WIAS Preprint No. 1328, (2008)

Fixed domain transformations and split-step finite difference schemes for nonlinear Black--Scholes equations for American options



Authors

  • Ankudinova, Julia
  • Ehrhardt, Matthias

2010 Mathematics Subject Classification

  • 91B26 35A35 65N99

Keywords

  • nonlinear Black-Scholes models, fixed domain transformation, split-step methods, American options

Abstract

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection equations, where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself.
In this chapter we will be concerned with several models from the most relevant class of nonlinear Black-Scholes equations for American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives.
We will analytically approach the option price by following the ideas proposed by Ševčovič and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. Finally, we will present the results of a split-step finite difference schemes for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.

Appeared in

  • Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science, 2008, ISBN: 978-1-60456-931-5.

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