Variational Analysis for the Black and Scholes Equation with Stochastic Volatility

Yves Achdou; Nicoletta Tchou

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 3, page 373-395
  • ISSN: 0764-583X

Abstract

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We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.

How to cite

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Achdou, Yves, and Tchou, Nicoletta. "Variational Analysis for the Black and Scholes Equation with Stochastic Volatility." ESAIM: Mathematical Modelling and Numerical Analysis 36.3 (2010): 373-395. <http://eudml.org/doc/194109>.

@article{Achdou2010,
abstract = { We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods. },
author = {Achdou, Yves, Tchou, Nicoletta},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Degenerate parabolic equations; european options; weighted Sobolev spaces; finite element and finite difference method.; degenerate parabolic equations; mean reverting Orstein-Uhlenbeck process; finite element method; finite difference method},
language = {eng},
month = {3},
number = {3},
pages = {373-395},
publisher = {EDP Sciences},
title = {Variational Analysis for the Black and Scholes Equation with Stochastic Volatility},
url = {http://eudml.org/doc/194109},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Achdou, Yves
AU - Tchou, Nicoletta
TI - Variational Analysis for the Black and Scholes Equation with Stochastic Volatility
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 3
SP - 373
EP - 395
AB - We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.
LA - eng
KW - Degenerate parabolic equations; european options; weighted Sobolev spaces; finite element and finite difference method.; degenerate parabolic equations; mean reverting Orstein-Uhlenbeck process; finite element method; finite difference method
UR - http://eudml.org/doc/194109
ER -

References

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