Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
256 J.-M. Brun and B. Mohammadi [Cia78] Ph. Ciarlet. The finite element method for elliptic problems. North- Holland, 1978. [Cou89] J. Cousteix. Turbulence et couche limite. Cepadues publishers, 1989. [Fin00] J. Finnigan. Turbulence in plant canopies. Annu. Rev. Fluid Mech., 32:519–571, 2000. [Gri01] A. Griewank. Computational differentiation. Springer, New york, 2001. [HM97] F. Hecht and B. Mohammadi. Mesh adaptation by metric control for multi-scale phenomena and turbulence. AIAA paper 1997-0859, 1997. [IMSH06] B. Ivorra, D. E. Hertzog, B. Mohammadi, and J. G. Santiago. Semideterministic and genetic algorithms for global optimization of microfluidic protein-folding devices. Internat. J. Numer. Methods Engrg., 66(2):319–333, 2006. [Ivo06] B. Ivorra. Semi-deterministic global optimization. PhD thesis, University of Montpellier, 2006. [MP94] B. Mohammadi and O. Pironneau. Analysis of the k-epsilon turbulence model. Wiley, 1994. [MP01] B. Mohammadi and O. Pironneau. Applied shape optimization for fluids. Oxford University Press, 2001. [MP06] B. Mohammadi and G. Puigt. Wall functions in computational fluid dynamics. Comput. & Fluids, 40(3):2101–2124, 2006. [MS03] B. Mohammadi and J. H. Saiac. Pratique de la simulation numérique. Dunod, Paris, 2003. [RT81] M. R. Raupach and A. S. Thom. Turbulence in and above plant canopies. Annu. Rev. Fluid Mech., 13:97–129, 1981. [Sim97] J. Simpson. Gravity currents in the environment and laboratory. Cambridge University Press, 2nd edition, 1997. [Sum71] B. Sumner. A modeling study of several aspects of canopy flow. Monthly Weather Review, 99(6):485–493, 1971. [VP05] K. Veroy and A. Patera. Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: Rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Methods Fluids, 47(2):773–788, 2005.
Calibration of Lévy Processes with American Options Yves Achdou 1 UFR Mathématiques, Université Paris 7, Case 7012, FR-75251 PARIS Cedex 05, France and Laboratoire Jacques-Louis Lions, Université Paris 6, France achdou@math.jussieu.fr Summary. We study options on financial assets whose discounted prices are exponential of Lévy processes. The price of an American vanilla option as a function of the maturity and the strike satisfies a linear complementarity problem involving a non-local partial integro-differential operator. It leads to a variational inequality in a suitable weighted Sobolev space. Calibrating the Lévy process may be done by solving an inverse least square problem where the state variable satisfies the previously mentioned variational inequality. We first assume that the volatility is positive: after carefully studying the direct problem, we propose necessary optimality conditions for the least square inverse problem. We also consider the direct problem when the volatility is zero. 1 Introduction Black–Scholes’ model [BS73, Mer73] is a continuous time model involving a risky asset (the underlying asset) whose price at time τ is S τ and a risk-free asset whose price at time τ is S 0 τ = e rτ , r ≥ 0. It assumes that the price of the risky asset satisfies the following stochastic differential equation: dS τ = S τ (rdτ + σdW τ ), (1) where W τ is a standard Brownian motion on the probability space (Ω,A, P ∗ ) (the probability P ∗ is called the risk-neutral probability). An American vanilla call (resp. put) option on the risky asset is a contract giving its owner the right to buy (resp. sell) a share at a fixed price x at any time before a maturity date t. The price x is called the strike. Exercising the option yields a payoff P ◦ (S) =(S − x) + (resp. P ◦ (S) =(S − x) − ) for the call (resp. put) option, when the price of the underlying asset is S. 1 I wish to dedicate this work to O. Pironneau with all my friendship. I have been working with Olivier for almost fifteen years now, and for me, it has always been an exciting intellectual and human experience.
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256 J.-M. Brun <strong>and</strong> B. Mohammadi<br />
[Cia78] Ph. Ciarlet. The finite element method for elliptic problems. North-<br />
Holl<strong>and</strong>, 1978.<br />
[Cou89] J. Cousteix. Turbulence et couche limite. Cepadues publishers, 1989.<br />
[Fin00] J. Finnigan. Turbulence in plant canopies. Annu. Rev. Fluid Mech.,<br />
32:519–571, 2000.<br />
[Gri01] A. Griewank. Computational differentiation. Springer, New york, 2001.<br />
[HM97] F. Hecht <strong>and</strong> B. Mohammadi. Mesh adaptation by metric control for<br />
multi-scale phenomena <strong>and</strong> turbulence. AIAA paper 1997-0859, 1997.<br />
[IMSH06] B. Ivorra, D. E. Hertzog, B. Mohammadi, <strong>and</strong> J. G. Santiago. Semideterministic<br />
<strong>and</strong> genetic algorithms for global optimization of microfluidic<br />
protein-folding devices. Internat. J. Numer. Methods Engrg.,<br />
66(2):319–333, 2006.<br />
[Ivo06] B. Ivorra. Semi-deterministic global optimization. PhD thesis, University<br />
of Montpellier, 2006.<br />
[MP94] B. Mohammadi <strong>and</strong> O. Pironneau. Analysis of the k-epsilon turbulence<br />
model. Wiley, 1994.<br />
[MP01] B. Mohammadi <strong>and</strong> O. Pironneau. Applied shape optimization for fluids.<br />
Oxford University Press, 2001.<br />
[MP06] B. Mohammadi <strong>and</strong> G. Puigt. Wall functions in computational fluid<br />
dynamics. Comput. & Fluids, 40(3):2101–2124, 2006.<br />
[MS03] B. Mohammadi <strong>and</strong> J. H. Saiac. Pratique de la simulation numérique.<br />
Dunod, Paris, 2003.<br />
[RT81] M. R. Raupach <strong>and</strong> A. S. Thom. Turbulence in <strong>and</strong> above plant<br />
canopies. Annu. Rev. Fluid Mech., 13:97–129, 1981.<br />
[Sim97] J. Simpson. Gravity currents in the environment <strong>and</strong> laboratory. Cambridge<br />
University Press, 2nd edition, 1997.<br />
[Sum71] B. Sumner. A modeling study of several aspects of canopy flow. Monthly<br />
Weather Review, 99(6):485–493, 1971.<br />
[VP05] K. Veroy <strong>and</strong> A. Patera. Certified real-time solution of the parametrized<br />
steady incompressible Navier–Stokes equations: Rigorous reduced-basis<br />
a posteriori error bounds. Internat. J. Numer. Methods Fluids,<br />
47(2):773–788, 2005.