A finite volume approach for contingent claims valuation

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Abstract
This paper presents a finite volume approach for solving two‐dimensional contingent claims valuation problems. The contingent claims PDEs are in non‐divergence form. The finite volume method is more flexible than finite difference schemes which are often described in the finance literature and frequently used in practice. Moreover, the finite volume method naturally handles cases where the underlying partial differential equation becomes convection dominated or degenerate. A compact method is developed which uses a high‐order flux limiter for the convection terms. This paper will demonstrate how a variety of two‐dimensional valuation problems can all be solved using the same approach. The generality of the approach is in part due to the fact that changes caused by different model specifications are localized. Constraints on the solution are treated in a uniform manner using a penalty method. A variety of illustrative example computations are presented.

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