Universal Smiles

Abstract

The SABR model has two variables, the forward asset price and the local volatility . A singular perturbation analysis has shown that the marginal density Q(T, F) defined by can be found through by solving a one-dimensional (1D) effective forward equation of the form where and This reduces the valuation of European options to one spatial dimension from two. Recently, similar asymptotic analyses have shown that for all commonly used stochastic volatility models, the marginal density Q(Tex, F) can be obtained through O(ε2) by solving the same 1D effective forward equation. The only differences are in theformulas for the coefficients b and c in terms of each model's fundamental parameters. These stochastic volatility models include the Heston and generalized Heston models, the mean-reverting SABR (λ-SABR) and dynamic SABR models, the exponential volatility models, ZABR-like models, cross-FX SABR models, and the SABR models for baskets and spreads. Here we analyze the above “universal” effective forward equation, and obtain explicit asymptotic formulas for the implied volatilities of European options for all the above models. These new formulas reduce to the original SABR implied vol formulas under moderate conditions, but are much more accurate in extreme situations.

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