# The fundamental theorem of asset pricing under transaction costs

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## Abstract

This paper proves the fundamental theorem of asset pricing with transaction costs, when bid and ask prices follow locally bounded càdlàg (right-continuous, left-limited) processes.

The *robust no free lunch with vanishing risk condition* (RNFLVR) for simple strategies is equivalent to the existence of a *strictly consistent price system* (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The RNFLVR condition implies that admissible strategies are predictable processes of finite variation.

The Appendix develops an extension of the familiar Stieltjes integral for càdlàg integrands and finite-variation integrators, which is central to modelling transaction costs with discontinuous prices.

## Keywords

Arbitrage Fundamental theorem of asset pricing Transaction costs Admissible strategies Finite variation## Mathematics Subject Classification

91B28 62P05 26A45 60H05## JEL Classification

G12## Notes

### Acknowledgements

We are indebted to Walter Schachermayer for many stimulating discussions which inspired this paper. We also thank Friedrich Hubalek for useful comments, and Martin Schweizer for a careful reading, which helped correct some gaps in a earlier version.

Partially supported by NSF (DMS-0807994 and DMS-1109047), SFI (07/MI/008, 07/SK/M1189, 08/SRC/FMC1389), the ERC (278295), FP7 (RG-248896), the Hungarian Science Foundation (OTKA, F 049094), and the Austrian Science Fund (FWF, P 19456), at the Financial and Actuarial Mathematics Research Unit of Vienna University of Technology.

## References

- 1.Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ.
**81**, 637–654 (1973) CrossRefGoogle Scholar - 2.Campi, L., Schachermayer, W.: A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch.
**10**, 579–596 (2006) MathSciNetMATHCrossRefGoogle Scholar - 3.Cherny, A.: General arbitrage pricing model. II. Transaction costs. In: Séminaire de Probabilités XL. Lecture Notes in Math., vol. 1899, pp. 447–461. Springer, Berlin (2007) CrossRefGoogle Scholar
- 4.Choulli, T., Stricker, Ch.: Séparation d’une sur- et d’une sousmartingale par une martingale. In: Séminaire de Probabilités, XXXII. Lecture Notes in Math., vol. 1686, pp. 67–72. Springer, Berlin (1998) CrossRefGoogle Scholar
- 5.Dalang, R.C., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep.
**29**, 185–201 (1990) MathSciNetMATHGoogle Scholar - 6.Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann.
**300**, 463–520 (1994) MathSciNetMATHCrossRefGoogle Scholar - 7.Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann.
**312**, 215–250 (1998) MathSciNetMATHCrossRefGoogle Scholar - 8.Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin, (2006) MATHGoogle Scholar
- 9.Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978) MATHGoogle Scholar
- 10.Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72. North-Holland, Amsterdam (1982) MATHGoogle Scholar
- 11.Dybvig, P.H., Ross, S.A.: Arbitrage. The New Palgrave: A Dictionary of Economics
**1**, 100–106 (1987) Google Scholar - 12.Galtchouk, L.I.: Optional martingales. Math. USSR Sb.
**40**, 435–468 (1981) CrossRefGoogle Scholar - 13.Galtchouk, L.I.: Stochastic integrals with respect to optional semimartingales and random measures. Teor. Veroâtn. Ee Primen.
**29**, 93–107 (1984) Google Scholar - 14.Guasoni, P.: Optimal investment with transaction costs and without semimartingales. Ann. Appl. Probab.
**12**, 1227–1246 (2002) MathSciNetMATHCrossRefGoogle Scholar - 15.Guasoni, P., Rásonyi, M., Schachermayer, W.: The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance
**6**, 157–191 (2010) MATHCrossRefGoogle Scholar - 16.Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory
**20**, 381–408 (1979) MathSciNetMATHCrossRefGoogle Scholar - 17.Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl.
**11**, 215–260 (1981) MathSciNetMATHCrossRefGoogle Scholar - 18.Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (2003) MATHGoogle Scholar
- 19.Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory
**66**, 178–197 (1995) MathSciNetMATHCrossRefGoogle Scholar - 20.Kabanov, Yu.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch.
**3**, 237–248 (1999) MathSciNetMATHCrossRefGoogle Scholar - 21.Kabanov, Yu.M., Last, G.: Hedging under transaction costs in currency markets: a continuous-time model. Math. Finance
**12**, 63–70 (2002) MathSciNetMATHCrossRefGoogle Scholar - 22.Kabanov, Yu.M., Stricker, Ch.: The Harrison-Pliska arbitrage pricing theorem under transaction costs. J. Math. Econ.
**35**, 185–196 (2001) MathSciNetMATHCrossRefGoogle Scholar - 23.Kabanov, Yu.M., Stricker, Ch.: A teachers’ note on no-arbitrage criteria. In: Séminaire de Probabilités, XXXV. Lecture Notes in Math., vol. 1755, pp. 149–152. Springer, Berlin (2001) CrossRefGoogle Scholar
- 24.Kabanov, Yu.M., Rásonyi, M., Stricker, Ch.: No-arbitrage criteria for financial markets with efficient friction. Finance Stoch.
**6**, 371–382 (2002) MathSciNetMATHCrossRefGoogle Scholar - 25.Kramkov, D.O.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields
**105**, 459–479 (1996) MathSciNetMATHCrossRefGoogle Scholar - 26.Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ.
**8**, 15–35 (1981) MathSciNetMATHCrossRefGoogle Scholar - 27.Lenglart, E.: Tribus de Meyer et théorie des processus. In: Séminaire de Probabilité, XIV. Lecture Notes in Math., vol. 784, pp. 500–546. Springer, Berlin (1980) Google Scholar
- 28.Rásonyi, M.: A remark on the superhedging theorem under transaction costs. In: Séminaire de Probabilités XXXVII. Lecture Notes in Math., vol. 1832, pp. 394–398. Springer, Berlin (2003) CrossRefGoogle Scholar
- 29.Ross, S.A.: Return, risk and arbitrage. In: Friend, I., Bicksler, J. (eds.) Risk and Return in Finance. Ballinger, Cambridge (1977) Google Scholar
- 30.Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976) MATHGoogle Scholar
- 31.Schachermayer, W.: Martingale measures for discrete-time processes with infinite horizon. Math. Finance
**4**, 25–55 (1994) MathSciNetMATHCrossRefGoogle Scholar - 32.Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance
**14**, 19–48 (2004) MathSciNetMATHCrossRefGoogle Scholar - 33.Yan, J.-A.: Caractérisation d’une classe d’ensembles convexes de
*L*^{1}ou*H*^{1}. In: Séminaire de Probabilité, XIV. Lecture Notes in Math., vol. 784, pp. 220–222. Springer, Berlin (1980) Google Scholar