Denis Belomestny and Markus Reiβ
different maturities. By basing our estimation on option data we draw infer-
ence on the underlying risk neutral price process, which in general cannot
be determined from historical price data due to the incompleteness of the
Levy market.
The observed option prices will be slightly unprecise due to bid-ask
spreads or other market frictions. In the ideal case of precise observations
for all possible strike prices the state price density and hence the Levy triplet
can be uniquely identified using the formula by Breeden and Litzenberger
(1978). Under the realistic model of finitely many noisy observations we
cannot hope to determine the triplet correctly, we should rather try to
provide an estimator which is as good as possible for the given accuracy
of the data. This optimality property is usually assessed by the minimax
paradigm, which measures the inherent complexity of the statistical problem
class. One of the main results of the present paper is a lower bound, showing
that already in the simple exponential Levy model the estimation problem
is in general severely ill-posed, that is, the estimation error for any part of
the Levy triplet as a function of the accuracy of the observations will only
converge with a logarithmic rate for any conceivable estimation procedure.
On the other hand, we propose an explicit construction of an estimator
that attains this optimal minimax rate. The procedure is based on the in-
version of the explicit pricing formula via Fourier transforms by Carr and
Madan (1999) and a regularisation in the spectral domain. Using the Fast
Fourier Transformation, the procedure is easy to implement and yields good
results in simulations in view of the severe ill-posedness, see also the sup-
plement Belomestny and Reiβ (2006). Below, we discuss the features of our
method in detail and compare it with the penalized least squares approach
by Cont and Tankov (2004b). In comparison with standard statistical ill-
posed problems, the main challenges are the nonlinearity involved and the
complex interplay between the jump measure as nonparametric part and
the drift and diffusion coefficient as parametric parts.
The exponential Levy model reflects the assumption that the log returns
of the asset evolve independently and with identical distribution for the
same time steps, which is plausible for liquid markets and not too long time
horizons. This basic model class, first introduced by Merton (1976), has
been considered recently for a variety of pricing and optimisation problems
in finance, cf. the recent works by Kallsen (2000), Mordecki (2002), Emmer
and Kliippelberg (2004), Cont and Voltchkova (2005) and the references
therein.
When no model for the price process is specified, calibration from op-
tion data can be used to estimate the state price density, see Ait-Sahalia
and Duarte (2003). This density yields the distribution of the asset price
at the times of maturity, but does not provide any information on the evo-
lution of the price in time. A structural assumption on the price process
allows to find prices for path-dependent options or to perform a dynamic
risk management. In financial engineering information about the expected
time evolution is obtained by smoothing implied Black-Scholes volatility sur-