Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

(d) At any x ∈ R \{0}, respectively x ∈ R \{0, γT} in the case σ = 0, the
function O is twice differentiable with kO⁰⁰ k_L1(_R) 6 3. The first deriva-
tive O⁰ has a jump of height -1 at zero and, in the case σ = 0, a jump
of height +e^T(γ-λ) at γT .

(e) The Fourier transform of O satisfies

FO(v) = 1 ~ ^T(v~ i), v ∈ R .                (8)

v(v- i)

This identity extends to all complex values v with Im(v) ∈ [0, 1].

Remark that an interesting way to estimate γ and λ (but not ν) is
suggested by Proposition 1(d): a change point detection algorithm for jumps
in the derivative of O, as proposed by Goldenshluger, Tsybakov, and Zeevi
(2005), yields an estimate of γ and a subsequent estimate of the jump size
an estimate of λ.

We transform our observations (Yj) and predictors (Kj ) to

Oj := Yj/S ~ (1 ~ Kje^-rT/S)⁺ = O(xj) + δjεj,          (9)

xj :=log(Kj/S) ~rT,                                 (10)

where δj = S^-1σj. In practice, the design (xj) will be rather dense around
x = 0 and sparse for options further out of the money or in the money, cf.
Fengler (2005) for a study on the German DAX index.

In order to facilitate the subsequent analysis we make a mild moment
assumption on the price process, which guarantees by Proposition 1(b,c)
the exponential decay of O.

Assumption 1 We assume that C2 := E[e^2XT] is finite. This is equivalent
to postulating for the asset price a finite second moment: E[S_T² ] < ∞.

3. The method of estimation

3.1. Outline of the method

Since our asset follows an exponential Levy model, the jumps in the Levy
process appear exponentially transformed in the asset prices and it is intu-
itive that inference on the exponentially weighted jump measure

μ(x) := e^xν(x), x ∈ R,

will lead to spatially more homogeneous properties of the estimator than
for ν itself. Our calibration procedure relies essentially upon the formula

ψ(v) := T log(1 + iv(1 + iv)FO(v)´ = T log(φT(v ~ i))

= ~ ^σ2v- ⁺ i^{( σ 2 +} γ ) v ^{+ ( σ 2} /^{2 +} γ ~ ^λ ) ⁺ F μ⁽ v )^,      (¹¹)

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