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The numerical solution of nonlinear Black–Scholes equations (2008)
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BibTeX
@MISC{Ankudinova08thenumerical,
author = {Julia Ankudinova},
title = {The numerical solution of nonlinear Black–Scholes equations},
year = {2008}
}
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Abstract
Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor’s preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. In this work we will be concerned with several models from the most relevant class of nonlinear Black–Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives, where the nonlinearity results from the presence of transaction costs. In the European case we will consider a European Call option and analytically approach the option price by transforming the problem into a forward convection-diffusion equation with a nonlinear term. In case of American options we will consider an American Call option and transform this free boundary problem into a fully nonlinear parabolic equation defined on a fixed domain following Sevčovič’s idea [72]. Finally, we will present the numerical results of different discretization schemes for European and American options for various volatility models including Leland’s model, Barles’ and Soner’s model and the Risk Adjusted
Keyphrases
nonlinear black scholes equation option price numerical solution fakult ii mathematik und naturwissenschaften american option technische universit berlin stock price transaction cost numerical result forward convection-diffusion equation leland model nonlinear term free boundary problem various volatility model illiquid market european call option large investor preference several model american call option risk adjusted relevant class accurate value nonlinearity result different factor fixed domain realistic assumption nonlinear parabolic equation unprotected portfolio different discretization scheme sev ovi idea european case
