Summary
We develop certain cubature (quadrature) rules for expectations of diffusion processes in ℝN 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bally V, Pagès G (2003) A Quantization Algorithm for Solving Discrete Time Multidimensional Optimal Stopping Problems. Bernoulli 6(9):1–47.
Bally V, Pagès G (2003) Error Analysis of the Quantization Algorithm for Obstacle Problems. Stochastic Processes and their Applications 106(1):1–40.
Dayanik S, Karatzas I (2003) On the Optimal Stopping Problem for Onedimensional Diffusions. Stochastic Processes & Applications 107:173–212.
Fournie E, Lasry JM, Lebuchoux J, Lions P-L, Touzi N (1999) Some Applications of Malliavin Calculus to Monte Carlo Methods in Finance. Finance and Stochastics 3:391–412.
. Glasserman P (2004) Monte Carlo Methods in Financial Engineering. Springer Verlag.
El Karoui N, Karatzas I (1995) On the Optimal Stopping Problem Associated with an American Put-Option. IMA Lecture Notes in Mathematics & Applicatioins 65:35–46.
. Kusuoka S (2001) Approximation of Expectation of Diffusion Process and Mathematical Finance. In: Advanced Studies in Pure Mathematics 31:147–165. Taniguchi Conference on Mathematics Nara’98.
. Kusuoka S (2004) Approximation of Expectation of Diffusion Processes Based on Lie Algebra and Malliavin Calculus. In: Advances in Mathematical Economics 6:69–83. Springer-Verlag.
. Lyasoff A (2007) Dynamic Integration of Interpolating Functions and Some Concrete Optimal Stopping Problems. The Mathematica Journal 10(4) (to appear).
Longstaff FA, Schwartz RS (2001) Valuing American Options By Simulation: A simple Least-Square Approach. Review of Financial Studies 14:113–147.
Lyons T, Victoir N (2004) Cubature on Wiener Space. Proceedings of the Royal Society Lond. ser. A 460:169–198.
Peskir G, Shiryaev A (2006) Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics - ETH Zürich, Birkhäuser.
. Press WH, Teukolsky SA, Vetterling WT, Flannery B (1992) Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. Cambridge University Press.
Rogers LCG (2002) Monte Carlo Valuation of American Options. Mathematical Finance 12(3):271–286.
. Stroud AH (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lyasoff, A. (2008). Spline Cubatures for Expectations of Diffusion Processes and Optimal Stopping in Higher Dimensions (with Computational Finance in View). In: Sarychev, A., Shiryaev, A., Guerra, M., Grossinho, M.d.R. (eds) Mathematical Control Theory and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69532-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-69532-5_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69531-8
Online ISBN: 978-3-540-69532-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)