Exponential Lévy models are popular for option pricing, as their jump component captures empirical features of financial data that the Black-Scholes model cannot. Belomestny and Reiß introduced a spectral calibration approach for a single maturity, assuming a constant Lévy triplet on the whole interval. Tendijck and Koorevaar extended this to a time-inhomogeneous approach estimating a distinct triplet on each maturity interval from European put and call prices. Koorevaar further established asymptotic normality and confidence intervals for each of the estimated triplets.
This thesis extends that pointwise normality result to a functional CLT in L²(K) (where K ⊂ ℝ is compact) for the estimated Lévy density. We first derive a candidate covariance kernel of the exponentially-tilted estimation error, and identify a central structural obstruction: the rescaled kernel converges to an oscillatory kernel that is not integrable in ℝ². Hence, the associated covariance operator is not nuclear in L²(ℝ), and a Giné-León Hilbert space CLT cannot be obtained. We therefore restrict the domain to a compact set and modulate the error by a cosine factor to remove the oscillations, which yields a Giné-León CLT for the linear part of the error. The bias and remainder terms vanish under appropriate scaling, yielding a functional Central Limit Theorem for the full cosine-modulated error.
As an application, this result is transferred through the Gil-Pelaez formula to obtain a convergence-in-distribution result for the pricing error of a digital call option where the error enters through estimation of the Lévy density. This enables the computation of finite-sample confidence intervals that bridge the gap between the theory and practice. Finally, possible extensions are discussed. ... Read more