# American Option Prices as Unique Viscosity Solutions to Degenerated Hamilton-Jacobi-Bellman Equations

1 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : In this paper we show that the American price of standard (bounded) options in the Black-Scholes one-dimensional model, which is classically given by the value function of an optimal stopping problem, is also the value function of a degenerate stochastic control problem. As a byproduct we get that the American price $u^{*}$ is the unique bounded and continuous viscosity solution of the fully non-linear parabolic equation $-\frac{% \partial u^{*}}{\partial t}\left( t,x\right) =\left( Au^{*}\right) ^{+}\left( t,x\right),t~0, u^{*}\left( T,x\right) =\varphi \left( x\right)$ where $A$ is the infinitesimal generator of the Black-Scholes model, $T$ the maturity and $\varphi$ the payoff of the option.
Keywords :
Type de document :
Rapport
[Research Report] RR-3934, INRIA. 2000
Domaine :

https://hal.inria.fr/inria-00072718
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 10:41:51
Dernière modification le : mercredi 29 novembre 2017 - 15:52:40
Document(s) archivé(s) le : jeudi 24 mars 2011 - 12:11:15

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• HAL Id : inria-00072718, version 1

### Citation

Claude Martini. American Option Prices as Unique Viscosity Solutions to Degenerated Hamilton-Jacobi-Bellman Equations. [Research Report] RR-3934, INRIA. 2000. 〈inria-00072718〉

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