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Calibration of Lévy Processes with American Options 261
The fact that the discounted price is a martingale is equivalent to E ∗ (e Xτ )=1,
i.e.
∫
e z ν(dz) < ∞ and β = − σ2
|z|>1
2
∫R
− (e z − 1 − z1 |z|≤1 )ν(dz).
We will also assume that ∫ |z|>1 e2z ν(dz) < ∞, so the discounted price is a
square integrable martingale.
We note B the integral operator:
∫ (
(Bv)(S) = v(Se z ) − v(S) − S(e z − 1) ∂ )
∂S v(S) ν(dz).
R
Consider an American option with payoff P ◦ and maturity t: in [BL84],
Bensoussan and Lions assumed σ>0 and studied the variational inequality
stemming from the complementarity problem P (t, S) =P ◦ (S), and for
τ0,
∂P
S 2 ∂ 2 P
∂P
∂τ (τ,S)+σ2 (τ,S)+rS
2 ∂S2 ∂S (τ,S) − rP(τ,S)+(BP)(τ,S) ≤ 0, (6)
P (τ,S) ≥ P ◦ (S), (7)
and
⎛
⎝ ∂P
⎞
S 2 ∂ 2 P
∂P
∂τ (τ,S)+σ2 (τ,S)+rS
2 ∂S2 ∂S (τ,S) ⎠ (P (τ,S) − P ◦ (S)) = 0, (8)
−rP(τ,S)+(BP)(τ,S)
in suitable Sobolev spaces with decaying weights near +∞ and 0. They
proved that the price of the American option is P τ = P (τ,S τ ). Other approaches
with viscosity solutions are possible, see [Pha98], especially in the
case σ = 0. One advantage of the variational methods is that they provide
stability estimates. For numerical methods for options on Lévy driven assets,
see [MvPS04, MSW04, MNS03, AP05a, CV04, CV03].
In what follows, we assume that the Lévy measure has a density, ν(dz) =
k(z)dz. The main goal of the present work is to study a least-square method
for calibrating the volatility σ and the jump density k in order to recover the
prices of a family of American options available on the market.
We shall focus on a family of vanilla put options indexed by i ∈ I, with
maturities t i and strikes x i . One observes S ◦ the price of the risky asset and
the prices ( ¯P i ) i∈I of the above-mentioned family of options. We call T the
maximal maturity: T =max i∈I t i .
The first idea is to try to minimize the functional (σ, k) ↦→ ∑ i∈I ω i| ¯P i −
P i (0,S ◦ )| 2 + J R (σ, k) fork and σ in a suitable set,where