Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

faces, e.g. Fengler (2005). For the generalised Black-Scholes model Dupire’s
formula permits the calibration from option prices, see e.g. Jackson, Süli,
and Howison (1999) for a numerical approach and Crepey (2003) for a the-
oretical study. The calibration of parametric exponential Levy models has
been studied for example by Eberlein, Keller, and Prause (1998) and Carr,
Geman, Madan, and Yor (2002).

After introducing the financial and statistical model in Section 2, the
estimation method is developed in Section 3. The main theoretical results
are formulated in Section 4. We conclude in Section 5. The proofs of the
upper and lower bounds are deferred to Sections 6 and 7, respectively.

2. The model

2.1. The exponential Lévy model and option prices

Since we base our calibration on option prices, we place ourselves imme-
diately in a risk neutral world, modeled by a filtered probability space
(Ω, F, Q, (F_t)), on which the price process (S_t,t > 0) of an asset after
discounting forms a martingale. As is standard in the calibration literature,
the martingale measure Q is assumed to be settled by the market and to be
identical for all options under consideration.

We suppose that under Q the process S_t follows the Levy model (1),
where S > 0 is the present value of the asset and r > 0 is the riskless
interest rate, which is assumed to be known and constant. An excellent
reference for this model in finance is the monograph by Cont and Tankov
(2004a). In this paper we shall only consider Levy processes X with a jump
component of finite intensity and absolutely continuous jump distribution.
Extensions to the infinite intensity case can be found in Belomestny and
Reiβ (2005). The characteristic function of XT is then given by the Levy-
Khintchine representation

σ > 0 is called volatility, γ ∈ R drift and the non-negative function ν ∈
L¹(R) is the jump density with intensity λ := kνk_L1(_R) .

A risk neutral price at time t = 0 for a European call option with strike
K and maturity Γ is given by

C ( K,Γ ) = e^-rT E[( ST — K )⁺],                  (3)

where (A)⁺ := max(A, 0). By the independence of increments in X the
martingale condition on e^-rt St may be equivalently characterized by

∀t > 0: E[e^x] = 1 ^^ — + γ + I (e^x — 1)ν(x)dx = 0.
2      _-∞

Observe that we have imposed implicitly the exponential moment condition
₀^∞ (e^x — 1)ν (x) dx < ∞ to ensure the existence of E[St]. Another conse-
quence is that the characteristic function φT is defined on the whole strip

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