Exact Simulation Of The Sabr Model

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 beta = 1, or when beta < 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 beta < 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%.