Spectral calibration of exponential Lévy Models [1]

Denis Belomestny and Markus Reiβ

e^-T^σ2u2m¹ /²e^-(T^σ2u2/^m)m (Tσ2/m)^2mu^2m-2 |u - i|^2m-2 du

= ε^-2δ²m^-m-3Γ(m) < ε^-2δ²m^-m-3(m - 1)^m-1 /²e^{1 -m} ~ ε^-2m^-3-se^-m

Consequently, the Kullback-Leibler divergence remains small when choosing
m > 2log(ε^-1), but m < log(ε^- 1), which gives δ ~ log(ε^- 1)^-s/². From the
basic general lower bound result we therefore obtain the asymptotic lower
bound for μ.

7.4. Lower bound for σ², γ and λ in the case σ > 0

Since the proof is very similar to the preceding calculations, we only give
the perturbations of the basic triplet T₀ = (σ₀, γ₀, μ0) with σ0 > 0 which are
least favourable. More details can be found in Belomestny and Reiβ (2005).

For γ we leave σ₀ fixed and use a perturbation of the form

γ 1 := Y0 + δ,   Fμ 1(u) := Fμ₀(u) — δi(u — i)e

For λ we keep σ₀ and γ₀ fixed and consider

Fμ 1(u) := Fμo(u) + δe-^-u^m(^u

For σ² we leave γ₀ invariant and consider the perturbation

2m 2m

σ2 := σ2 + 2δ, Fμ 1(u):= Fμ₀(u)+ δ(u — i)²e ^u /U

Each time m is chosen to be of order log(ε^-1) and the value of δ > 0 results
from the smoothness class considered.

Acknowledgements

We thank Peter Tankov for intense discussions on different calibration ap-
proaches and for sharing his experience in assessing their practical perfor-
mances. We are grateful for the comments and questions by two anonymous
referees which have led to considerable improvements.

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