There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these models volatility is not constant anymore, but depends on some extra variables. It can be, for example, transaction costs, a risk from a portfolio, preferences of a large trader, etc. In this thesis we focus on these models. In the first chapter we introduce some important theory of financial derivatives. The second chapter is devoted to the volatility models. We derive three models concerning transaction costs (RAPM, Leland's  and Barles-Soner's model) and Frey's model which assumes a large (dominant) trader on the market. In the third and in the forth chapter we derive portfolio and make numerical experiments with a free boundary. We use the first order additive and the second order Strang splitting methods. We also use approximations of Barles-Soner's model using the identity function and introduce an approximation with the logarithm function of Barles-Soner's model. These models we finally compare with models where the volatility includes constant transaction costs.

<p>In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.</p>

In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.