A Low-Bias Simulation Scheme for the Sabr Stochastic Volatility Model

Abstract

The Stochastic Alpha Beta Rho Stochastic Volatility (SABR-SV) model is widely used in the financial industry for the pricing of fixed income instruments. In this paper we develop a low-bias simulation scheme for the SABR-SV model, which deals efficiently with (undesired) possible negative values in the asset price process, the martingale property of the discrete scheme and the discretization bias of commonly used Euler discretization schemes. The proposed algorithm is based the analytic properties of the governing distribution. Experiments with realistic model parameters show that this scheme is robust for interest rate valuation.

On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options

“…Milstein-based Monte Carlo simulation is employed here as a reference for our one time-step simulation. In [24] , the authors have shown that the Milstein scheme is relatively stable when applied for the SABR model compared to the Euler or log-Euler schemes. In Fig.…”

“…However some recent developments make this computation affordable. In [24] , Chen et al proposed a forward asset simulation based on a combination of moment-matching (Quadratic Gaussian) and enhanced direct inversion procedures. We employ this technique also here.…”

“…The asset simulation is performed either by a moment-matched quadratic Gaussian approximation or by an enhanced direct inversion. The authors in [24] proposed a threshold value to choose the suitable technique among these two. The moment-matching approach relies on the fact that, for S (0) 0, the distribution function in Eq.…”

“…The moment-matching approximation is safely and accurately applicable when 0 < ψ ≤ 2 and μ ≥ 0 [24] . Otherwise, a direct inversion procedure must be employed.…”

“…As pointed out in [24,26] , the exact simulation of the asset price can, in certain stochastic volatility models, give rise to loss of the martingale property (because of the approximation of a continuous process by its discrete equivalent). This is especially seen, when the size of the time-step is large and for certain configurations of the SABR parameters, like small β values, close-to-zero initial asset values S 0 , high vol-vol parameter α or large initial volatility σ 0 .…”