New Results on Deterministic Pricing of Financial Derivatives

Monte Carlo simulation is widely used to price complex nancial instruments. Recent theoretical results and extensive computer testing indicate that deterministic methods may be far superior in speed and conndence. Simulations using the Sobol or Faure points are examples of deterministic methods. For the sake of brevity, we refer to a deterministic method using the name of the sequence of points which the method uses, e.g., Sobol method. In this paper we test the generalized Faure sequence due to Tezuka T95]. We also test a modiied Sobol method; this includes further improvements from those in PT95]. We compare these two low discrepancy deterministic methods with basic Monte Carlo. We summarize our conclusions regarding the valuation of a Collateralized Mortgage Obligation which we divide into three groups. Similar results hold for other nancial instruments such as asian options. Deterministic methods beat Monte Carlo by a wide margin. In particular, Both the generalized Faure and modiied Sobol methods converge signiicantly faster than Monte Carlo. The generalized Faure method always converges at least as fast as the modiied Sobol method and frequently faster. The Monte Carlo method is sensitive to the initial seed. Deterministic methods outperform Monte Carlo for a small number of sample points. In particular, Deterministic methods attain small error with a small number of points.