# Consistent price systems and face-lifting pricing under transaction costs

## Author Info

• Paolo Guasoni
• Mikl\'os R\'asonyi
• Walter Schachermayer
Registered author(s):

## Abstract

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion. Using the constructed price systems, we show, under very general assumptions, the following face-lifting'' result: the asymptotic superreplication price of a European contingent claim $g(S_T)$ equals $\hat{g}(S_0)$, where $\hat{g}$ is the concave envelope of $g$ and $S_t$ is the price of the asset at time $t$. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.

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File URL: http://arxiv.org/pdf/0803.4416

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 0803.4416.

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Publication status: Published in Annals of Applied Probability 2008, Vol. 18, No. 2, 491-520
Handle: RePEc:arx:papers:0803.4416

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Web page: http://arxiv.org/

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## References

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1. repec:fth:inseep:9513 is not listed on IDEAS
2. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-313, September.
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5. HuyËn Pham & Nizar Touzi & Jaksa Cvitanic, 1999. "A closed-form solution to the problem of super-replication under transaction costs," Finance and Stochastics, Springer, vol. 3(1), pages 35-54.
6. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, vol. 10(4), pages 579-596, December.
7. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
8. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
9. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
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11. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 217-230, August.
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## Citations

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Cited by:
1. Biagini, Francesca & Fink, Holger & Klüppelberg, Claudia, 2013. "A fractional credit model with long range dependent default rate," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1319-1347.
2. Emmanuel Denis & Yuri Kabanov, 2012. "Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs," Finance and Stochastics, Springer, vol. 16(1), pages 135-154, January.
3. Attila Herczegh & Vilmos Prokaj & Mikl\'os R\'asonyi, 2013. "Diversity and no arbitrage," Papers 1301.4173, arXiv.org.
4. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
5. Christian Bender & Mikko S. Pakkanen & Hasanjan Sayit, 2013. "Sticky continuous processes have consistent price systems," Papers 1310.7857, arXiv.org, revised Nov 2013.
6. Paolo Guasoni & Miklós Rásonyi & Walter Schachermayer, 2010. "The fundamental theorem of asset pricing for continuous processes under small transaction costs," Annals of Finance, Springer, vol. 6(2), pages 157-191, March.
7. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2013. "On the existence of shadow prices," Finance and Stochastics, Springer, vol. 17(4), pages 801-818, October.
8. Erhan Bayraktar & Hasanjan Sayit, 2010. "On the stickiness property," Quantitative Finance, Taylor & Francis Journals, vol. 10(10), pages 1109-1112.
9. Erhan Bayraktar & Xiang Yu, 2013. "On the Market Viability under Proportional Transaction Costs," Papers 1312.3917, arXiv.org.
10. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633, arXiv.org, revised Jan 2013.
11. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
12. S. Gerhold & J. Muhle-Karbe & W. Schachermayer, 2013. "The dual optimizer for the growth-optimal portfolio under transaction costs," Finance and Stochastics, Springer, vol. 17(2), pages 325-354, April.
13. Yan Dolinsky & Halil Soner, 2013. "Duality and convergence for binomial markets with friction," Finance and Stochastics, Springer, vol. 17(3), pages 447-475, July.
14. Christian Bender, 2012. "Simple arbitrage," Papers 1210.5391, arXiv.org.
15. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.
16. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
17. Christian Bender & Mikko S. Pakkanen & Hasanjan Sayit, 2013. "Sticky continuous processes have consistent price systems," CREATES Research Papers 2013-38, School of Economics and Management, University of Aarhus.
18. Paolo Guasoni & Johannes Muhle-Karbe, 2012. "Portfolio Choice with Transaction Costs: a User's Guide," Papers 1207.7330, arXiv.org.
19. Peter Bank & Selim G\"okay, 2013. "Superreplication when trading at market indifference prices," Papers 1310.3113, arXiv.org.
20. Nils Chr. Framstad, 2012. "On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes," Papers 1206.5756, arXiv.org, revised Jun 2012.

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