An Algorithm for Simulating Bermudan Option Prices on Simulated Asset Prices

Abstract

European options and other derivatives that do not allow early exercise can be priced efficiently by recursion methods that start at known payoffs on the expiration date and work backwards in time to value the derivative at the initial date. Path-dependent derivatives like mortgage-backed securities (MBS) and many interest rate derivatives require forward-looking Monte Carlo simulations. But there are increasing numbers of more complex securities that feature both early exercise and path-dependence, such as callable MBS and Bermudan options on Bermudan swaptions, where pricing requires looking in both directions. The typical case is a contract in which, at discrete dates, the holder must decide between immediate exercise or continuation of a derivative whose underlying must be valued by Monte Carlo simulation. This decision is presented repeatedly along each simulated path for the underlying interest rate, and the continuation value in each case must be estimated by a sub-simulation of future rate paths starting from that point. This leads to a great increase in the total number of simulations required to achieve accurate pricing. It also leads to a less well-known problem, that the sampling error in the Monte Carlo simulation effectively becomes incorporated in the rate volatility of the underlying paths, which induces an upward bias in the model values. In this paper, Huge and Rom-Poulsen present a technique for minimizing the problem. Sampling error is reduced by projecting „crude Monte Carlo” values onto a set of basis functions to extract the expected value with much less noise. A series of examples illustrates clearly the nature of the problem and the effectiveness of their solution. The result is far more efficient model estimation for this challenging class of derivatives.