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Option Pricing Model Based on a Markov-modulated Diffusion with Jumps

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  • Nikita Ratanov

Abstract

The paper proposes a class of financial market models which are based on inhomogeneous telegraph processes and jump diffusions with alternating volatilities. It is assumed that the jumps occur when the tendencies and volatilities are switching. We argue that such a model captures well the stock price dynamics under periodic financial cycles. The distribution of this process is described in detail. For this model we obtain the structure of the set of martingale measures. This incomplete model can be completed by adding another asset based on the same sources of randomness. Explicit closed-form formulae for prices of the standard European options are obtained for the completed market model.

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File URL: http://arxiv.org/pdf/0812.0761
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Paper provided by arXiv.org in its series Papers with number 0812.0761.

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Date of creation: Dec 2008
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Handle: RePEc:arx:papers:0812.0761

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  1. Alessandro De Gregorio & Stefano Iacus, 2007. "Change point estimation for the telegraph process observed at discrete times," UNIMI - Research Papers in Economics, Business, and Statistics, Universitá degli Studi di Milano unimi-1053, Universitá degli Studi di Milano.
  2. Miquel Montero, 2007. "Renewal equations for option pricing," Papers 0711.2624, arXiv.org, revised Jun 2008.
  3. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  4. X. Guo, 2001. "Information and option pricings," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 1(1), pages 38-44.
  5. Robert J. Elliott & John van der Hoek, 1997. "An application of hidden Markov models to asset allocation problems (*)," Finance and Stochastics, Springer, vol. 1(3), pages 229-238.
  6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  7. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  8. 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.
  9. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, Econometric Society, vol. 41(1), pages 135-55, January.
  10. Mazza, Christian & Rulliere, Didier, 2004. "A link between wave governed random motions and ruin processes," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 205-222, October.
  11. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
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