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Telegraph Processes with Random Jumps and Complete Market Models

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  • Nikita Ratanov

    (Universidad del Rosario)

Abstract

We propose a new generalisation of jump-telegraph process with variable velocities and jumps. Amplitude of the jumps and velocity values are random, and they depend on the time spent by the process in the previous state of the underlying Markov process. This construction is applied to markets modelling. The distribution densities and the moments satisfy some integral equations of the Volterra type. We use them for characterisation of the equivalent risk-neutral measure and for the expression of historical volatility in various settings. The fundamental equation is derived by similar arguments. Historical volatilities are computed numerically.

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  • Nikita Ratanov, 2015. "Telegraph Processes with Random Jumps and Complete Market Models," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 677-695, September.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:3:d:10.1007_s11009-013-9388-x
    DOI: 10.1007/s11009-013-9388-x
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    References listed on IDEAS

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    1. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
    2. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    4. Gerald Cheang & Carl Chiarella, 2011. "A Modern View on Merton's Jump-Diffusion Model," Research Paper Series 287, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Nikita Ratanov, 2008. "Option Pricing Model Based on a Markov-modulated Diffusion with Jumps," Papers 0812.0761, arXiv.org.
    6. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
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    Cited by:

    1. Ratanov, Nikita, 2015. "Hypo-exponential distributions and compound Poisson processes with alternating parameters," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 71-78.
    2. Antonio Di Crescenzo & Barbara Martinucci & Shelemyahu Zacks, 2018. "Telegraph Process with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 333-352, March.
    3. Antonio Crescenzo & Barbara Martinucci & Paola Paraggio & Shelemyahu Zacks, 2021. "Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 837-858, September.
    4. Claudio Macci & Barbara Martinucci & Enrica Pirozzi, 2021. "Asymptotic Results for the Absorption Time of Telegraph Processes with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1077-1096, September.
    5. Nikita Ratanov, 2016. "Option Pricing Under Jump-Diffusion Processes with Regime Switching," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 829-845, September.
    6. Johan GB Beumee & Chris Cormack & Peyman Khorsand & Manish Patel, 2014. "Path Diffusion, Part I," Papers 1406.0077, arXiv.org.

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