Time-dependent barrier options and boundary crossing probabilities

The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black-Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.