Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

11

Table 1. The minimax rates vq,σ_max for the different parameters q.

Since the underlying Levy triplet is only identifiable if O (x) is known for
all x ∈ R, we consider the asymptotics of a growing number of observations
with

and A := min(xN, -x1) → ∞. (21)

In contrast to standard regression estimates we shall always track ex-
plicitly the dependence on the level (δk ) of the noise in the observations,
which is usually rather small for observed option prices. The subsequent
analysis can certainly be improved for a concrete design (xj ) and concrete
noise levels (δj), but for revealing the main features it is more transparent
and concise to state the results in terms of the abstract noise level

ε := ∆³/² + ∆¹ /² ∣∣J∏_t.

comprising the level of the numerical interpolation error and of the stochas-
tic error simultaneously. Here and in the sequel we use the norms ∣δ∣_l∞ :=
sup_k δk and ∣δ∣_l²2 := ^P_kδ_k².

We now state the main results about the risk upper bounds of the estima-
tors obtained by the basic procedure, given the specific choices in Section
6.1, and about the risk lower bounds valid for any estimation procedure
whatsoever. The proofs are given in Sections 6 and 7 for the upper and
lower bounds, respectively.

Theorem 1. Assume e A < ∆² and ∆∣∣J∣∣2₂ < ∣∣^∣∣2∞. For any σ > σ_max
we choose

U-_σ := σ—¹(2log(ε^-1)/T¢1 /2, Uɔ := ε^-2/(^2s+⁵),           (23)

in the cases σ_max > 0 and σ_max = 0, respectively. Then every estimator
q ∈ {σ²,γ,λ,μ} for the corresponding parameter q satisfies the following
asymptotic risk bound:

^suP      Et [∣∣q - q∣∣ ²]1 /² . v_i,-^,

T∈Gs (R,σ

max)

where ∣∣∙∣∣ denotes the absolute value for q ∈ {σ², γ, λ} and the L²(R)-norm
for q = μ and the rate v_q,_σm_ax is given in Table 1.

The two assumptions in the theorem are not very severe: because of
the exponential decay of O the width A of the design only needs to grow
logarithmically and the error levels (δk ) need only be square summable after

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