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Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view. (English) Zbl 1337.60090

Summary: We study the distribution \(\mathbb{E}_{x}[\exp(-q\int_{0}^{t} 1_{(a,b)}(X_{s})ds); X_{t} \in dy]\), where \(-\infty \leq a < b < \infty\) and where \(q, t > 0\) and \(x \in \mathbb{R}\), 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.

MSC:

60G51 Processes with independent increments; Lévy processes
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories
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