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

For a spectrally negative Lévy process $X$, we study the following distribution: $\mathbb{E}_x \left[ \mathrm{e}^{- q \int_0^t \mathbf{1}_{(a,b)} (X_s) \mathrm{d}s } ; X_t \in \mathrm{d}y \right],$ where $-\infty \leq a < b < \infty$, and where $q,t>0$ and $x \in \mathbb{R}$. 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.

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