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Numerical Methods for Non-Linear Black-Scholes Equations

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  • Pascal Heider

Abstract

In recent years non-linear Black-Scholes models have been used to build transaction costs, market liquidity or volatility uncertainty into the classical Black-Scholes concept. In this article we discuss the applicability of implicit numerical schemes for the valuation of contingent claims in these models. It is possible to derive sufficient conditions, which are required to ensure the convergence of the backward differentiation formula (BDF) and Crank-Nicolson scheme (CN) scheme to the unique viscosity solution. These stability conditions can be checked a priori and convergent schemes can be constructed for a large class of non-linear models and payoff profiles. However, if these conditions are not satisfied we show that the schemes are not convergent or produce spurious solutions. We study the practical implications of the derived stability criterions on relevant numerical examples.

Suggested Citation

  • Pascal Heider, 2010. "Numerical Methods for Non-Linear Black-Scholes Equations," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(1), pages 59-81.
    • Handle: RePEc:taf:apmtfi:v:17:y:2010:i:1:p:59-81
    DOI: 10.1080/13504860903075670
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    Cited by:

    1. Karol Duris & Shih-Hau Tan & Choi-Hong Lai & Daniel Sevcovic, 2015. "Comparison of the analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations," Papers 1511.05661, arXiv.org, revised Nov 2015.
    2. Lesmana, Donny Citra & Wang, Song, 2015. "Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 318-330.

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