Spectral calibration of exponential Lévy Models [1]

Denis Belomestny and Markus Reiβ

{z ∈ C | Im(z) ∈ [-1, 0]} in the complex plane, which will be important
later. We reduce the number of parameters by introducing the negative
log-forward moneyness

x := log(K/S) - rT,

such that the call price in terms of x is given by

C(x, T) = S E[(e^XT -e^x)⁺].

The analogous formula for the price of a put option is P(x, T ) = S E[(e^x -
e^XT )⁺] and the well-known put-call parity is easily established:

C(x,T) - P(x, T) = S E[e^XT - e^x] = S(1 -e^x).           (5)

2.2. The observations

We focus on the calibration from options with a fixed maturity T > 0 and
mention the straight-forward extension to several maturities in Section 3.1.
The prices of N call options (or by the put-call parity (5) alternatively put
options) are observed at different strikes Kj , j = 1, . . . , N, corrupted by
noise:

Yj =C(Kj,T)+σjεj, j= 1,...,N.                (6)

We assume the observational noise (εj ) to consist of independent centred
random variables with E[ε_j²] = 1 and sup_j E[ε_j⁴] < ∞. The noise levels (σj )
are assumed to be positive and known.

For observational noise with a known and smooth correlation structure
the calibration problem becomes more stable. As long as no empirically val-
idated model for the observational noise exists, we work under the assump-
tion of independent perturbations which is canonical and least favourable.

As we need to employ Fourier techniques, we introduce the function

_{O(x) :=} ⁽S^-1C(x, T), x>0,                      ₍₇₎

S^-1P(x, T), x < 0

in the spirit of Carr and Madan (1999). O records normalised call prices
for x > 0 and normalised put prices for x 6 0. The following important
properties of O are easily obtained from the put-call parity (5) and the
martingale condition (4), see Belomestny and Reiβ (2005) for the exact
derivation.

Proposition 1.

(a) We have O(x) = S^-1C(x, T) - (1 - e^x)⁺ for all x ∈ R.

(b) O(x) ∈ [0, 1 ∧ e^x] holds for all x ∈ R.

(c) If Cα := E[e^αXT] is finite for some α > 1, then O(x) 6 Cαe^(1-α)x holds
for all x > 0.

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