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A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization
Author(s):
Yves
Achdou;
Bruno
Franchi;
Nicoletta
Tchou.
Journal:
Math. Comp.
74
(2005),
1291-1322.
MSC (2000):
Primary 35K65, 65M15, 65M60, 65N30
Posted:
October 5, 2004
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Abstract:
This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which 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 analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.
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Additional Information:
Yves
Achdou
Affiliation:
UFR Mathématiques, Université Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France; and Laboratoire J.L. Lions, Université Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France
Email:
achdou@math.jussieu.fr
Bruno
Franchi
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy
Email:
bfranchi@dm.unibo.it
Nicoletta
Tchou
Affiliation:
IRMAR, Université de Rennes 1, Rennes, France
Email:
nicoletta.tchou@univ-rennes1.fr
DOI:
10.1090/S0025-5718-04-01714-4
PII:
S 0025-5718(04)01714-4
Keywords:
Finance,
degenerate parabolic operator,
finite elements
Received by editor(s):
April 16, 2003
Received by editor(s) in revised form:
March 3, 2004
Posted:
October 5, 2004
Additional Notes:
The second author was partially supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy, project ``Analysis in metric spaces''.
Copyright of article:
Copyright
2004,
American Mathematical Society
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