Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

setup. Note the general order in which the (asymptotic) quality of estima-
tion decreases: σ2, γ, λ and finally μ, which is related to the domination
property formulated in Alt-Sahalia and Jacod (2004). For small values of σ
and finite samples the performance is not so bad, compare the simulations
in Section 3.3; it just needs a lot more observations to improve on that.

At first sight the rates for the parametric estimation part are astonishing.
They are worse than in usual semi-parametric problems which also indicates
that misspecified parametric models will give unreliable estimates for the
volatility and jump intensity. In the case σ = 0, however, these rates are
easily understood when employing the language of distributions. With δ₀
denoting the Dirac distribution in zero and δ₀⁰ its derivative we have

log(ψτ(u)) = TF(γδ0 + ν - λδ0¢(u).

Estimating the density of XT and similarly its characteristic function from
the noisy observations of O amounts roughly to differentiate the observed
function twice, cf. Alt-Sahalia and Duarte (2003) and the remark after equa-
tion (34) below. This gives the minimax rate for ν and μ as that of esti-
mating the second derivative of a regression function of regularity s + 2.
For the parameter λ it suffices to estimate the jump in the antiderivative
of F^{- 1}(log(φ_τ)), which corresponds to a pointwise estimation problem in
the first derivative of a regression function, while for γ the analogy is the
estimation of the regression function itself at zero. This explains also why
in the class G_s we have measured the regularity not only in L² , but also
uniformly. In fact, if we only assume an L²-Sobolev condition, then the
same lower bound techniques will yield slower rates for the parameters, as
is typical for pointwise estimation problems.

Observe that the estimation of the jump density at zero is only possible
by imposing a certain regularity there, otherwise it is clearly not possible
to detect jumps of height zero.

5. Conclusion

We have developed an estimation procedure for the nonparametric calibra-
tion of exponential Levy models which is mathematically satisfying because
of its minimax properties and which yields a straight-forward algorithm for
the implementation. The corresponding lower bound results show that the
calibration is in general a hard problem to solve, at least if high accuracy is
desired. Nevertheless, the estimation procedure is well suited to gain general
insight into the size of the parameters and the structure of the jump den-
sity. Even if reasonable parametric models exist that can be better fitted, a
goodness-of-fit test based on our nonparametric approach should always be
used to check against model misspecification.

Our procedure can be adapted to different models as long as the inverse
transformation from the option prices to the characteristic function can be
calculated and the unknown quantities can be determined from the struc-
ture of the characteristic function, cf. the treatment of unbounded jump

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