The valuation of path-dependent options in finance creates many interesting mathematical challenges. Among them are a large Delta and Gamma near the expiry leading to a big error in pricing those exotic options as well as European vanilla options. Also, the higher
order corrections of the asymptotic prices of the derivatives in some stochastic volatility models are difficult to be evaluated. In this paper we use the method of matched asymptotic expansions to obtain more practical values of lookback and barrier option prices near the expiry. Our results verify that matching asymptotics is a useful tool for PDE methods in path-dependent option pricing.
