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 kbk k = 1 and |xk+1


xk-1 | 6 2:


Fbk∣∣l2 = 2∏kb∖∖bkkb2 6 (4π∆)1 /2,   ∣∣Fbk∣∣ 6 \\bk\\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((u)) — λ + Re(L(u) + R(u))) wjU(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 Eh∣∣     R(u)wσU (u) dui.

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

∣/ (u)wσ(u)du∣ = 2πf μ(s)(x)F-1(wU(u)/(iu)s)(x)dx

-U                  -∞


6 U-(s+3) kμ(s) U∣∣F(w~σ(u)/us) ∣∣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

Eh∣Z-UL(u)wσU(u)du∣ i = ∣∣Z-U


- i) E[F (O) - O)(u)]wσU (u)


ψτ ( u — i )


2
d u


+ Varh Γu u(u i)C>lu) w(u) dui =: Lb + L.

-J-U       Vt ( u — i )          -l

The bias term is easily bounded by Proposition 2, using the uniform bound
on
U s+3 wσU (u)/us :

U

|Lb| 6 ∣∣F(Oi-O)IU / φτ(u — i)-1(u4 + u2)1 /2wσ(u)|du

-U

U2

< 2U-(s+3)      eτɪu +2tkL1 |u|s+2 du.

-U



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