Pricing knockout options with a general boundary

In modern financial markets, various option contracts have been introduced and traded. Among them, there is a kind of options whose contract is nullified whenever the underlying asset price reaches a predetermined ‘knockout’ price level. Some mathematical models have been developed for the knockout options as an extension of the Black–Scholes model, under a rather restrictive assumption that the knockout boundary of the asset price process is an exponential function of the time to expiration. This paper provides approximate but closed-form pricing formulas for European down-and-out call and up-and-out put options with a general knockout boundary. These formulas are consistent with the exact pricing formula developed by Merton for the exponential boundary case. Our formulas have a considerable advantage of the computation time over the finite-difference method as well as Monte Carlo simulation. A set of numerical tests shows that the formulas are sufficiently accurate for practical applications.