Denis Belomestny and Markus Reiβ
everywhere twice differentiable, but more regularity will persist for regular
jump densities. Note that for a financial Levy model it is quite reasonable
to assume that the Levy measure is absolutely continuous and has even
a smooth density (at least off the origin). Prices are conceived by a large
number of agents on the market who in addition all share some uncertainty
about possible jump sizes, which smears out possible point masses.
As usually, the estimation procedure is specified by certain parameters.
The stabilisation of the logarithm by the function κ(v) is enforced mainly
for theoretical reasons to prevent explosions in the logarithm due to large
deviations, its practical importance is minor. For the weights wσU , wγU , wλU
it suffices to use weight functions satisfying (29) below for some large smax
like in standard nonparametrics where the order of the kernel must only be
sufficiently large, see Belomestny and Reiβ (2006) for an example. Like for
classical kernel estimators, their choice is not very critical. We are thus left
with only one important tuning parameter, the spectral cut-off frequency U .
In Theorem 1 an asymptotically optimal choice is given, while Belomestny
and Reiβ (2006) discuss some methods to determine U directly from the
data. Note, however, that a proper mathematical analysis for these com-
pletely data-driven (i.e., unsupervised) choices of U seems challenging due
to the underlying nonlinear ’change point detection’-structure, for which a
data-driven algorithm even in the idealized linear setting of Goldenshluger,
Tsybakov, and Zeevi (2005) is not yet available.
While the spectral calibration method is here only applied to the non-
parametric estimation of the Levy triplet in an exponential Levy model, it is
more generally applicable. Suppose we prescribe a finite-dimensional para-
metric model for the Levy measure. Then we can follow steps (a) through (c)
and fit the remainder term in step (d) to the parameters by a least-squares
criterion. In comparison with the classical least-squares approach this has
the advantage of yielding faster algorithms, which are also more robust due
to the variance reduction caused by the spectral cut-off. Moreover, many
more financial models have been propagated where the option price and
the model parameters are linked by a relationship in the spectral domain,
cf. Duffie, Filipovic, and Schachermayer (2003) and the references therein.
Although each model needs to be analyzed in detail, the general principles
of the spectral calibration method will apply.
Let us finally make a comparison with the nonlinear penalized least-
squares (PLS) approach by Cont and Tankov (2004b) for the same cali-
bration problem. There an exponential Levy model is selected as a prior
and exponential Levy models are considered that are obtained by a mar-
tingale measure equivalent to the prior. For each model the sum of squares
of the distances between observed and model option prices is penalized by
the relative entropy with respect to the prior. The estimated triplet is ob-
tained by minimizing this penalized least squares criterion. In practice, the
Levy measure is approximated by a finite-dimensional collection of point
measures and the minimizer is found by an iterative descent algorithm.