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On perpetual American put valuation and first-passage in a regime-switching model with jumps. (English) Zbl 1164.60066
The authors introduce a continuous-time irreducible Markov process \(Z\) with finite state space and a regime-switching phase-type Lévy process \(X\), controlled by \(Z\). They consider financial markets with bond and stock, where stock is an exponent of \(X\). This consideration is motivated by the observation that Lévy processes have been successfully calibrated to options with single, short time maturities whereas regime-switching models fit well longer dated options. Under this model, for the case of the dense class of phase-type jumps, explicit and analytically tractable results are obtained for the value function of a perpetual American put option and for the corresponding optimal exercise strategy. It is proved that the optimal stopping time takes the form of the first-passage problem of \(X\) under some level \(k(Z_t)\), and the corresponding first-passage problem is solved. The solution is supported by path transformation and a new matrix Wiener-Hopf factorization result for this class of processes.

MSC:
60K15 Markov renewal processes, semi-Markov processes
91B28 Finance etc. (MSC2000)
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