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Characterizing Attainable Claims: A New Proof

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Michael R. Tehranchi
Michael R. Tehranchi*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: m.tehranchi@statslab.cam.ac.uk
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Abstract

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This short note offers a new proof of the following fact: in a discrete-time arbitrage-free market model, a contingent claim is attainable if and only if its expected value is the same under all equivalent martingale measures. The proof is based on Rogers' (1994) proof of the Dalang-Morton-Willinger (1990) theorem.

Keywords

Fundamental theorem of asset pricingattainable contingent claim

MSC classification

Secondary: 60G42: Martingales with discrete parameter 91B25: Asset pricing models
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1013 - 1022
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Dalang, R. C., Morton, A. and Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Reports 29, 185–201.CrossRefGoogle Scholar
[2] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.CrossRefGoogle Scholar
[3] Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer, Berlin.Google Scholar
[4] Harrison, J. M. and Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408.Google Scholar
[5] Harrison, J. M. and Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260.Google Scholar
[6] Jacka, S. D. (1992). A martingale representation result and an application to incomplete financial markets. Math. Finance 2, 23–34.Google Scholar
[7] Jacod, J. and Shiryaev, A. N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2, 259–273.CrossRefGoogle Scholar
[8] Kabanov, Y. and Stricker, C. (2001). A teachers' note on no-arbitrage criteria. In Séminaire de Probabilités XXXV (Lecture Notes Math. 1755), Springer, Berlin, pp. 149–152.Google Scholar
[9] Rogers, L. C. G. (1994). Equivalent martingale measures and no-arbitrage. Stoch. Stoch. Reports 51, 41–49.CrossRefGoogle Scholar
[10] Rogers, L. C. G. (1998). The origins of risk-neutral pricing and the Black–Scholes formula. In Handbook of Risk Management and Analysis, ed. Alexander, C. O., John Wiley, Chichester, pp. 81–94.Google Scholar
[11] Ross, S. A. (1978). A simple approach to the valuation of risky streams. J. Business 51, 453–475.CrossRefGoogle Scholar