Probability Theory
Application of large deviation methods to the pricing of index options in finance
[Méthodes de grandes déviations et pricing d'options sur indice]

Présenté par : Paul Malliavin

Avellaneda, Marco1 ; Boyer-Olson, Dash1 ; Busca, Jérôme2 ; Friz, Peter1
1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA
2 CNRS, Ceremade, Université Paris Dauphine, pl. du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 263-266.

Nous montrons une formule asymptotique donnant la volatilité implicite d'une option sur indice à partir des volatilités des actifs sous-jacents. La démonstration repose sur les estimations de densités de diffusion en temps petit du type grandes déviation de Varadhan (Comm. Pure Appl. Math. 20 (1967)). On pourra trouver une version détaillée de ces résultats dans l'article (RISK 15 (10) (2002)).

We develop an asymptotic formula for calculating the implied volatility of European index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for evaluating the multivariate probability distribution function for stock prices, which is based on large-deviation estimates of diffusion processes densities by Varadhan (Comm. Pure Appl. Math. 20 (1967)). A detailed version of these results can be found in (RISK 15 (10) (2002)).

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00032-3
Avellaneda, Marco 1 ; Boyer-Olson, Dash 1 ; Busca, Jérôme 2 ; Friz, Peter 1

1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA
2 CNRS, Ceremade, Université Paris Dauphine, pl. du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France 
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Avellaneda, Marco; Boyer-Olson, Dash; Busca, Jérôme; Friz, Peter. Application of large deviation methods to the pricing of index options in finance. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 263-266. doi : 10.1016/S1631-073X(03)00032-3. http://www.numdam.org/articles/10.1016/S1631-073X(03)00032-3/

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