Processus de Lévy en Finance - Laboratoire de Probabilités et ...

2.5. REGULARIZING THE CALIBRATION PROBLEM 85
with characteristic triplet (A, ν, γ h ) with respect to the truncation function h, define
( ) ∫ 1 ∞
f(β, Q) := γ h +
2 + β A +
−∞
( ) ( 1
= γ h +
2 + β A +
∫ ∞
+
−∞
∫ ∞
−∞
{
(e x − 1)e β(ex −1) − h(x) −
{
}
(e x − 1)e β(ex−1) − h(x) ν(dx)
)
h 2 (x)ν(dx)
( ) }
1
2 + β h 2 (x) ν(dx). (2.25)
Since (e x − 1)e β(ex −1) − x − ( 1
2 + β) x 2 = o(|x| 2 ) and the integrand in the last term of (2.25) is
bounded on (−∞, B], by Proposition 1.7, for every β, lim n f(β, P n ) = f(β, P ).
The support of ν Pn
is bounded from above by B, and the dominated convergence theorem
allows to compute the derivative of f(β, P n ) by interchanging the derivative and the integral:
f ′ β (β, P n) = A Pn +
∫ ∞
−∞
(e x − 1) 2 e β(ex −1) ν Pn (dx) > 0.
Therefore, β n is the unique solution of f(β, P n ) = 0. Let β ∗ be the solution of f(β, P ) = 0. The
support of ν P is also bounded from above by B and f ′ β (β∗ , P ) > 0. This means that there exist
ε > 0 and finite constants β − < β ∗ and β + > β ∗ such that f(β − , P ) < −ε and f(β + , P ) > ε.
One can then find N such that for all n ≥ N, f(β − , P n ) < −ε/2 and f(β + , P n ) > ε/2, which
means that β n ∈ [β − , β + ] and the sequence {β n } is bounded.
To show that the sequence of relative entropies is bounded, observe that for |x| ≤ 1,
∣
∣e β(ex−1) − 1 − βx∣ ≤ βe β(e−1)+1 (1 + βe)|x| 2
and that for x ≤ B,
∣
∣e β(ex −1) − 1 − βx1 |x|≤1
∣ ∣∣ ≤ βe
β(e B +1) + 1 + βB.
The uniform boundedness of the sequence of relative entropies I(Q n |P n ) now follows from
Proposition 1.6 and Equation (2.15).
2.5 Regularizing the calibration problem
As observed in Section 2.2, problem (2.11) is ill-posed and hard to solve numerically. In particular,
its solutions, when they exist, may not be stable with respect to perturbations of market
data. If we do not know the prices C M exactly but only know the perturbed prices C δ M that