Stochastic volatility, smile & asymptotics

Abstract

We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull and White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is guaranteed by a theorem of Renault and Touzi, for which we present a simplified proof. Motivated by the robustness of the smile effect to specific modelling of the unobserved volatility process, we introduce a methodology that does not depend on a particular stochastic volatility model. We start with the Black-Scholes pricing PDE with a random volatility coefficient. We identify and exploit distinct time scales of fluctuation for the stock price and volatility processes yielding an asymptotic approximation that is a Black-Scholes type price or hedging ratio plus a Gaussian random variable quantifying the risk from the uncertainty in the volatility. These lead us to translate volatility risk into pricing and hedging bands for the derivative securities, without needing to estimate the market's value of risk or to specify a parametric model for the volatility process. For some special cases, we can give explicit formulas. We outline how this method can be used to save on the cost of hedging in a random volatility environment, and run simulations demonstrating its effectiveness. The theory needs estimation of a few statistics of the volatility process, and we run experiments to obtain approximations to these from simulated stock price and smile curve data.