Denis Belomestny and Markus Reiβ
which is a simple consequence of the formulae (2) and (8). Note that the
function ψ is up to a shift in the argument the cumulant-generating function
of the Levy process and a continuous version of the logarithm must be taken
such that ψ(0) = 0, which is implied by the martingale condition.
Formula (11) shows that the Levy triplet is uniquely identifiable given
the observation of the whole option price function O without noise: Fμ(v)
tends to zero as |v | → ∞ due to the Riemann-Lebesgue Lemma such that
ψ is the sum of a quadratic polynomial and a function vanishing at infinity.
Then σ2 , γ, λ are identifiable as coefficients in the polynomial for arguments
tending to infinity. The function Fμ(v) is obtained as the difference between
ψ and the polynomial.
This identification procedure, however, is not stable such that the prob-
lem becomes ill-posed. Still, a properly refined application of this approach
combined with a spectral regularisation method will equip us with esti-
mators for the whole triplet T = (σ2 ,γ,μ) (we parametrize Levy triplets
equivalently with μ or ν).
The model (11) has a structure similar to the well-known partial linear
models, but in fact there is one substantial difference: the function Fμ is
not supposed to be smooth, but instead it is decaying for high frequencies
because we work in the spectral domain. This is also why we shall regularize
the problem by cutting off frequencies |v | higher than a certain threshold
level U , which depends on the noise level and the smoothness assumptions
on the unknown jump density. Let us present the basic estimation procedure.
Further details are specified in Section 6.1, while a more elaborate numerical
implementation is presented in (Belomestny and Reiβ 2006).
(a) We approximate the function O by building a function O ∈ L1, approx-
imating the true function O, based on the observations (Oj ). It suffices
to interpolate the data points (Oj ) linearly, but in simulations it turns
out that some smoothing procedure is preferable, cf. the discussion in
Section 3.2.
(b) For κ(v) ∈ (0, 1), specified later in (27), we calculate
ψ(v) := T logκ(v)(1 + iv(1+ iv)FO(v)´ , v ∈ R, (12)
where the trimmed log-function log>κ : C \ {0} → C is given by
log(z), |z| ≥ κ
g>κ log(κ z/|z|), |z| < κ
and log(• ) is taken in such a way that ψ(v) is continuous with ψ(0) = 0
(almost surely the argument of the logarithm in (12) does not vanish).
If we observe option prices for different maturities Tk, we perform the
steps (a) and (b) for each Tk separately and aggregate at this point the
different estimators for ψ to obtain one estimator with less variance, e.g.
by taking a weighted average. Similarly, estimators obtained on different
days can be aggregated at this stage.