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Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk

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Jocelyne Bion-Nadal
Abstract

We introduce, in continuous time, an axiomatic approach to assign to any financial position a dynamic ask (resp. bid) price process. Taking into account both transaction costs and liquidity risk this leads to the convexity (resp. concavity) of the ask (resp. bid) price. Time consistency is a crucial property for dynamic pricing. Generalizing the result of Jouini and Kallal, we prove that the No Free Lunch condition for a time consistent dynamic pricing procedure (TCPP) is equivalent to the existence of an equivalent probability measure $R$ that transforms a process between the bid process and the ask process of any financial instrument into a martingale. Furthermore we prove that the ask price process associated with any financial instrument is then a $R$-supermartingale process which has a cadlag modification. Finally we show that time consistent dynamic pricing allows both to extend the dynamics of some reference assets and to be consistent with any observed bid ask spreads that one wants to take into account. It then provides new bounds reducing the bid ask spreads for the other financial instruments.

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File URL: http://arxiv.org/abs/math/0703074
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Paper provided by arXiv.org in its series Quantitative Finance Papers with number math/0703074.

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Date of creation: Mar 2007
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Handle: RePEc:arx:papers:math/0703074

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, vol. 9(2), pages 269-298, 04. [Downloadable!] (restricted)
  2. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany. [Downloadable!]
  3. Jocelyne Bion-Nadal, 2006. "Time Consistent Dynamic Risk Processes, Cadlag Modification," Quantitative Finance Papers math/0607212, arXiv.org. [Downloadable!]
  4. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October. [Downloadable!] (restricted)
  5. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2006. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 10(3), pages 427-448, September. [Downloadable!] (restricted)
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