Singular perturbations for boundary value problems arising from exotic options

We study the pricing of three exotic derivative securities (barrier, lookback, and passport options) which can be characterized by boundary value PDE problems in the context of popular Markovian stochastic volatility models of stock prices. By extending the fast mean-reverting asymptotic analysis in [J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, London, 2000], the usual "Greek" correction to the Black-Scholes prices of these contracts is further corrected by a boundary integral term that is rapidly computed numerically. In the case of the passport option, the asymptotic method is effective in accounting for stochastic volatility effects in a simple and robust fashion even in the presence of a highly nonlinear embedded stochastic control problem.