Asymptotic Approximations to CEV and SABR Models

Abstract

The problem of pricing, hedging and calibrating equity derivatives in a fast and consistent fashion is considered when the underlying asset does not follow the standard Black-Scholes model but instead the CEV or SABR models. The underlying process in the CEV model has volatility as a deterministic function of the asset price while in the SABR model the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem. For these models, default may be possible and the original ray approximation is not valid near the default boundary so a modifiedd asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model and the SABR results. The applicability of the results is illustrated by deriving new analytical approximations for vanilla options based on the CEV and SABR models. The accuracy of the results is demonstrated numerically.

Keywords: asymptotic approximations, perturbation methods, deterministic volatility, stochastic volatility, CEV model, SABR model

JEL Classification: C00, G12