Utilizing Monte Carlo Simulation for Valuation: the Case of Barrier Options under Stochastic Interest Rates

Abstract

In this chapter we propose aMonte Carlo approach for pricing barrier options when analytical pricing formulas are unavailable. Barrier options are among the most commonly used options in the financial market and our approach should therefore be of interest. Based on numerical examples presented in the chapter, it seems likely that our approach reduces the computation time by a factor between 236,000 and 94,000,000.1 To put this in perspective, assume we have a computer that uses one second to estimate the price using our proposed pricing algorithm. To obtain comparable price estimates using standard Monte Carlo simulations would require a computation time between three days and three years! Plain vanilla put and call options give the owner the right to sell or buy an asset at a pre specified price at some future point in time. Barrier options are either of knock-out or knock-in type. If the price of the underlying asset crosses some barrier H, a knock-out option becomes worthless, i.e., the option contract is canceled. For a knock-in option the option is invoked when the underlying asset crosses the barrier. Thus, a knock-in option expires worthless if the price of the underlying asset never crosses the barrier during the option’s life. Analytical pricing formulas for barrier options are readily available when the value of the underlying asset follows a geometric Brownian motion and interest rates are deterministic. Here we have in mind a situation where the underlying asset follows a process that precludes the derivation of an analytical pricing formula for the option. In particular we focus on the situation where interest rates are stochastic. Other situations could be where the underlying asset follows more complicated price processes. We propose a Monte Carlo approach to value the barrier option. Estimating the market value of barrier options by Monte Carlo simulations is know to be rather time consuming (see e.g., Broadie et al. (1997)). First, a relatively high number of simulations is needed in order to reduce the standard error of the price estimates. Second, since the barrier option is path-dependent, the whole price path for the underlying asset is needed to determine whether the barrier H has been crossed or not. Approximating the price path with few monitoring points, results in biased estimates. In fact, somewhat surprisingly many monitoring points is needed to get unbiased price estimates. Both these facts make estimation of barrier option