Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

17

= 1 κ(u)^-2(u⁴ + u²)F(O) - O)(u)|².

Together with the previous result this gives for all u ∈ R the assertion of

the lemma.

ut

We shall frequently use the following norm bounds for the B-splines (bk),

which follow from kb_k k_∞ = 1 and |x_k+1

xk-1 | 6 2∆:

∣Fbk∣∣_l2 = 2∏kb∖∖b_kkb2 6 (4π∆)¹ /²,   ∣∣Fbk∣∣∞ 6 \\b_k\\_Lɪ 6 2∆.     (35)

We decompose σ² in terms of L and R from (33) and (34):

^U
σ2 = /

-U

2

-2(u² — 1) + γ + Re(Fμ(u)) — λ + Re(L(u) + R(u))) w^jU(u) du

^U

= σ² + /   Re (Fμ(u) + L(u) + R(u)) wσ (u)du,

-U

which yields

(36)

∣ U                                      ∣²                ∣ U                                  ∣²

E[ ∣σ² — σ²1²] 6 3∣∕ F μ ( u ) wU ( u )d u ∣ +3 E ∣/   L ( u ) wU ( u )d u ∣

U²

+ 3 E^h∣∣     R(u)w_σ^U (u) du^∣∣ ⁱ.

Let us consider the three terms in the sum separately. The nuisance of
Fμ causes a deterministic error which can be bounded using ( iu ) ^sFμ (u ) =
F μ (^s )(u) and the Plancherel isometry by:

∣/ Fμ(u)wσ(u)du∣ = 2π∣ f μ(^s)(x)F^-1(wU(u)/(iu)^s)(x)dx∣

-U                  -∞

6 U^-(^s+³) kμ(^s) ∣U∣∣F(w~σ(u)/u^s) ∣∣_b 1.

(37)

The linear error term can be split into a bias and a variance part (Var[Z] :=

E[|Z - E[Z]|²]):

U                   ²U

E^h∣∣^Z_-UL(u)w_σ^U(u)du∣^{∣ i} = ∣^∣Z_-U

- i) E[F (O⁾ - O)(u)]w_σ^U (u)

ψτ ( u — i )

2
d u∣