Partial Differential Equations - Modelling and ... - ResearchGate
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276 Y. Achdou ∂u ∂t ∈ L2 ((0,T)×R + ), such that u(t =0)=u ◦ and for all v ∈ K with bounded support in x, 〈 〉 ∂u ∂t + Au + rx, v − u ≥ 0, for a.a. t>0. (57) There exists ˇX >0 such that u(t, x) =0, ∀t ∈ [0,T], x ≥ ˇX, (58) The function u is non-increasing with respect to x and non-decreasing with respect to t and there exists a non-decreasing continuous function γ :(0,T] → (S, ˇX), such that for all t ∈ (0,T), {x >0 s.t. u(t, x) =u ◦ (x)} =[γ(t), +∞). References [Ach05] Y. Achdou. An inverse problem for a parabolic variational inequality arising in volatility calibration with American options. SIAM J. Control Optim., 43(5):1583–1615 (electronic), 2005. [Ach06] Y. Achdou. An inverse problem for a parabolic variational inequality with an integro-differential operator arising in the calibration of Lévy processes with American options. Submitted, 2006. [Ada75] R. A. Adams. Sobolev spaces, volume 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich Publishers], New York, 1975. [AP05a] Y. Achdou and O. Pironneau. Computational methods for option pricing, volume 30 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. [AP05b] Y. Achdou and O. Pironneau. Numerical procedure for calibration of volatility with American options. Appl. Math. Finance, 12(3):201–241, 2005. [BL84] A. Bensoussan and J.-L. Lions. Impulse control and quasivariational inequalities. µ. Gauthier-Villars, Montrouge, 1984. Translated from the French by J. M. Cole. [BS73] F. Black and M. S. Scholes. The pricing of options and corporate liabilities,. Journal of Political Economy,, 81:637–654, 1973. [CT04] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. [CV03] R. Cont and E. Voltchkova. Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. Rapport Interne 513, CMAP, Ecole Polytechnique, 2003. [CV04] R. Cont and E. Voltchkova. Integro-differential equations for option prices in exponential Lévy models. Rapport Interne 547, CMAP, Ecole Polytechnique, 2004. [Dup97] B. Dupire. Pricing and hedging with smiles. In Mathematics of derivative securities (Cambridge, 1995), pages 103–111. Cambridge Univ. Press, Cambridge, 1997.
Calibration of Lévy Processes with American Options 277 [Hin01] [IK00] [JLL90] [Lio69] [Lio73] [LL97] [Mer73] [MNS03] [MP84] [MSW04] [MvPS04] [Pha98] [Pir84] M. Hintermüller. Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization. M2AN Math. Model. Numer. Anal., 35(1):129–152, 2001. K. Ito and K. Kunisch. Optimal control of elliptic variational inequalities. Appl. Math. Optim., 41(3):343–364, 2000. P. Jaillet, D. Lamberton, and B. Lapeyre. Variational inequalities and the pricing of American options. Acta Appl. Math., 21(3):263–289, 1990. J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969. J.-L. Lions. Perturbations singulières dans les problèmes aux limites et en contrôle optimal, volume 323 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, 1973. D. Lamberton and B. Lapeyre. Introduction au calcul stochastique appliqué à la finance. Ellipses, 1997. R. C. Merton. Theory of rational option pricing. Bell J. Econom. and Management Sci., 4:141–183, 1973. A.-M. Matache, P.-A. Nitsche, and C. Schwab. Wavelet Galerkin pricing of American options on Lévy driven assets. 2003. Research Report SAM 2003-06. F. Mignot and J.-P. Puel. Contrôle optimal d’un système gouverné par une inéquation variationnelle parabolique. C. R. Acad. Sci. Paris Sér. I Math., 298(12):277–280, 1984. A.-M. Matache, C. Schwab, and T. P. Wihler. Fast numerical solution of parabolic integro-differential equations with applications in finance. Technical report, IMA University of Minnesota, 2004. Reseach report No. 1954. A.-M. Matache, T. von Petersdoff, and C. Schwab. Fast deterministic pricing of Lévy driven assets. Mathematical Modelling and Numerical Analysis, 38(1):37–72, 2004. H. Pham. Optimal stopping of controlled jump-diffusion processes: A viscosity solution approach. Journal of Mathematical Systems, 8(1):1– 27, 1998. O. Pironneau. Optimal shape design for elliptic systems. Springer Series in Computational Physics. Springer-Verlag, New York, 1984.
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276 Y. Achdou<br />
∂u<br />
∂t ∈ L2 ((0,T)×R + ), such that u(t =0)=u ◦ <strong>and</strong> for all v ∈ K with bounded<br />
support in x,<br />
〈 〉<br />
∂u<br />
∂t + Au + rx, v − u ≥ 0, for a.a. t>0. (57)<br />
There exists ˇX >0 such that<br />
u(t, x) =0, ∀t ∈ [0,T], x ≥ ˇX, (58)<br />
The function u is non-increasing with respect to x <strong>and</strong> non-decreasing with<br />
respect to t <strong>and</strong> there exists a non-decreasing continuous function γ :(0,T] →<br />
(S, ˇX), such that for all t ∈ (0,T), {x >0 s.t. u(t, x) =u ◦ (x)} =[γ(t), +∞).<br />
References<br />
[Ach05] Y. Achdou. An inverse problem for a parabolic variational inequality<br />
arising in volatility calibration with American options. SIAM J. Control<br />
Optim., 43(5):1583–1615 (electronic), 2005.<br />
[Ach06] Y. Achdou. An inverse problem for a parabolic variational inequality<br />
with an integro-differential operator arising in the calibration of Lévy<br />
processes with American options. Submitted, 2006.<br />
[Ada75] R. A. Adams. Sobolev spaces, volume 65 of Pure <strong>and</strong> Applied Mathematics.<br />
Academic Press [A subsidiary of Harcourt Brace Jovanovich<br />
Publishers], New York, 1975.<br />
[AP05a] Y. Achdou <strong>and</strong> O. Pironneau. Computational methods for option pricing,<br />
volume 30 of Frontiers in Applied Mathematics. Society for Industrial<br />
<strong>and</strong> Applied Mathematics (SIAM), Philadelphia, PA, 2005.<br />
[AP05b]<br />
Y. Achdou <strong>and</strong> O. Pironneau. Numerical procedure for calibration of<br />
volatility with American options. Appl. Math. Finance, 12(3):201–241,<br />
2005.<br />
[BL84] A. Bensoussan <strong>and</strong> J.-L. Lions. Impulse control <strong>and</strong> quasivariational<br />
inequalities. µ. Gauthier-Villars, Montrouge, 1984. Translated from the<br />
French by J. M. Cole.<br />
[BS73]<br />
F. Black <strong>and</strong> M. S. Scholes. The pricing of options <strong>and</strong> corporate liabilities,.<br />
Journal of Political Economy,, 81:637–654, 1973.<br />
[CT04] R. Cont <strong>and</strong> P. Tankov. Financial modelling with jump processes.<br />
Chapman & Hall/CRC Financial Mathematics Series. Chapman &<br />
Hall/CRC, Boca Raton, FL, 2004.<br />
[CV03]<br />
R. Cont <strong>and</strong> E. Voltchkova. Finite difference methods for option pricing<br />
in jump-diffusion <strong>and</strong> exponential Lévy models. Rapport Interne 513,<br />
CMAP, Ecole Polytechnique, 2003.<br />
[CV04] R. Cont <strong>and</strong> E. Voltchkova. Integro-differential equations for option<br />
prices in exponential Lévy models. Rapport Interne 547, CMAP, Ecole<br />
Polytechnique, 2004.<br />
[Dup97]<br />
B. Dupire. Pricing <strong>and</strong> hedging with smiles. In Mathematics of derivative<br />
securities (Cambridge, 1995), pages 103–111. Cambridge Univ. Press,<br />
Cambridge, 1997.
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