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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  1. Y. Achdou and B. Franchi (in preparation).  
  2. H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Masson (1983).  
  3. T. Cazenave and A. Haraux, An introduction to semilinear evolution equations. The Clarendon Press Oxford University Press, New York (1998). Translated from the 1990 French original by Y. Martel and revised by the authors.  
  4. J. Douglas and T.F. Russell, Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element methods or finite difference method. SIAM J. Numer. Anal.19 (1982) 871-885.  
  5. J.-P. Fouque, G. Papanicolaou and K. Ronnie Sircar. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge (2000).  
  6. B. Franchi, R. Serapioni and F. Serra Cassano. Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math.22 (1996) 859-890.  
  7. B. Franchi and M.C. Tesi, A finite element approximation for a class of degenerate elliptic equations. Math. Comp.69 (2000) 41-63.  
  8. K.O. Friedrichs, The identity of weak and strong extensions of differential operators. Trans. Amer. Math. Soc.55 (1944) 132-151.  
  9. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. I and II. Dunod, Paris (1968).  
  10. A. Pazy, Semi-groups of linear operators and applications to partial differential equations. Appl. Math. Sci.. 44, Springer Verlag (1983).  
  11. O. Pironneau and F. Hecht, FREEFEM. www.ann.jussieu.fr  
  12. O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math.8 (2000) 25-35.  
  13. M.H. Protter and H.F. Weinberger, Maximum principles in differential equations. Springer-Verlag, New York (1984). Corrected reprint of the 1967 original.  
  14. E. Stein and J. Stein, Stock price distributions with stochastic volatility: an analytic approach. The review of financial studies4 (1991) 727-752.  
  15. H.A Van Der Vorst, Bi-cgstab: a fast and smoothly converging variant of bi-cg for the solution of nonlinear systems. SIAM J. Sci. Statist. Comput.13 (1992) 631-644.  
  16. P. Willmott, J. Dewynne and J. Howison, Option pricing: mathematical models and computations. Oxford financial press (1993).  

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