A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

Abstract

We study the SABR stochastic volatility model with the volatility-of-volatility parameter ν. We provide a method to expand the price CSABR(S, K, ν, σ, τ ) of a European call in this model as a Taylor series in ν, CSABR(S, K, ν, σ, τ ) = CBS(S,K, σ, τ ) νC1 ν2C2 . . . νkCk O(νk 1), where CBS is the usual Black-Scholes pricing formula. As an illustration of the method, we compute the terms C1 and C2. The formulas for the correction terms C1 and C2 are fully explicit and do not require any integration or any special functions (unlike CBS). We also provide approximate formulas for the implied volatility and for the hedging parameter
Δ. Formulas for other greeks can be obtained similarly. We provide numerical tests to show the accuracy of our formula for small ν and τ being not too large.