Asymptotic Approximations to Deterministic and Stochastic Volatility Models

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 stochastic CEV (constant elasticity of variance) or SABR (stochastic alpha beta rho) model. The underlying process in the SABR model has 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 the SABR model, default may be possible, and the original ray approximation is not valid near the default boundary, so a modified asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model, which has deterministic volatility, as a special case of 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.

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