Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.