European and American options: The semi-Markov case |
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| Authors: | Guglielmo D’Amico Jacques Janssen Raimondo Manca |
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| Institution: | 1. Dipartimento di Scienze del Farmaco, Universitá “G.D’Annunzio” di Chieti, Italy;2. Jacan & Euria Université de Bretagne Occidentale Brest, France;3. Dipartimento di Matematica per le Decisioni Economiche, Finanziarie ed Assicurative Universitá “La Sapienza” di Roma, Italy;1. Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal;2. King Abdullah University of Science and Technology (KAUST), CSMSE Division, Thuwal 23955-6900, Saudi Arabia;3. KAUST SRI, Uncertainty Quantification Center in Computational Science and Engineering, Saudi Arabia;1. Department of Computer Science, Ryerson University, Toronto, Canada;2. Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada;3. Pos-Graduation Program of Electrical Engineering, Federal University of Pará, Brazil;4. Faculty of Computer Engineering, Federal University of Pará, Brazil;1. University of Ioannina, Department of Mathematics, 451 10 Ioannina, Greece;2. University of Missouri-Columbia, Department of Statistics, Columbia, USA |
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| Abstract: | In this paper, we assume that the log return of the underlying asset follows a semi-Markov process, then from the knowledge of the kernel we derive an explicit expression for the value of the option and for the bare risk in the case of the European call (put) option and, by means of a recursive system, we derive the value and the bare risk in the case of the American option. The prices and risks we obtained depend explicitly on the waiting-time distributions of the asset and they are duration dependent. The link with models based on Markov Chains and Continuous Time Random Walks is debated. |
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