Partial Differential Equations - Modelling and ... - ResearchGate

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.