[Expansions asymptotiques pour équations paraboliques dégénérées]
On démontre des résultats de convergence asymptotique pour certaines expansions analytiques de solutions d'équations aux dérivés partielles dégénérées avec des applications aux mathématiques financières. En particulier, on combine des estimations d'erreur à temps petit, globales dans l'espace, obtenues précédemment dans le cas uniformément parabolique, avec quelques bornes a priori sur de « courts cylindres », et on attend la convergence asymptotique à temps petit de la solution approchée dans le cas parabolique dégénéré.
We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.
Accepté le :
Publié le :
@article{CRMATH_2014__352_12_1011_0,
author = {Pagliarani, Stefano and Pascucci, Andrea},
title = {Asymptotic expansions for degenerate parabolic equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {1011--1016},
publisher = {Elsevier},
volume = {352},
number = {12},
year = {2014},
doi = {10.1016/j.crma.2014.09.024},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.09.024/}
}
TY - JOUR AU - Pagliarani, Stefano AU - Pascucci, Andrea TI - Asymptotic expansions for degenerate parabolic equations JO - Comptes Rendus. Mathématique PY - 2014 SP - 1011 EP - 1016 VL - 352 IS - 12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.09.024/ DO - 10.1016/j.crma.2014.09.024 LA - en ID - CRMATH_2014__352_12_1011_0 ER -
%0 Journal Article %A Pagliarani, Stefano %A Pascucci, Andrea %T Asymptotic expansions for degenerate parabolic equations %J Comptes Rendus. Mathématique %D 2014 %P 1011-1016 %V 352 %N 12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.09.024/ %R 10.1016/j.crma.2014.09.024 %G en %F CRMATH_2014__352_12_1011_0
Pagliarani, Stefano; Pascucci, Andrea. Asymptotic expansions for degenerate parabolic equations. Comptes Rendus. Mathématique, Tome 352 (2014) no. 12, pp. 1011-1016. doi : 10.1016/j.crma.2014.09.024. http://www.numdam.org/articles/10.1016/j.crma.2014.09.024/
[1] Pricing options under stochastic volatility: a power series approach, Finance Stoch., Volume 13 (2009), pp. 269-303
[2] Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 890-896
[3] Expansion formulas for European options in a local volatility model, Int. J. Theor. Appl. Finance, Volume 13 (2010), pp. 603-634
[4] Time dependent Heston model, SIAM J. Financ. Math., Volume 1 (2010), pp. 289-325
[5] A jump to default extended CEV model: an application of Bessel processes, Finance Stoch., Volume 10 (2006), pp. 303-330
[6] Approximate solutions to second order parabolic equations. I: analytic estimates, J. Math. Phys., Volume 51 (2010), p. 103502, 26
[7] The small-time smile and term structure of implied volatility under the Heston model, SIAM J. Financ. Math., Volume 3 (2012), pp. 690-708
[8] Singular perturbations in option pricing, SIAM J. Appl. Math., Volume 63 (2003), pp. 1648-1665 (electronic)
[9] Equivalent Black volatilities, Appl. Math. Finance, Volume 6 (1999), pp. 147-159
[10] Managing smile risk, Wilmott (2002), pp. 84-108
[11] A General Asymptotic Implied Volatility for Stochastic Volatility Models, Frontiers in Quantitative Finance, Wiley, 2008
[12] A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., Volume 6 (1993), pp. 327-343
[13] M. Lorig, S. Pagliarani, A. Pascucci, Analytical expansions for parabolic equations, Preprint.
[14] A Taylor series approach to pricing and implied vol for LSV models, J. Risk (2014) (in press)
[15] Analytical approximation of the transition density in a local volatility model, Cent. Eur. J. Math., Volume 10 (2012) no. 1, pp. 250-270
[16] Adjoint expansions in local Lévy models, SIAM J. Financ. Math., Volume 4 (2013), pp. 265-296
[17] Estimates near the boundary for solutions of second order parabolic equations, Berlin (1998), pp. 637-647 (electronic)
[18] Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab., Volume 15 (1987) no. 1, pp. 1-39
[19] Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin–Watanabe, Probab. Theory Relat. Fields, Volume 92 (1992), pp. 275-311
Cité par Sources :
