Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

> 2 exp(-τ^σ2axv² - 4TR´ =: _κ(v). (27)

The only reason for the factor 1/2 is the mathematical tractability giving
later the bound of Lemma 1.

Concerning the choice of the weight functions, we take advantage of the
smoothness s of μ by taking functions w such that Fw has s vanishing mo-
ments. Equivalently expressed in the spectral domain, the weight functions
w(u) grow with frequencies ∣u∣ like ∣u∣^s to profit from the decay of ∣Fμ(u) |.
Hence, we define for all U > 0 families of weight functions by rescaling:
w_σ^U (u) = U ^-³w_σ¹ (u/U), w_γ^U (u) = U ^-²w_γ¹ (u/U), w_λ^U (u) = U^-¹w_λ¹ (u/U),
(28)

where the functions w_σ¹ , w_γ¹ , w_λ¹ satisfy conditions (17) and (18) as well as

F(w_σ¹ (u)/u^s), F(w_γ¹ (u)/u^s), F(w_λ¹ (u)/u^s) ∈ L¹(R). (29)

In addition the support of the weight functions w_σ¹ , w_γ¹ , w_λ¹ is assumed to
be contained in [-1, 1]. Note that the property F (w(u)/u^s) ∈ L¹(R) means
in particular that w(u)/u^s is continuous and bounded such that
∣wσ^U(u)∣ .U^-⁽^s⁺³⁾∣u∣^s, ∣wγ^U(u)∣ .U^-⁽^s⁺²⁾∣u∣^s, ∣wλ^U(u)∣ .U^-⁽^s⁺¹⁾∣u∣^s.

(30)

6.2. A numerical approximation result

Proposition 2. Under the hypothesis e^-A . ∆² we obtain uniformly over
all Lévy triplets satisfying Assumption 1

sup ∣E[FO(u) - FO(u)] ∣ = sup ∣FO_l (u) -FO(u) ∣ . ∆². (31)

u∈R u∈R

Proof. By standard Fourier estimates the assertion follows once we have
proved kOl - OkL1 . ∆² .

Note that O - β0 is twice differentiable except at the points xj₀-1 , 0, xj₀and possibly γT by Proposition 1(d). Moreover, O - β0 has a derivative
near zero which is uniformly bounded by a constant C0 , which follows from
the L¹-estimate for O⁰⁰ in Proposition 1(d).

Starting with the case σ> 0, we obtain the classical quadrature estimate
for the trapezoidal rule using the mean value theorem:

∣Oι(x) -O(x)∣

dx 6 kO⁰⁰kL1∆² +2C0∆².

By Assumption 1 and Proposition 1(b,c) the extrapolation error is bounded

[x0,x1]∪[xN,xN +1]

∣E[O(x) - O(x)] ∣ dx 6 4C₂∆e^-⁽^a^-^δ).

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