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stochastic differential equation; stochastic volatility; price of a general option; price of the European call option; Monte Carlo approximations
This paper continues the research started in [J. Štěpán and P. Dostál: The ${\mathrm d}X(t) = Xb(X){\mathrm d}t + X\sigma (X) {\mathrm d}W$ 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 $s\ge 0,$ where $Y$ is the exponential of Wiener process and $\tau (s)=\int \tilde{\sigma }^{-2}(Y(u))\, {\mathrm d}u$. Both methods are compared for the European option and the special choice $\tilde{\sigma }(y)=\sigma _2I_{(-\infty ,y_0]}(y)+\sigma _1I_{(y_0,\infty )}(y).$
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