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, risks from an unprotected portfolio, large investor’s preferences or illiquid markets (which may have an impact on the stock price), the volatility, the drift and the option price itself. In this paper we will focus on 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 due to transaction costs. We will analytically approach the option price by transforming the problem for a European Call option into a convection-diffusion equation with a nonlinear term and the free boundary problem for an American Call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following Ševčovič’s idea. Finally, we will present the results of different numerical discretization schemes for European options for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.