Comptes Rendus
no. 8
Probability Theory
Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model
[Comportement asymptotique de la distribution du prix de l'action dans les modèles à volatilité stochastique : le modèle de Hull–White]

Présenté par : Jean-Pierre Kahane

Archil Gulisashvili1 ; Elias M. Stein2
1 Department of Mathematics, Ohio University, Athens, OH 45701, USA
2 Department of Mathematics, Princeton University, Princeton, NJ 08540, USA
Comptes Rendus. Mathématique, Volume 343 (2006) no. 8, pp. 519-523.

La présente Note étudie le comportement asymptotique de la densité de distribution du processus du prix de l'action dans le modèle de Hull–White. On determine la partie principale dans le développement asymptotique en zéro et en l'infini pour une telle densité et on estime l'erreur correspondante. Des problèmes similaires se résolvent pour les moyennes temporelles du processus de volatilité qui sont aussi intéressants dans l'étude des options asiatiques.

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.09.029

Archil Gulisashvili 1 ; Elias M. Stein 2

1 Department of Mathematics, Ohio University, Athens, OH 45701, USA
2 Department of Mathematics, Princeton University, Princeton, NJ 08540, USA 
@article{CRMATH_2006__343_8_519_0,
     author = {Archil Gulisashvili and Elias M. Stein},
     title = {Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the {Hull{\textendash}White} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {519--523},
     publisher = {Elsevier},
     volume = {343},
     number = {8},
     year = {2006},
     doi = {10.1016/j.crma.2006.09.029},
     language = {en},
}
TY  - JOUR
AU  - Archil Gulisashvili
AU  - Elias M. Stein
TI  - Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 519
EP  - 523
VL  - 343
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2006.09.029
LA  - en
ID  - CRMATH_2006__343_8_519_0
ER  - 
%0 Journal Article
%A Archil Gulisashvili
%A Elias M. Stein
%T Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model
%J Comptes Rendus. Mathématique
%D 2006
%P 519-523
%V 343
%N 8
%I Elsevier
%R 10.1016/j.crma.2006.09.029
%G en
%F CRMATH_2006__343_8_519_0
Archil Gulisashvili; Elias M. Stein. Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model. Comptes Rendus. Mathématique, Volume 343 (2006) no. 8, pp. 519-523. doi : 10.1016/j.crma.2006.09.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.029/

[1] L. Alili; J.C. Gruet An explanation of a generalized Boujerol's identity in terms of hyperbolic geometry (M. Yor, ed.), Exponential Functionals and Principal Values Related to Brownian Motion, Biblioteca de la Revista Matemática Ibero-Americana, Revista Matemática Iberoamericana, Madrid, 1997, pp. 15-33

[2] D. Dufresne The integral of geometric Brownian motion, Adv. Appl. Probab., Volume 33 (2001), pp. 223-241

[3] J.-P. Fouque; G. Papanicolaou; K.R. Sircar Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000

[4] J. Hull; A. White The pricing of options on assets with stochastic volatilities, J. Finance, Volume 42 (1987), pp. 281-300

[5] H. Matsumoto; M. Yor On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts, Adv. Appl. Probab., Volume 35 (2003), pp. 184-206

[6] M. Schröder On the integral of geometric Brownian motion, Adv. Appl. Probab., Volume 35 (2003), pp. 159-183

[7] E.M. Stein; J. Stein Stock price distributions with stochastic volatility: An analytic approach, Rev. Financial Stud., Volume 4 (1991), pp. 727-752

[8] M. Yor On some exponential functionals of Brownian motion, Adv. Appl. Probab., Volume 24 (1992), pp. 509-531

[9] M. Yor Exponential Functionals of Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2001

Cité par Sources :

Commentaires - Politique