# Article

 Title: The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. II (English) Author: Štěpán, Josef Author: Dostál, Petr Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 39 Issue: 6 Year: 2003 Pages: [681]-701 Summary lang: English . Category: math . Summary: 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 $\tilde{\sigma }(y)=\sigma _2I_{(-\infty ,y_0]}(y)+\sigma _1I_{(y_0,\infty )}(y).$ (English) Keyword: stochastic differential equation Keyword: stochastic volatility Keyword: price of a general option Keyword: price of the European call option Keyword: Monte Carlo approximations MSC: 60H10 MSC: 65C30 MSC: 91B28 MSC: 91G80 idZBL: Zbl 1249.60128 idMR: MR2035644 . Date available: 2009-09-24T19:57:58Z Last updated: 2015-03-24 Stable URL: http://hdl.handle.net/10338.dmlcz/135565 . Related article: http://dml.cz/handle/10338.dmlcz/135564 . Reference: [1] Billingsley P.: Convergence of Probability Measures.Wiley, New York – Chichester – Weinheim 1999 Zbl 0944.60003, MR 1700749 Reference: [2] Geman H., Madan D. B., Yor M.: Stochastic volatility, jumps and hidden time changes.Finance and Stochastics 6 (2002), 63–90 Zbl 1006.60026, MR 1885584, 10.1007/s780-002-8401-3 Reference: [3] Kallenberg O.: Foundations of Modern Probability.Springer–Verlag, New York – Berlin – Heidelberg 1997 Zbl 0996.60001, MR 1464694 Reference: [4] Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales.Volume 2: Itô Calculus. Cambridge University Press, Cambridge 2000 Zbl 0977.60005, MR 1780932 Reference: [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 2035643 .

## Files

Files Size Format View
Kybernetika_39-2003-6_2.pdf 2.982Mb application/pdf View/Open

Partner of