Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T08:47:28.763Z Has data issue: false hasContentIssue false
  • English
  • Français

The Markov Chain Market

Published online by Cambridge University Press:  17 April 2015

Ragnar Norberg
Ragnar Norberg*
Affiliation:
London School of Economics and Political Science, Department of Statistics, Houghton Street, London WC2A 2AE, United Kingdom, E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a financial market driven by a continuous time homogeneous Markov chain. Conditions for absence of arbitrage and for completeness are spelled out, non-arbitrage pricing of derivatives is discussed, and details are worked out for some cases. Closed form expressions are obtained for interest rate derivatives. Computations typically amount to solving a set of first order partial differential equations. An excursion into risk minimization in the incomplete case illustrates the matrix techniques that are instrumental in the model.

Keywords

Continuous time Markov chainsMartingale analysisArbitrage pricing theoryRisk minimizationUnit linked insurance
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 2 , November 2003 , pp. 265 - 287
Copyright
Copyright © ASTIN Bulletin 2003

References

Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. Springer-Verlag.CrossRefGoogle Scholar
Björk, T., Kabanov, Y. and Runggaldier, W. (1997) Bond market structures in the presence of marked point processes. Mathematical Finance 7, 211–239.CrossRefGoogle Scholar
Björk, T. (1998) Arbitrage Theory in Continuous Time, Oxford University Press.CrossRefGoogle Scholar
Cox, J., Ross, S. and Rubinstein, M. (1979) Option pricing: A simplified approach. J. of Financial Economics 7, 229–263.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1994) A general version of the fundamental theorem on asset pricing. Mathematische Annalen 300, 463–520.CrossRefGoogle Scholar
Eberlein, E. and Raible, S. (1999) Term structure models driven by general Levy processes. Mathematical Finance 9, 31–53.CrossRefGoogle Scholar
Elliott, R.J. and Kopp, P.E. (1998) Mathematics of financial markets, Springer-Verlag.Google Scholar
Föllmer, H. and Sondermann, D. (1986) Hedging of non-redundant claims. In Contributions to Mathematical Economics in Honor ofGerard Debreu, 205–223, eds. Hildebrand, W., Mas-Collel, A., North-Holland.Google Scholar
Gantmacher, F.R. (1959) Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
Harrison, J.M. and Kreps, D.M. (1979) Martingales and arbitrage in multi-period securities markets. J. Economic Theory 20, 381–408.CrossRefGoogle Scholar
Harrison, J.M. and Pliska, S. (1981) Martingales and stochastic integrals in the theory of continuous trading. J. Stoch. Proc. and Appl. 11, 215–260.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. (1975) A first Course in Stochastic Processes, 2nd. ed., Academic Press.Google Scholar
Merton, R.C. (1976) Option pricing when underlying stock returns are discontinuous. J. Financial Economics 3, 125–144.CrossRefGoogle Scholar
Møller, T. (1998) Risk minimizing hedging strategies for unit-linked life insurance. ASTIN Bulletin 28, 17–47.CrossRefGoogle Scholar
Norberg, R. (1995) A time-continuous Markov chain interest model with applications to insurance. J. Appl. Stoch. Models and Data Anal., 245–256.CrossRefGoogle Scholar
Norberg, R. (1999) On the Vandermonde matrix and its role in mathematical finance. Working paper No. 162, Laboratory of Actuarial Math., Univ. Copenhagen.Google Scholar
Norberg, R. (2002) Anomalous PDEs in Markov chains: domains of validity and numerical solutions. Research Report, Department of Statistics, London School of Economics: http:// stats.lse.ac.uk/norberg. (Submitted to Finance and Stochastics).Google Scholar
Pliska, S.R. (1997) Introduction to Mathematical Finance, Blackwell Publishers.Google Scholar