Kybernetika 39 no. 6, 681-701, 2003

The dX(t)=Xb(X)dt+Xσ(X)dW equation and financial mathematics. II

Josef Štěpán and Petr Dostál


This paper continues the research started in [J. Štěpán and P. Dostál: 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 $\tilde\sigma(y)=\sigma_2I_{(-\infty,y_0]}(y)+\sigma_1I_{(y_0,\infty)}(y).$