Kybernetika

This paper continues the research started in [J. \v{S}t\v{e}p\'{a}n and P. Dost\'{a}l: The ${\rm d}X(t) = Xb(X){\rm d}t + X\sigma(X) {\rm d}W$ equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price $X(t)$ born by the above semilinear SDE with $\sigma(x,t)=\tilde\sigma(x(t)),$ we suggest two methods how to compute the price of a~general option $g(X(T))$. The first, a~more universal one, is based on a~Monte Carlo procedure while the second one provides explicit formulas. We in this case need an~information on the two dimensional distributions of ${\mathcal L}(Y(s), \tau(s))$ for $s\geq 0,$ where $Y$ is the exponential of Wiener process and $\tau(s)=\int\tilde\sigma^{-2}(Y(u))\, {\rm d}u$. Both methods are compared for the European option and the special choice $\tilde\sigma(y)=\sigma_2I_{(-\infty,y_0]}(y)+\sigma_1I_{(y_0,\infty)}(y).$