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The Minimal Entropy Martingale Measure for Exponential Markov Chains

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Young Lee  and
Thorsten Rheinländer
Young Lee*
Affiliation:
Deutsche Bank AG and London School of Economics
Thorsten Rheinländer*
Affiliation:
Vienna University of Technology
*
∗ Postal address: Deutsche Bank AG, 1 Great Winchester Street, London EC2N 2DB, UK. Email address: young.lee@db.com
∗∗ Postal address: Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner Hauptstraße 8/105-1, 1040 Vienna, Austria. Email address: rheinlan@fam.tuwien.ac.at
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Abstract

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In this article we investigate the minimal entropy martingale measure for continuous-time Markov chains. The conditions for absence of arbitrage and existence of the minimal entropy martingale measure are discussed. Under this measure, expressions for the transition intensities are obtained. Differential equations for the arbitrage-free price are derived.

Keywords

Continuous-time Markov chainrelative entropymartingale measure

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 344 - 358
Copyright
© Applied Probability Trust 

Footnotes

The views expressed in this paper are those of the author and do not necessarily reflect the position of Deutsche Bank AG.

References

Aase, K. K. (1988). Contingent claims valuation when the security price is a combination of an Itô process and a random point process. Stoch. Process. Appl. 28, 185–220.Google Scholar
Benth, F. E. and Meyer-Brandis, T. (2005). The density process of the minimal entropy martingale measure in a stochastic volatility model with Jumps. Finance Stoch. 9, 563–575.Google Scholar
Björk, T., Kabanov, Y. and Runggaldier, W. (1997). Bond market structure in the presence of marked point processes. Math. Finance 7, 211–239.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2001). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.Google Scholar
Delbaen, F. et al. (2002). Exponential hedging and entropic penalties. Math. Finance 12, 99–123.Google Scholar
Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel, 2nd edn. Hermann, Paris.Google Scholar
Fritelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10, 39–52.Google Scholar
Grandits, P. and Rheinländer, T. (2002). On the minimal entropy martingale measure. Ann. Prob. 30, 1003–1038.CrossRefGoogle Scholar
Hubalek, F. and Sgarra, C. (2006). Esscher transforms and the minimal entropy martingale measure for exponential Lévy models. Quant. Finance 6, 125–145.CrossRefGoogle Scholar
Hull, J. and White, A. (1988). The pricing of options on assets with stochastic volatility. J. Finance 42, 281–300.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line. Springer, New York.Google Scholar
Lee, Y. and Rheinländer, T. (2012). Optimal martingale measures for defaultable assets. Stoch. Process. Appl. 122, 2870–2884.Google Scholar
Lépingle, D. and Mémin, J. (1978). Sur l'intégrabilité uniforme des martingales exponentielles. Z. Wahrscheinlichkeitsth. 42, 175–203.Google Scholar
Merton, R. C. (1976). Option pricing when underlying asset returns are discontinuous. J. Financial Econom. 3, 125–144.Google Scholar
Miyahara, Y. (2000). Minimal relative entropy martingale measure of birth and death process. Discussion paper, Nagoya City University.Google Scholar
Norberg, R. (2003). The Markov chain market. ASTIN Bull. 33, 265–287.CrossRefGoogle Scholar
Norberg, R. (2005). Anomalous PDEs in Markov chains: domains of validity and numerical solutions. Finance Stoch. 9, 519–537.Google Scholar
Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Rheinländer, T. and Steiger, G. (2006). The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models. Ann. Appl. Prob. 16, 1319–1351.Google Scholar
Rheinländer, T. and Sexton, J. (2011). Hedging Derivatives (Advanced Ser. Statist. Sci. Appl. Prob. 15). World Scientific Publishing, London.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1, 2nd edn. Cambridge University Press.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
Steiger, G. (2005). The optimal martingale measure for investors with exponential utility function. Doctoral Thesis, ETH Zürich.Google Scholar