Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

(c) With an estimate ψ of ψ at hand, we obtain estimators for the para-
metric part (σ² , γ, λ) by an averaging procedure taking into account
the polynomial structure in (11). Upon fixing the spectral cut-off value
U > 0, we set

where the weight functions w_σ^U , w_γ^U and w_λ^U satisfy

U₂                U                 U

- —u- wσU ( u )d u =1,     uwY^ ( u )d u =1,     wU ( u )d u = 1;

U               -U            -U

^U

w_σ^U (u) du =

—U

The estimate of the coefficients can be understood as an orthogonal
projection estimate with respect to a weighted L²-scalar product.

(d) Finally, we define the estimator for μ as the inverse Fourier transform
of the remainder:

^μ(x) := F ¹ h(^ψ(• ) + ^σ2-⁽• ^— i)^{2 —} i^γ(• ^— i) + ʌ´ 1[—u,u]⁽• )i ^(χ). ⁽¹⁹⁾

Then the identity Fμ(0) = — ɪ — γ + λ shows that the estimated triplet
still satisfies the martingale condition (4).

3.2. Discussion of the method

First note that the computational complexity of the estimation procedure
is very low. Step (a) is a standard interpolation or regression estimation
procedure, which is well established and fast. The only time consuming
steps are the three integrations in step (c) and the (fast) Fourier transforms
in steps (a) and (d).

In step (a) a reasonable approximation of FO based on discrete data
must be found. Asymptotically, it suffices to use simple linear interpola-
tion because all regularisation takes place later in the spectral domain by
damping high frequencies. Depending on the observation design (xj) and
the noise levels (δj ), it may nevertheless pay off to invest more in obtaining
a good approximating function O. When the distance ∆j = xj — xj_-1 be-
tween the transformed prices is rather large compared to the noise level δj ,
the numerical approximation error prevails and higher order interpolation
schemes might significantly reduce the total error ∣O — O∣ if O is smooth.
In Proposition 2 below, we only take advantage of the fact that O is almost

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