Spectral calibration of exponential Lévy Models [1]

Denis Belomestny and Markus Reiβ

densities in Belomestny and Reiβ (2005). In fact, the inhomogeneity in em-
pirical jump densities across maturities (see Cont and Tankov (2004a) and
Belomestny and Reiβ (2006)) suggests that the exponential Levy model
should be extended, for example by abandoning homogeneity in time. In
conclusion we believe that the question of calibration for models in finan-
cial mathematics should be addressed with the same rigour and intensity as
other primary questions like pricing, hedging and risk management.

6. Proof of the upper bounds

All calculations take place in the setting of Section 4. To facilitate the
calculations we introduce the exponentially increasing function

E(x) := ^{e -1}, x > 0, and set E(0) := 1.              (25)

6.1. Specification of the method

In step (a) we interpolate the data (Oj) by setting

N

O(x) = βo(x) + X Ojbj(x), x ∈ R,

j=1

where (bj ) are linear splines and the function β0 is added to take care of
the jump in the derivative of O at zero: β₀⁰ (0+) - β₀⁰ (0-) = -1. We choose
bk, k = 1, . . . , N, as the linear B-spline with knots at xk-1, xk, xk+1 and
β₀ as the linear spline with knots at xj_0-1, 0, xj₀ and with β₀(xj_-1) =
β0(xj) = 0, β0 (0) = xj-1xj /(xj-1 - xj), where the index j0 is defined
by xj₀ -1 < 0 < xj₀ (excluding the improbable case xj = 0). To ease the
mathematical treatment of the extrapolation error, we assume that all data
is contained in the interval (-A - ∆, A+∆). Adding the extrapolated design
points x0 = -A - ∆ and xN+1 = A + ∆, we set O(x0) = O(xN +1) = 0. As
bias we encounter the following linear interpolation of O

N

Ol(x):= E[O)(x)] = X O(x_j)b_j(x)+ β₀(x), x ∈ R .       (26)

j=1

More generally, we merely need to ensure for step (a) that the results of
Proposition 2 and estimate (35) are satisfied.

We have enforced ∣ψ_τ(v)| > log(κ(v)) in (12) to prevent unboundedness
in the case of large stochastic errors. For Levy triplets in Gs(R,σ_max) a
reasonable choice for κ(v) can be obtained from the following calculation
using the identity σ~ + γ + Fμ(0) = λ derived from the martingale condition
(4):

ɪ ∖φτ(v - i) | = 1 exp (-T^σ2v² - TFμ(0)+ T Re(Fμ(v))´

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