Matthias Ehrhardt (WIAS Berlin)

"On the numerical solution of nonlinear Black-Scholes equations"

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 Sevcovic's idea.

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
pricing methodology.


References:

M. Ehrhardt (ed.),
"Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing",
Nova Science Publishers, Inc.,Hauppauge, 2008.

J. Ankudinova and M. Ehrhardt,
"The numerical solution of nonlinear Black-Scholes equations",
Comput. Math. Appl., Vol. 56, Number 3, (2008), 799-812.

M. Ehrhardt and R.E. Mickens,
"A fast, stable and accurate numerical method for the Black-Scholes equation of American options",
Int. J. Theoret. Appl. Finance, Vol. 11, Issue 5, (2008), 471-501.

D. Sevcovic,
"An iterative algorithm for evaluating approximations to the
optimal exercise boundary for a nonlinear Black-Scholes equation",
Canad. Appl. Math. Quarterly 15 (2007), 77-79.