Discrete Artificial Boundary Conditions for the Black-Scholes Equation of American Options
Please always quote using this URN:urn:nbn:de:0296-matheon-3200
- In this work we derive an exact discrete artificial boundary condition for the Crank-Nicolson scheme for solving the Black-Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the artificial boundary condition on a purely discrete level. Since the exact discrete artificial boundary condition includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate artificial boundary conditions with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally we illustrate the efficiency of the proposed method on several examples and compare it to previously obtained discretized artificial boundary conditions.
Author: | Matthias Ehrhardt, Ronald E. Mickens |
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URN: | urn:nbn:de:0296-matheon-3200 |
Referee: | Volker Mehrmann |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2006/10/02 |
Release Date: | 2006/01/31 |
Tag: | |
Institute: | Technische Universität Berlin |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Axx General topics / 35A35 Theoretical approximation to solutions (For numerical analysis, see 65Mxx, 65Nxx) |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N99 None of the above, but in this section | |
Preprint Number: | 310 |