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Calibration of Lévy Processes with American Options 271
(T i ,x i ); we call ū i =¯p i −x i +S, i ∈ I. The parameters of the Lévy process, i.e.
the volatility σ, the exponent α and the function ψ will be found as solutions
of a least square problem, where the functional to be minimized is the sum of
a suitable Tychonoff regularization functional J R (σ, α, ψ) andof
J(u) = ∑ ω i (u(T i ,x i ) − ū i ) 2 ,
i∈I
where ω i are positive weights, and u = u(σ, α, ψ) is a solution of (VIP), with
T =max i∈I T i .
We aim at finding some necessary optimality conditions satisfied by the
solutions of the least square problem. The main difficulty comes from the fact
that the derivability of the functional J(u) with respect to the parameter
(σ, α, ψ) is not guaranteed. To obtain some necessary optimality conditions,
we shall consider first a least square problem where u is the solution of the
penalized problem (37) rather than (VIP), obtain necessary optimality conditions
for this new problem, then have the penalty parameter ε tend to 0 and
pass to the limit in the optimality conditions. Such a program has already been
applied in [Ach05] for calibrating the local volatility with American options,
see also [AP05b, AP05a] for a related numerical method and results. The idea
originally comes from Hintermüller [Hin01] and Ito and Künisch [IK00], who
applied a similar program for elliptic variational inequalities. At this point, we
should also mention Mignot and Puel [MP84] who applied an elegant method
for finding optimality conditions for a special control problem for a parabolic
variational inequality.
4.2 Preliminary Technical Results
With the aim of finding optimality conditions for the least square problem
(not completely defined yet), we first state some results concerning the adjoint
of B.
Under the assumptions of Proposition 2, it can be checked that the operator
B T defined by
∫ (
B T u(x) = k(z) x(e z − 1) ∂u
)
∂x (x) − e2z u(xe z )+(2e z − 1)u(x) dz (40)
z∈R
is a continuous operator
⎧
⎪⎨ from V s to V s−2α , if α> 1 2 ,
from V
⎪⎩
s to V s−1 , if α< 1 2 ,
from V s to V s−1−ε , for any ε>0, if α = 1 2 .
If α> 1 2 , then for all u, v ∈ V α , 〈B T u, v〉 = 〈Bv,u〉. This identity holds for
all u, v ∈ V s with s> 1 2 if α ≤ 1 2 .