12
Denis Belomestny and Markus Reiβ
renormalisation. The latter condition can certainly be further relaxed since
this term is caused by a rough bound on the quadratic remainder term.
For the lower bounds we appeal to the equivalence between the regression
and the Gaussian white noise model, as established by Brown and Low
(1996), and consider merely the idealized observation model
dZ(x) = O(x)dx+εdW(x), x ∈ R, (24)
with the noise level asymptotics ε → 0, a two-sided Brownian motion W
and with O = OT denoting the option price function from (7) for the given
triplet T . Here, the noise level ε corresponds exactly to the regression er-
ror ∆1 / 2 и l∞ . Due to Assumption 1 the option price functions O decrease
exponentially and the results by Brown and Low (1996) remain valid for
unbounded intervals. This simplification avoids tedious numerical approxi-
mations in the proofs.
Theorem 2. Let s ∈ N, R > 0 and σmax > 0 be given. For the observation
model (24) and any quantity q ∈ {σ2 ,γ,λ,μ} the following asymptotic risk
lower bounds hold:
inf sup Eτ[∣∣q - q∣∣2]1 /2 & m,-
g, T∈Gs ( R,σmaχ )
where ∣∣∙∏ denotes the absolute value for q ∈ {σ2, γ, λ} and the L2 (R)-
norm for q = μ, the infimum is always taken over all estimators, that is all
measurable functions of the observation Z, and the rate vq,σmax is given in
Table 1. Hence, our estimators are rate-optimal.
4.2. Discussion of the results
As we want to identify the Levy triplet exactly in the limit, we have to
assume the asymptotics ∆ → 0 and A → ∞ in the upper bound result. The
numerical interpolation error term ∆3/2 contained in ε can be made smaller
by using higher-order schemes, see Section 3.2. On the other hand, the
statistical error term ∆1 /2∣∣J∣∣∕∞ cannot be avoided as proved by the lower
bound. Another way to study the calibration problem is to keep the number
N of observations fixed and just to consider the asymptotics ∣δ ∣l∞ → 0. In
this case the original Levy triplet is not identifiable and the triplet of interest
has to be properly defined in the set of triplets giving rise to the uncorrupted
option prices, cf. Cont and Tankov (2005) for a minimum relative entropy
approach.
We observe that for σ > 0 the rate corresponds to a severely ill-posed
problem, while for known σ = 0 the rates are much better, but still ill-posed
compared to those obtained in classical nonparametric regression. The se-
vere ill-posedness in the case σ > 0 is due to an underlying deconvolution
problem with the Gaussian kernel of variance σ2 : the law of the diffusion
part of XT is convolved with that of the compound Poisson part to give
the density of XT . This type of estimation problem has been studied thor-
oughly by Butucea and Matias (2005) in an idealized density estimation