and corresponding sensitivities, in particular the exotic options that cannot be valued with closed-form solutions. We focus on to the (computational) efficiency of the simulation schemes: though the Broadie and Kaya (2006) paper provided an exact simulation method for the Heston dynamics, we argue why its practical use might be limited. Instead we consider efficient approximations of the exact scheme, which try to exploit certain distributional features of the underlying variance process. The resulting methods are fast, highly accurate and easy to implement. We conclude by numerically comparing our new schemes to the exact scheme of Broadie and Kaya, the almost exact scheme of Smith, the Kahl-Jäckel scheme, the Full Truncation scheme of Lord et al. and the Quadratic Exponential scheme of Andersen.

Calibration and simulation of Heston model

“…as noted by van Haastrecht and Pelsser [40] does not entail any discretization error in the stock price direction. On the other hand, the scheme usually shows biases in the Euler discretization of the variance process (and thus in resulting stock prices).…”

“…This is probably due to the fact that the IJK scheme in Kahl-Jäckel [42] was not stated explicitly for Heston model. The wrongly interpreted scheme published in Haastrecht and Pelsser [40] can be found in some other papers as well.…”

“…0 and so the choice of the fix functions may seem almost indifferent, but the contrary is true. While some schemes are extremely biased for practical sizes of the time step, others turn out to be almost bias-free for not "too extreme parameter configurations" as pointed out in Van Haastrecht and Pelsser [40].…”

“…Proposed schemes then use precalculated values of the inverse of the gamma distribution and the compound Poisson random variables are approximated using a mixture of exponential distribution and quadratic Gaussian distribution. There are some similarities to the non-central chi-squared inversion scheme presented by van Haastrech and Pelsser [40]. None of these methods are, however, widely accepted by practitioners and further numerical analysis of these schemes is an ongoing research.…”

“…In [40], van Haastrecht and Pelser commented on Kahl-Jäckel scheme implemented in the following way. Conditional on time s a discretization of the log Stock price process S for t > s (with ıt WD t s) reads exactly…”