Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view

Abstract

We study the distribution E_x[exp(-q∫₀^t 1_(a,b)(X_s)ds); X_t ∈ dy], where -∞ ≤ a < b < ∞, and where q, t > 0 and x ∈ R for a spectrally negative Lévy process X. More precisely, we identify the Laplace transform with respect to t of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.