| 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 |
| . |
