Spectral calibration of exponential Lévy Models [1]



Spectral calibration of exponential Levy models ??

15


> 2 exp(-τσ2axv2 - 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 us to profit from the decay of Fμ(u) |.
Hence, we define for all
U > 0 families of weight functions by rescaling:
wσU (u) = U -3wσ1 (u/U), wγU (u) = U -2wγ1 (u/U), wλU (u) = U-1wλ1 (u/U),
(28)

where the functions wσ1 , wγ1 , wλ1 satisfy conditions (17) and (18) as well as

F(wσ1 (u)/us), F(wγ1 (u)/us), F(wλ1 (u)/us) L1(R).        (29)

In addition the support of the weight functions wσ1 , wγ1 , wλ1 is assumed to
be contained in [
-1, 1]. Note that the property F (w(u)/us) L1(R) means
in particular that
w(u)/us is continuous and bounded such that
wσU(u) .U-(s+3)us, wγU(u) .U-(s+2)us,   wλU(u) .U-(s+1)us.

(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 FOl (u) -FO(u) . 2.     (31)

uR                        uR

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, xj0
and 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:

xN

x1


Oι(x) -O(x)


dx 6 kO00kL12 +2C02.


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

by


/

[x0,x1][xN,xN +1]


E[O(x) - O(x)] dx 6 4C2∆e-(a-δ).



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