Spline Cubatures for Expectations of Diffusion Processes and Optimal Stopping in Higher Dimensions (with Computational Finance in View)

We develop certain cubature (quadrature) rules for expectations of diffusion processes in ℝ N that are analogous to the well known spline interpolation quadratures for ordinary integrals. By incorporating such rules in appropriate backward induction procedures, we develop new numerical algorithms for solving free-boundary (optimal stopping) problems, or ordinary fixed-boundary problems. The algorithms developed in the paper are directly applicable to pricing contingent claims of both American and European types on multiple underlying assets.

[1]  Savas Dayanik,et al.  On the optimal stopping problem for one-dimensional diffusions , 2003 .

[2]  Andrew Lyasoff,et al.  Dynamic Integration of Interpolating Functions and Some Concrete Optimal Stopping Problems , 2008 .

[3]  A. Stroud Approximate calculation of multiple integrals , 1973 .

[4]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal Stopping and Free-Boundary Problems , 2006 .

[5]  S. Kusuoka Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus , 2003 .

[6]  S. Kusuoka Approximation of Expectation of Diffusion Process and Mathematical Finance , 2001 .

[7]  Pierre-Louis Lions,et al.  Applications of Malliavin calculus to Monte Carlo methods in finance , 1999, Finance Stochastics.

[8]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[9]  Edward C. Waymire,et al.  A self-similar invariance of critical binary Galton-Watson trees , 2000 .

[10]  G. Pagès,et al.  A quantization algorithm for solving multidimensional discrete-time optimal stopping problems , 2003 .

[11]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[12]  Terry Lyons,et al.  Cubature on Wiener space , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  G. Pagès,et al.  Error analysis of the optimal quantization algorithm for obstacle problems , 2003 .

[14]  L. Rogers Monte Carlo valuation of American options , 2002 .