Fast strong approximation Monte Carlo schemes for stochastic volatility models

Abstract

Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein-Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.Stochastic volatility models, Stochastic numerical integration, Strong approximation error, Hyperbolic Ornstein-Uhlenbeck process, Hyperbolic volatility,

Calibration and simulation of Heston model

“…Another mistake is in the last term, where there is 1 2 instead of 1 4 compared to our equation. b n is v p n for Heston in the original article by Kahl and Jäckel [42]. This implies b.s/ D v.s/ 1=2 , which cancels out with v.s/ 1=2 and leaves in the notation of Haastrecht and Pelsser.…”

“…These two were causing difficulties because of the representation of their rather high values for finer discretizations. Also IJK scheme by Kahl and Jäckel [42], which is supposed to be a drop-in method for simple schemes, is tested with parameters obtained from the market and it appears not to be useful for pricing options using Monte Carlo simulations. We also did experiments and proposed a scheme compromising speed and accuracy based on the observed performances of the so far known schemes and we seem to have out-performed the other simple schemes when using the Monte Carlo simulations for pricing options.…”

“…This implies b.s/ D v.s/ 1=2 , which cancels out with v.s/ 1=2 and leaves in the notation of Haastrecht and Pelsser. 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.…”

“…Authors Kahl and Jäckel [42] claim the IJK scheme is not superior in convergence order to the log-Euler scheme, but they praise the fact that it excels over other equally simple methods by showing a superior convergence behaviour that is faster by a multiplicative factor.…”

“…This however does not match our implementation of the scheme. The Heston log price S comes from the formula by Kahl and Jäckel [42], which is exactly…”