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The fundamental theorem of asset pricing under proportional transaction costs

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  • Alet Roux

Abstract

We extend the fundamental theorem of asset pricing to a model where the risky stock is subject to proportional transaction costs in the form of bid-ask spreads and the bank account has different interest rates for borrowing and lending. We show that such a model is free of arbitrage if and only if one can embed in it a friction-free model that is itself free of arbitrage, in the sense that there exists an artificial friction-free price for the stock between its bid and ask prices and an artificial interest rate between the borrowing and lending interest rates such that, if one discounts this stock price by this interest rate, then the resulting process is a martingale under some non-degenerate probability measure. Restricting ourselves to the simple case of a finite number of time steps and a finite number of possible outcomes for the stock price, the proof follows by combining classical arguments based on finite-dimensional separation theorems with duality results from linear optimisation.

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  • Alet Roux, 2007. "The fundamental theorem of asset pricing under proportional transaction costs," Papers 0710.2758, arXiv.org.
    • Handle: RePEc:arx:papers:0710.2758
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    1. Napp, C., 2003. "The Dalang-Morton-Willinger theorem under cone constraints," Journal of Mathematical Economics, Elsevier, vol. 39(1-2), pages 111-126, February.
    2. A. Bensoussan & H. Julien, 2000. "On the Pricing of Contingent Claims with Frictions," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 89-108, April.
    3. 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.
    4. Gondzio, Jacek & Kouwenberg, Roy & Vorst, Ton, 2003. "Hedging options under transaction costs and stochastic volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1045-1068, April.
    5. Igor V. Evstigneev & Klaus Schürger & Michael I. Taksar, 2004. "On The Fundamental Theorem Of Asset Pricing: Random Constraints And Bang‐Bang No‐Arbitrage Criteria," Mathematical Finance, Wiley Blackwell, vol. 14(2), pages 201-221, April.
    6. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    7. Edirisinghe, Chanaka & Naik, Vasanttilak & Uppal, Raman, 1993. "Optimal Replication of Options with Transactions Costs and Trading Restrictions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(1), pages 117-138, March.
    8. Perrakis, Stylianos & Lefoll, Jean, 1997. "Derivative Asset Pricing with Transaction Costs: An Extension," Computational Economics, Springer;Society for Computational Economics, vol. 10(4), pages 359-376, November.
    9. He, Hua & Pages, Henri F, 1993. "Labor Income, Borrowing Constraints, and Equilibrium Asset Prices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(4), pages 663-696, October.
    10. Clotilde Napp, 2001. "Pricing Issues with Investment Flows," Post-Print halshs-00151401, HAL.
    11. Alexander Melnikov & Yury Petrachenko, 2005. "On option pricing in binomial market with transaction costs," Finance and Stochastics, Springer, vol. 9(1), pages 141-149, January.
    12. Sid Browne, 1995. "Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 937-958, November.
    13. (*), Thaleia Zariphopoulou & George M. Constantinides, 1999. "Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences," Finance and Stochastics, Springer, vol. 3(3), pages 345-369.
    14. A. Bizid & E. Jouini, 2001. "Incomplete Markets And Short-Sales Constraints: An Equilibrium Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(02), pages 211-243.
    15. Jouini, Elyes, 2000. "Price functionals with bid-ask spreads: an axiomatic approach," Journal of Mathematical Economics, Elsevier, vol. 34(4), pages 547-558, December.
    16. repec:dau:papers:123456789/5630 is not listed on IDEAS
    17. Wendell H. Fleming & Thaleia Zariphopoulou, 1991. "An Optimal Investment/Consumption Model with Borrowing," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 802-822, November.
    18. George M. Constantinides & Thaleia Zariphopoulou, 2001. "Bounds on Derivative Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 331-346, July.
    19. Napp, Clotilde, 2001. "Pricing issues with investment flows Applications to market models with frictions," Journal of Mathematical Economics, Elsevier, vol. 35(3), pages 383-408, June.
    20. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    21. Elyégs Jouini & Hédi Kallal, 1995. "Arbitrage In Securities Markets With Short‐Sales Constraints," Mathematical Finance, Wiley Blackwell, vol. 5(3), pages 197-232, July.
    22. Freddy Delbaen & Yuri M. Kabanov & Esko Valkeila, 2002. "Hedging under Transaction Costs in Currency Markets: a Discrete‐Time Model," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 45-61, January.
    23. Perrakis, Stylianos & Lefoll, Jean, 2000. "Option pricing and replication with transaction costs and dividends," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1527-1561, October.
    24. Albanese, Claudio & Tompaidis, Stathis, 2008. "Small transaction cost asymptotics and dynamic hedging," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1404-1414, March.
    25. Bernard Bensaid & Jean‐Philippe Lesne & Henri Pagès & José Scheinkman, 1992. "Derivative Asset Pricing With Transaction Costs1," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 63-86, April.
    26. Laurence Carassus & Elyès Jouini, 1998. "Investment and Arbitrage Opportunities with Short Sales Constraints," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 169-178, July.
    27. repec:dau:papers:123456789/5599 is not listed on IDEAS
    28. Tepla, Lucie, 2000. "Optimal portfolio policies with borrowing and shortsale constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1623-1639, October.
    29. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    30. repec:crs:wpaper:9514 is not listed on IDEAS
    31. Cuoco, Domenico, 1997. "Optimal Consumption and Equilibrium Prices with Portfolio Constraints and Stochastic Income," Journal of Economic Theory, Elsevier, vol. 72(1), pages 33-73, January.
    32. Shunming Zhang & Chunlei Xu & Xiaotie Deng, 2002. "Dynamic Arbitrage‐Free Asset Pricing with Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 89-97, January.
    33. Prasad Chalasani & Somesh Jha, 2001. "Randomized Stopping Times and American Option Pricing with Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 33-77, January.
    34. Pham, Huyen & Touzi, Nizar, 1999. "The fundamental theorem of asset pricing with cone constraints," Journal of Mathematical Economics, Elsevier, vol. 31(2), pages 265-279, March.
    35. Laurence Carassus & Huye^n Pham & Nizar Touzi, 2001. "No Arbitrage in Discrete Time Under Portfolio Constraints," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 315-329, July.
    36. Jér^me Detemple & Marcel Rindisbacher, 2005. "Closed‐Form Solutions For Optimal Portfolio Selection With Stochastic Interest Rate And Investment Constraints," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 539-568, October.
    37. Jörn Sass, 2005. "Portfolio optimization under transaction costs in the CRR model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(2), pages 239-259, June.
    38. (**), Christophe Stricker & (*), Miklós Rásonyi & Yuri Kabanov, 2002. "No-arbitrage criteria for financial markets with efficient friction," Finance and Stochastics, Springer, vol. 6(3), pages 371-382.
    39. Nicole El Karoui & Monique Jeanblanc-Picqué, 1998. "Optimization of consumption with labor income," Finance and Stochastics, Springer, vol. 2(4), pages 409-440.
    40. Kabanov, Yu. M. & Stricker, Ch., 2001. "The Harrison-Pliska arbitrage pricing theorem under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 185-196, April.
    41. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    42. Boyle, Phelim P & Vorst, Ton, 1992. "Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-293, March.
    43. Perrakis, Stylianos & Lefoll, Jean, 2004. "The American put under transactions costs," Journal of Economic Dynamics and Control, Elsevier, vol. 28(5), pages 915-935, February.
    44. repec:crs:wpaper:9513 is not listed on IDEAS
    45. repec:jbs:wpaper:062005 is not listed on IDEAS
    46. repec:dau:papers:123456789/5594 is not listed on IDEAS
    47. Fulvio Ortu, 2001. "Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 24(2), pages 79-105, November.
    48. repec:dau:papers:123456789/5604 is not listed on IDEAS
    49. M. Dempster & I. Evstigneev & M. Taksar, 2006. "Asset Pricing and Hedging in Financial Markets with Transaction Costs: An Approach Based on the Von Neumann–Gale Model," Annals of Finance, Springer, vol. 2(4), pages 327-355, October.
    50. Lukasz Stettner, 2000. "Option Pricing in Discrete‐Time Incomplete Market Models," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 305-321, April.
    51. Ioannis Karatzas & (*), S. G. Kou, 1998. "Hedging American contingent claims with constrained portfolios," Finance and Stochastics, Springer, vol. 2(3), pages 215-258.
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