Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance (2007)
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| Venue: | JOURNAL OF COMPUTATIONAL FINANCE |
| Citations: | 30 - 12 self |
BibTeX
@ARTICLE{Forsyth07numericalmethods,
author = {P. A. Forsyth and G. Labahn},
title = {Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance},
journal = {JOURNAL OF COMPUTATIONAL FINANCE},
year = {2007}
}
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Abstract
Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton-Jacobi-Bellman (HJB) or Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB type equations, we can guarantee convergence of a Newton-type (Policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newton-type iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees.
Keyphrases
controlled hamilton-jacobibellman pdes numerical method stock borrowing fee relevant solution nonlinear discretized algebraic equation newton-type iteration cannot hjbi case specific interesting case unequal borrowing lending cost iteration scheme optimal control problem viscosity solution many nonlinear option pricing problem discretization method option pricing piecewise constant control approximation hjb type equation jump process general approach numerical example
