Geodesic
The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. II
Kybernetika, Tome 39 (2003) no. 6, pp. 681-701

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR   Zbl

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 $\tilde{\sigma }(y)=\sigma _2I_{(-\infty ,y_0]}(y)+\sigma _1I_{(y_0,\infty )}(y).$
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 $\tilde{\sigma }(y)=\sigma _2I_{(-\infty ,y_0]}(y)+\sigma _1I_{(y_0,\infty )}(y).$
Classification : 60H10, 65C30, 91B28, 91G80
Keywords: stochastic differential equation; stochastic volatility; price of a general option; price of the European call option; Monte Carlo approximations 
Štěpán, Josef; Dostál, Petr. The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. II. Kybernetika, Tome 39 (2003) no. 6, pp. 681-701. http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a1/
@article{KYB_2003_39_6_a1,
     author = {\v{S}t\v{e}p\'an, Josef and Dost\'al, Petr},
     title = {The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. {II}},
     journal = {Kybernetika},
     pages = {681--701},
     year = {2003},
     volume = {39},
     number = {6},
     mrnumber = {2035644},
     zbl = {1249.60128},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a1/}
}
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
SP  - 681
EP  - 701
VL  - 39
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a1/
LA  - en
ID  - KYB_2003_39_6_a1
ER  - 
%0 Journal Article
%A Štěpán, Josef
%A Dostál, Petr
%T The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. II
%J Kybernetika
%D 2003
%P 681-701
%V 39
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a1/
%G en
%F KYB_2003_39_6_a1

[1] Billingsley P.: Convergence of Probability Measures. Wiley, New York – Chichester – Weinheim 1999 | MR | Zbl

[2] Geman H., Madan D. B., Yor M.: Stochastic volatility, jumps and hidden time changes. Finance and Stochastics 6 (2002), 63–90 | DOI | MR | Zbl

[3] Kallenberg O.: Foundations of Modern Probability. Springer–Verlag, New York – Berlin – Heidelberg 1997 | MR | Zbl

[4] Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales. Volume 2: Itô Calculus. Cambridge University Press, Cambridge 2000 | MR | Zbl

[5] Štěpán J., Dostál P.: The ${\mathrm d}X(t)=Xb(X){\mathrm d}t+X\sigma (X)\,{\mathrm d}W$ equation and financial mathematics I. Kybernetika 39 (2003), 653–680 | MR