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Comparison of the analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations

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  • Karol Duris
  • Shih-Hau Tan
  • Choi-Hong Lai
  • Daniel Sevcovic

Abstract

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Different linearization techniques such as Newton's method and analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black-Scholes equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters.

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  • 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.
    • Handle: RePEc:arx:papers:1511.05661
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    References listed on IDEAS

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    1. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
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    6. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
    7. Pascal Heider, 2010. "Numerical Methods for Non-Linear Black-Scholes Equations," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(1), pages 59-81.
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