Journal of Applied Probability

Option pricing driven by a telegraph process with random jumps

Oscar López and Nikita Ratanov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
J. Appl. Probab., Volume 49, Number 3 (2012), 838-849.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1346955337

Digital Object Identifier
doi:10.1239/jap/1346955337

Mathematical Reviews number (MathSciNet)
MR3012103

Zentralblatt MATH identifier
1260.91230

Subjects
Primary: 91B28
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes

Keywords
Jump-telegraph process equivalent martingale measure option pricing hedging

Citation

López, Oscar; Ratanov, Nikita. Option pricing driven by a telegraph process with random jumps. J. Appl. Probab. 49 (2012), no. 3, 838--849. doi:10.1239/jap/1346955337. https://projecteuclid.org/euclid.jap/1346955337


Export citation

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. M. Avellaneda, World Scientific, River Edge, NJ, pp. 159–171.
  • Di Crescenzo, A. and Pellerey, F. (2002). On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171–184.
  • 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.
  • Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets. Springer, New York.
  • 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.
  • Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.
  • Guo, X. (2001). Information and option pricings. Quant. Finance 1, 38–44.
  • Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.
  • Mandelbrot, B. (1963). The variation of certain speculative prices. J. Business 36, 394–419.
  • Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144.
  • Naik, V. and Lee, M. (1990). General equilibrium pricing of options on the market portfolio with discontinuous returns. Rev. Financial Studies 3, 493–521.
  • 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.
  • Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.
  • 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.
  • Ratanov, N. (2007). A jump telegraph model for option pricing. Quant. Finance 7, 575–583.
  • Ratanov, N. (2010). Option pricing model based on a Markov-modulated diffusion with jumps. Braz. J. Prob. Statist. 24, 413–431.
  • Ratanov, N. and Melnikov, A. (2008). On financial markets based on telegraph processes. Stochastics 80, 247–268.
  • Weiss, G. H. (2002). Some applications of persistent random walks and the telegrapher's equation. Physica A 311, 381–410.
  • Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.