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Finance and Stochastics

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01-10-2014

FTAP in finite discrete time with transaction costs by utility maximization

Authors: Jörn Sass, Martin Smaga

Published in: Finance and Stochastics | Issue 4/2014

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Abstract

The aim of this paper is to prove the fundamental theorem of asset pricing (FTAP) in finite discrete time with proportional transaction costs by utility maximization. The idea goes back to L.C.G. Rogers’ proof of the classical FTAP for a model without transaction costs. We consider one risky asset and show that under the robust no-arbitrage condition, the investor can maximize his expected utility. Using the optimal portfolio, a consistent price system is derived.

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Given \(X_{T} \in\mathcal{A}^{0}_{T}\) such that E[U(X _T)⁻]=∞, we set E[U(X _T)]:=−∞.

Barron, E.N., Cardaliaguet, P., Jensen, R.: Conditional essential suprema with applications. Appl. Math. Optim. 48, 229–253 (2003) CrossRefMATHMathSciNet

Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math., vol. 580. Springer, Berlin (1977) MATH

Campi, L., Owen, M.P.: Multivariate utility maximization with proportional transaction costs. Finance Stoch. 15, 461–499 (2011) CrossRefMathSciNet

Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6, 113–165 (1996)

Czichowsky, C., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, shadow prices, and duality in discrete time. SIAM J. Financ. Math. 5, 258–277 (2014) CrossRefMathSciNet

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) CrossRefMATHMathSciNet

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

Deelstra, G., Pham, H., Touzi, N.: Dual formulation of the utility maximization problem under transaction costs. Ann. Appl. Probab. 11, 1353–1383 (2001) CrossRefMATHMathSciNet

Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multi-period securities markets. J. Econ. Theory 20, 381–408 (1979) CrossRefMATHMathSciNet

10.

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

11.

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

12.

Kabanov, Y., Rásonyi, M., Stricker, Ch.: On the closedness of sums of convex cones in L ⁰ and the robust no-arbitrage property. Finance Stoch. 7, 403–411 (2003) CrossRefMATHMathSciNet

13.

Kabanov, Y., Safarian, M.: Markets with Transaction Costs—Mathematical Theory. Springer Finance. Springer, Berlin (2010) CrossRef

14.

Pennanen, T.: Dual representation of superhedging costs in illiquid markets. Math. Financ. Econ. 5, 233–248 (2011) CrossRefMATHMathSciNet

15.

Rásonyi, M., Stettner, L.: On utility maximization in discrete-time financial market models. Ann. Appl. Probab. 15, 1367–1395 (2005) CrossRefMATHMathSciNet

16.

Rásonyi, M.: New methods in the arbitrage theory of financial markets with transaction costs. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, Ch. (eds.) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol. 1934, pp. 455–462. Springer, Berlin (2008) CrossRef

17.

Rásonyi, M.: Arbitrage with transaction costs revisited. In: Delbaen, F., Rásonyi, M., Stricker, Ch. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance, pp. 211–226. Springer, Berlin (2009) CrossRef

18.

Rogers, L.C.G.: Equivalent martingale measures and no-arbitrage. Stoch. Stoch. Rep. 51, 41–49 (1994) CrossRefMATH

19.

Rokhlin, D.B.: A theorem on martingale selection for relatively open convex set-valued random sequences. Math. Notes 81, 543–548 (2007) CrossRefMATHMathSciNet

20.

Rokhlin, D.B.: Constructive no-arbitrage criterion under transaction costs in the case of finite discrete time. Theory Probab. Appl. 52, 93–107 (2008) CrossRefMATHMathSciNet

21.

Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ. 11, 249–257 (1992) CrossRefMATHMathSciNet

22.

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

Title: FTAP in finite discrete time with transaction costs by utility maximization
Authors: Jörn Sass
Martin Smaga
Publication date: 01-10-2014
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 4/2014
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-014-0241-z

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Other articles of this Issue 4/2014

Asymptotic arbitrage with small transaction costs

Pricing vulnerable claims in a Lévy-driven model

Superreplication under model uncertainty in discrete time

Portfolio optimization under convex incentive schemes

Asian options and meromorphic Lévy processes

Optimal investment and contingent claim valuation in illiquid markets