Pricing Barrier Bond Options with One-factor Interest Rate Models

In this paper, we describe a numerical method to price barrier options on a zero-coupon bond. The method can be applied to one-factor short rate models where the transtion distribution function of the short rate is known and no explicit solutions for barrier bond options are available. We give computational examples with the Vasicek and Hull and White model. Barriers on the underlying zero-coupon bond are transformed to smooth time dependent barriers on the short rate. The first passage time densities of the short rate are numerically solved from integral equations. Given the prices of barrier options on a zero-coupon bond, one can also evaluate a continuous barrier caplet (floorlet). To evaluate the method, we price single and double knock-in barrier options on equity in the Black and Scholes' model. The prices computed by the method are compared to the prices from analytical formulae and the ordinary Monte Carlo simulation. It is shown that the method is more accurate and faster than the ordinary Monte Carlo simulation. Unlike lattice methods, this method avoids the problem of finding optimal positioning of barriers. When the barriers are time dependent and the underlying process is more general, we argue that the method has advantages over ordinary Monte Carlo simulation and lattice methods.