Using Monte Carlo Simulation and Importance Sampling to Rapidly Obtain Jump-Diffusion Prices of Continuous Barrier Options

The problem of pricing a continuous barrier option in a jump-difiusion model is studied. It is shown that via an efiective combination of importance sampling and analytic formulas that substantial speed ups can be achieved. These techniques are shown to be particularly efiective for computing deltas. The Merton jump-difiusion model is almost as venerable as the Black{ Scholes model, the flrst paper, (10), was published in 1976. It captures a feature of equity markets which is sorely lacking in the Black{Scholes model: crashes. It allows the pricing of vanilla call and put options via an inflnite sum. However, the pricing of exotic options is di-cult, and other than in special cases, numerical methods must be resorted to. Here, we introduce a new approach to pricing barrier options with Monte Carlo which is predicated on the use of importance sampling and analytic formulas when there are no jumps left. We focus here on the case of a down-and-out call option with strike above barrier for concreteness. The cases of up-and-outs and puts can be handled similarly. Knock-in options can be handled, as usual, via the observation that knock-in plus knock-out is the same as vanilla. Kou and Wang have studied the special case of barrier options for double exponential jumps, (7), and derived a formula in terms of the Laplace transform. Lewis has developed a Fourier transform approach, and obtained formulas in the special cases that the jump does not cross the barrier and when there is a unique jump size, (8). Penaud, (13), developed a lattice approach based on pre-computation of densities designed to be rapid for large portfolios of derivatives but slow for a single instrument. Here, we study the case where jumps are lognormal,