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Sticky Continuous Processes have Consistent Price Systems

Part of: Stochastic processes Mathematical finance

Published online by Cambridge University Press:  30 January 2018

Christian Bender ,
Mikko S. Pakkanen  and
Hasanjan Sayit
Christian Bender*
Affiliation:
Saarland University
Mikko S. Pakkanen*
Affiliation:
Aarhus University and CREATES
Hasanjan Sayit*
Affiliation:
Durham University
*
∗ Postal address: Department of Mathematics, Saarland University, Postfach 151150, D-66041 Saarbrücken, Germany. Email address: bender@math.uni-sb.de
∗∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. Email address: m.pakkanen@imperial.ac.uk
∗∗∗ Postal address: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: hasanjan.sayit@durham.ac.uk
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Abstract

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Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).

Keywords

Consistent price systemstickinessmartingalearbitragetransaction costs

MSC classification

Primary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 60G44: Martingales with continuous parameter
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 586 - 594
Copyright
© Applied Probability Trust 

References

Bayraktar, E. and Sayit, H. (2010). On the stickiness property. Quant. Finance 10, 1109–1112.CrossRefGoogle Scholar
Cherny, A. (2008). Brownian moving averages have conditional full support. Ann. Appl. Prob. 18, 1825–1830.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1994). “A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.CrossRefGoogle Scholar
Gasbarra, D., Sottinen, T. and van Zanten, H. (2011). Conditional full support of Gaussian processes with stationary increments. J. Appl. Prob. 48, 561–568.CrossRefGoogle Scholar
Guasoni, P. (2006). No arbitrage under transaction costs, with fractional Brownian motion and beyond.” Math. Finance 16, 569–582.CrossRefGoogle Scholar
Guasoni, P. and Rásonyi, M. (2015). Fragility of arbitrage and bubbles in local martingale diffusion models. Finance Stoch. 19, 215–231.CrossRefGoogle Scholar
Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs.” Ann. Appl. Prob. 18, 491–520.CrossRefGoogle Scholar
Guasoni, P., Rásonyi, M. and Schachermayer, W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6, 157–191.CrossRefGoogle Scholar
Herczegh, A., Prokaj, V. and Rásonyi, M. (2014). Diversity and no arbitrage. Stoch. Anal. Appl. 32, 876–888.CrossRefGoogle Scholar
Jouini, E. and Kallal, H. (1995). Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory 66, 178–197.CrossRefGoogle Scholar
Kabanov, Y. and Stricker, C. (2008). On martingale selectors of cone-valued processes. In Séminaire de Probabilités XLI (Lecture Notes Math. 1934), Springer, Berlin, pp. 439–442.CrossRefGoogle Scholar
Maris, F., Mbakop, E. and Sayit, H. (2011). “Consistent price systems for bounded processes. Commun. Stoch. Anal. 5, 633–645.Google Scholar
Pakkanen, M. S. (2010). Stochastic integrals and conditional full support. J. Appl. Prob. 47, 650–667.CrossRefGoogle Scholar
Pakkanen, M. S. (2011). Brownian semistationary processes and conditional full support. Internat. J. Theoret. Appl. Finance 14, 579–586.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Sayit, H. and Viens, F. (2011). Arbitrage-free models in markets with transaction costs. Electron. Commun. Prob. 16, 614–622.CrossRefGoogle Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, River Edge, NJ.CrossRefGoogle Scholar