The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics. II

Josef Štěpán; Petr Dostál

Kybernetika (2003)

  • Volume: 39, Issue: 6, page [681]-701
  • ISSN: 0023-5954

Abstract

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This paper continues the research started in [J. Štěpán and P. Dostál: The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price X ( t ) born by the above semilinear SDE with σ ( x , t ) = σ ˜ ( 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 ℒ ( Y ( s ) , τ ( s ) ) for s ≥ 0 , where Y is the exponential of Wiener process and τ ( s ) = ∫ σ ˜ - 2 ( Y ( u ) ) d u . Both methods are compared for the European option and the special choice σ ˜ ( y ) = σ 2 I ( - ∞ , y 0 ] ( y ) + σ 1 I ( y 0 , ∞ ) ( y ) .

How to cite

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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 -

References

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  1. Billingsley P., Convergence of Probability Measures, Wiley, New York – Chichester – Weinheim 1999 Zbl0944.60003MR1700749
  2. Geman H., Madan D. B., Yor M., 10.1007/s780-002-8401-3, Finance and Stochastics 6 (2002), 63–90 Zbl1006.60026MR1885584DOI10.1007/s780-002-8401-3
  3. Kallenberg O., Foundations of Modern Probability, Springer–Verlag, New York – Berlin – Heidelberg 1997 Zbl0996.60001MR1464694
  4. Rogers L.C.G., Williams D., Diffusions, Markov Processes and Martingales, Volume 2: Itô Calculus. Cambridge University Press, Cambridge 2000 Zbl0977.60005MR1780932
  5. Štěpán J., Dostál P., The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics I, Kybernetika 39 (2003), 653–680 MR2035643

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