Stochastic Volatility, Smile & Asymptotics

We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull & 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 & Touzi, for which we present a simplified proof. We also present new asymptotic formulas that describe the geometry of smile curves and can be used for interpolation of implied volatility data. Motivated by the robustness of the smile effect to specific modelling of the unobserved volatility process, we present a new approach to stochastic volatility modelling starting 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 variabl..