Štěpán, Josef, and Dostál, Petr. "The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. II." Kybernetika 39.6 (2003): [681]-701. <http://eudml.org/doc/33674>.
@article{Štěpán2003,
abstract = {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 $\tilde\{\sigma \}(y)=\sigma _2I_\{(-\infty ,y_0]\}(y)+\sigma _1I_\{(y_0,\infty )\}(y).$},
author = {Štěpán, Josef, Dostál, Petr},
journal = {Kybernetika},
keywords = {stochastic differential equation; stochastic volatility; price of a general option; price of the European call option; Monte Carlo approximations; stochastic differential equation; stochastic volatility; price; European call option; Monte Carlo approximation},
language = {eng},
number = {6},
pages = {[681]-701},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. II},
url = {http://eudml.org/doc/33674},
volume = {39},
year = {2003},
}
TY - JOUR
AU - Štěpán, Josef
AU - Dostál, Petr
TI - The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. II
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 6
SP - [681]
EP - 701
AB - 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 $\tilde{\sigma }(y)=\sigma _2I_{(-\infty ,y_0]}(y)+\sigma _1I_{(y_0,\infty )}(y).$
LA - eng
KW - stochastic differential equation; stochastic volatility; price of a general option; price of the European call option; Monte Carlo approximations; stochastic differential equation; stochastic volatility; price; European call option; Monte Carlo approximation
UR - http://eudml.org/doc/33674
ER -
