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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:
Related article:
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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


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