Exact Simulation of the SABR Model

Abstract

The stochastic alpha-beta-rho (SABR) model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketplace. However, no analytical solution to the SABR model exists that can be simulated directly. This paper explores the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method for the forward price and its volatility in two special but practically interesting cases, i.e., when the elasticity β 1, or when β < 1 and the price and volatility processes are instantaneously uncorrelated. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the noncentral chi-squared distribution functions, respectively. Two novel simulation schemes are proposed to achieve numerical accuracy, efficiency, and stability. One stems from numerical Laplace inversion and Asian option literature, and the other is based on recent developments in evaluating the noncentral chi-squared distribution functions in a robust way. Numerical examples demonstrate that our method is fast and accurate under various market environments. (ii) When β < 1 but the price and volatility processes are correlated, our simulation method becomes a semi-exact one. Numerical results suggest that it is still quite accurate when the time horizon is not long, e.g., no greater than one year. For long time horizons, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing under the SABR model, we propose a conditional simulation method, which reduces the variance of the plain simulation significantly, e.g., by more than 99%.

Hyperbolic normal stochastic volatility model

“…Therefore, one draw only requires one and a half (1.5) normal random variables, which is an unprecedented efficiency for any SV model simulation. Particularly, this method is more efficient than the exact SABR simulation of Cai et al (), although it is limited to the normal case. This study's method directly draws and , whereas Cai et al () first draw and , and subsequently .…”

“…Particularly, this method is more efficient than the exact SABR simulation of Cai et al (), although it is limited to the normal case. This study's method directly draws and , whereas Cai et al () first draw and , and subsequently . Although Equation states the transition from to , it can handle any time interval from to ().…”

“…The development of MC simulation methods of the SABR dynamics is relatively recent. While several efficient approximations (Chen, Oosterlee, & VanDerWeide, ; Leitao, Grzelak, & Oosterlee, ,b) have been proposed, an exact simulation method Cai et al () is available for the following three special cases: (a) , (b) , and (c) . The key element in this method is to simulate the time‐integrated variance , conditional on the terminal volatility .…”

“…Given the exact random numbers of and , in the normal SABR model is easily simulated as for a standard normal variable . Although a heavy Euler scheme is avoided, the method of Cai et al () still incurs a moderate computation cost due to the numerical inversion of the Laplace transform and root‐solving for the CDF inversion.…”

“…Although a heavy Euler scheme is avoided, the method of Cai et al (2017) still incurs a moderate computation cost due to the numerical inversion of the Laplace transform and root-solving for the CDF inversion. As opposed to the previous studies using  2 , this study's main result in Section 3.1, based on Alili and Gruet (1997), exploits  3 .…”