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Finance and Stochastics

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01.10.2012

The fundamental theorem of asset pricing under transaction costs

verfasst von: Paolo Guasoni, Emmanuel Lépinette, Miklós Rásonyi

Erschienen in: Finance and Stochastics | Ausgabe 4/2012

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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.

Vorheriger Artikel Polynomial processes and their applications to mathematical finance

Nächster Artikel Horizon dependence of utility optimizers in incomplete models

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Henceforth, θ _τ−,θ _τ,θ _τ+ denote respectively the left limit, the value, and the right limit of θ at τ.

Lenglart [27] and, independently, Galtchouk [12, 13] offer another definition of an integral with respect to a finite-variation process, whereby \(\int_{0}^{t} S\, d\theta=\int_{0}^{t} S\,d\theta_{+} - S_{t} (\theta _{t+}-\theta_{t})\). Their definition differs from the one introduced here, and does not fit the interpretation of ∫S dθ as a cost process. Take for example S=1_[0,1)+2_[1,∞) and θ=1_[1]. The asset jumps at time 1, and the strategy buys a share immediately before the jump, only to sell it right after the jump; hence the cost \(\int_{0}^{t} S\,d\theta\) should equal −1 for t>1, reflecting a gain of 1. Yet, under their definition, \(\int_{0}^{t} S\,d\theta \) is zero. By contrast, the predictable Stieltjes integral developed in this paper entails that ∫S dθ=1_[1]−1_(1,∞), in accordance with accounting rules.

Note that in the present paper there are also transaction costs represented by the process κ; hence the additional cost term ∫_[0,T] κ _u d∥θ∥_u appears, see Definition 4.4. Remember also that ∫_[0,T] S _t dθ _t (as well as ∫_[0,T] κ _u d∥θ∥_u) represent the cost of trading with strategy θ on the interval [0,T]; cf. Definition 4.4.

The clock σ _n ticks each time that the total variation ∥θ∥ increases by at least ε _m, stopping at ρ _m.

Set C ₀=∅, \(\pi'_{n}=\inf\{ t: (\omega,t)\in B\setminus C_{n}\}\), and

https://static-content.springer.com/image/art%3A10.1007%2Fs00780-012-0185-0/MediaObjects/780_2012_185_IEq377_HTML.gif

, which is predictable because B∖C _n is predictable and

https://static-content.springer.com/image/art%3A10.1007%2Fs00780-012-0185-0/MediaObjects/780_2012_185_IEq378_HTML.gif

, by Jacod and Shiryaev [18], I.2.38.

Note that a pathwise application of Corollary A.14 does not prove Proposition A.10, which requires the predictability of θ″.

Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) CrossRef

Campi, L., Schachermayer, W.: A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch. 10, 579–596 (2006) MathSciNetMATHCrossRef

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) CrossRef

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) CrossRef

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) MathSciNetMATH

Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetMATHCrossRef

Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998) MathSciNetMATHCrossRef

Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin, (2006) MATH

Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978) MATH

10.

Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72. North-Holland, Amsterdam (1982) MATH

11.

Dybvig, P.H., Ross, S.A.: Arbitrage. The New Palgrave: A Dictionary of Economics 1, 100–106 (1987)

12.

Galtchouk, L.I.: Optional martingales. Math. USSR Sb. 40, 435–468 (1981) CrossRef

13.

Galtchouk, L.I.: Stochastic integrals with respect to optional semimartingales and random measures. Teor. Veroâtn. Ee Primen. 29, 93–107 (1984)

14.

Guasoni, P.: Optimal investment with transaction costs and without semimartingales. Ann. Appl. Probab. 12, 1227–1246 (2002) MathSciNetMATHCrossRef

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) MATHCrossRef

16.

Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979) MathSciNetMATHCrossRef

17.

Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260 (1981) MathSciNetMATHCrossRef

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) MATH

19.

Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995) MathSciNetMATHCrossRef

20.

Kabanov, Yu.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3, 237–248 (1999) MathSciNetMATHCrossRef

21.

Kabanov, Yu.M., Last, G.: Hedging under transaction costs in currency markets: a continuous-time model. Math. Finance 12, 63–70 (2002) MathSciNetMATHCrossRef

22.

Kabanov, Yu.M., Stricker, Ch.: The Harrison-Pliska arbitrage pricing theorem under transaction costs. J. Math. Econ. 35, 185–196 (2001) MathSciNetMATHCrossRef

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) CrossRef

24.

Kabanov, Yu.M., Rásonyi, M., Stricker, Ch.: No-arbitrage criteria for financial markets with efficient friction. Finance Stoch. 6, 371–382 (2002) MathSciNetMATHCrossRef

25.

Kramkov, D.O.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105, 459–479 (1996) MathSciNetMATHCrossRef

26.

Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8, 15–35 (1981) MathSciNetMATHCrossRef

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)

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) CrossRef

29.

Ross, S.A.: Return, risk and arbitrage. In: Friend, I., Bicksler, J. (eds.) Risk and Return in Finance. Ballinger, Cambridge (1977)

30.

Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976) MATH

31.

Schachermayer, W.: Martingale measures for discrete-time processes with infinite horizon. Math. Finance 4, 25–55 (1994) MathSciNetMATHCrossRef

32.

Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004) MathSciNetMATHCrossRef

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)

Titel: The fundamental theorem of asset pricing under transaction costs
verfasst von: Paolo Guasoni
Emmanuel Lépinette
Miklós Rásonyi
Publikationsdatum: 01.10.2012
Verlag: Springer-Verlag
Erschienen in: Finance and Stochastics / Ausgabe 4/2012
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-012-0185-0

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