From the Implied Volatility Skew to a Robust Correction to Black-Scholes American Option Prices

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a xed boundaryvalue problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the e ect of the volatility skew.