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[(eXT -ex)+].
The analogous formula for the price of a put option is P(x, T ) = S E[(ex -
eXT )+] and the well-known put-call parity is easily established:
C(x,T) - P(x, T) = S E[eXT - ex] = S(1 -ex). (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[εj2] = 1 and supj E[εj4] < ∞. 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 - ex)+ for all x ∈ R.
(b) O(x) ∈ [0, 1 ∧ ex] holds for all x ∈ R.
(c) If Cα := E[eαXT] is finite for some α > 1, then O(x) 6 Cαe(1-α)x holds
for all x > 0.