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)).

Reçu le :
Accepté le :
Publié le :
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 
@article{CRMATH_2003__336_3_263_0,
     author = {Avellaneda, Marco and Boyer-Olson, Dash and Busca, J\'er\^ome and Friz, Peter},
     title = {Application of large deviation methods to the pricing of index options in finance},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {263--266},
     publisher = {Elsevier},
     volume = {336},
     number = {3},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00032-3},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00032-3/}
}
TY  - JOUR
AU  - Avellaneda, Marco
AU  - Boyer-Olson, Dash
AU  - Busca, Jérôme
AU  - Friz, Peter
TI  - Application of large deviation methods to the pricing of index options in finance
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 263
EP  - 266
VL  - 336
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00032-3/
DO  - 10.1016/S1631-073X(03)00032-3
LA  - en
ID  - CRMATH_2003__336_3_263_0
ER  - 
%0 Journal Article
%A Avellaneda, Marco
%A Boyer-Olson, Dash
%A Busca, Jérôme
%A Friz, Peter
%T Application of large deviation methods to the pricing of index options in finance
%J Comptes Rendus. Mathématique
%D 2003
%P 263-266
%V 336
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00032-3/
%R 10.1016/S1631-073X(03)00032-3
%G en
%F CRMATH_2003__336_3_263_0
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://archive.numdam.org/articles/10.1016/S1631-073X(03)00032-3/

[1] Avellaneda, M.; Boyer-Olson, D.; Busca, J.; Friz, P. Reconstruction of volatility; Pricing index options using the steepest-descent approximation, RISK Magazine, Volume 15 (2002) no. 10

[2] Berestycki, H.; Busca, J.; Florent, I. An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris, Sér. I, Volume 331 (2000) no. 12, pp. 965-969

[3] Berestycki, H.; Busca, J.; Florent, I. Asymptotics and calibration of local volatility models, Quantitative Finance, Volume 2 (2002) no. 1, pp. 61-69

[4] Britten-Jones; Neuberger, M.A. Option prices, implied prices processes, and stochastic volatility, J. Finance, Volume 55 (2000) no. 2, pp. 839-866

[5] Derman, E.; Kani, I. Riding on a smile, RISK, Volume 7 (1994) no. 2

[6] Derman, E.; Kani, I. Kamal, Trading and hedging of local volatility, J. Financial Engrg., Volume 6 (1997) no. 3, pp. 233-270

[7] Dupire, B. Pricing with a smile, RISK, Volume 7 (1994) no. 1

[8] J. Gatheral, Stochastic volatility and local volatility, in: Lecture Notes for Case Studies in Financial Modeling, M.S. Program in Math Finance, N.Y.U., 2001, http://www.math.nyu.edu/financial_mathematics

[9] J. Lim, Pricing and hedging options on baskets of stocks, Ph.D. Thesis, NYU, 2002

[10] Rubinstein, M. Implied binomial trees, J. Finance, Volume 49 (1994) no. 3, pp. 771-819

[11] Varadhan, S.R.S. On the behaviour of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., Volume 20 (1967)

Cité par Sources :