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Option Pricing Driven by a Telegraph Process with Random Jumps

Part of:Markov processes

Published online by Cambridge University Press:  04 February 2016

Oscar López and
Nikita Ratanov
Oscar López*
Affiliation:
Universidad del Rosario
Nikita Ratanov*
Affiliation:
Universidad del Rosario
*
∗ Postal address: Universidad del Rosario, Calle 12c, No. 4-69, Bogotá, Colombia.
∗ Postal address: Universidad del Rosario, Calle 12c, No. 4-69, Bogotá, Colombia.
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Abstract

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In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

Keywords

Jump-telegraph processequivalent martingale measureoption pricinghedging

MSC classification

Secondary:60J27: Continuous-time Markov processes on discrete state spaces60J75: Jump processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012, pp. 838 - 849
Copyright
© Applied Probability Trust 

References

Bladt, M. and Padilla, P. (2001). Nonlinear financial models: finite Markov modulation and its limits. In Quantitative Analysis in Financial Markets, Vol. III, ed. Avellaneda, M., World Scientific, River Edge, NJ, pp. 159–171.Google Scholar
Di Crescenzo, A. and Pellerey, F. (2002). On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171–184.Google Scholar
Di Masi, G. B., Kabanov, Y. M. and Runggaldier, W. J. (1994). Mean-variance hedging of options on stocks with Markov volatilities. Theory Prob. Appl. 39, 172–182.CrossRefGoogle Scholar
Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets. Springer, New York.Google Scholar
Elliott, R. J., Siu, T. K., Chan, L. and Lau, J. W. (2007). Pricing options under a generalized Markov-modulated Jump-diffusion model. Stoch. Anal. Appl. 25, 821–843.Google Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.CrossRefGoogle Scholar
Guo, X. (2001). Information and option pricings. Quant. Finance 1, 38–44.CrossRefGoogle Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.CrossRefGoogle Scholar
Mandelbrot, B. (1963). The variation of certain speculative prices. J. Business 36, 394–419.CrossRefGoogle Scholar
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144.Google Scholar
Naik, V. and Lee, M. (1990). General equilibrium pricing of options on the market portfolio with discontinuous returns. Rev. Financial Studies 3, 493–521.CrossRefGoogle Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.Google Scholar
Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Pogorui, A. A. and Rodríguez-Dagnino, R. M. (2009). Evolution process as an alternative to diffusion process and Black-Scholes formula. Random Operators Stoch. Equat. 17, 61–68.Google Scholar
Ratanov, N. (2007). A Jump telegraph model for option pricing. Quant. Finance 7, 575–583.Google Scholar
Ratanov, N. (2010). Option pricing model based on a Markov-modulated diffusion with Jumps. Braz. J. Prob. Statist. 24, 413–431.Google Scholar
Ratanov, N. and Melnikov, A. (2008). On financial markets based on telegraph processes. Stochastics 80, 247–268.CrossRefGoogle Scholar
Weiss, G. H. (2002). Some applications of persistent random walks and the telegrapher's equation. Physica A 311, 381–410.Google Scholar
Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.Google Scholar