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Arbitrage and Control Problems in Finance. Presentation

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Author Info
Elyès Jouini () (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - CNRS : UMR7534 - Université Paris Dauphine - Paris IX)

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Abstract

The theory of asset pricing takes its roots in the Arrow-Debreu model (see,for instance, Debreu 1959, Chap. 7), the Black and Scholes (1973) formula,and the Cox and Ross (1976) linear pricing model. This theory and its link to arbitrage has been formalized in a general framework by Harrison and Kreps (1979), Harrison and Pliska (1981, 1983), and Du¢e and Huang (1986). In these models, security markets are assumed to be frictionless: securities can be sold short in unlimited amounts, the borrowing and lending rates are equal, and there is no transaction cost. The main result is that the price process of traded securities is arbitrage free if and only if there exists some equivalent probability measure that transforms it into a martingale, when normalized by the numeraire. Contingent claims can then be priced by taking the expected value of their (normalized) payo§ with respect to any equivalent martingale measure. If this value is unique, the claim is said to be priced by arbitrage and it can be perfectly hedged (i.e. duplicated) by dynamic trading. When the markets are dynamically complete, there is only one such a and any contingent claim is priced by arbitrage. The of each state of the world for this probability measure can be interpreted as the state price of the economy (the prices of $1 tomorrow in that state of the world) as well as the marginal utilities (for consumption in that state of the world) of rational agents maximizing their expected utility.

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Paper provided by HAL in its series Post-Print with number halshs-00167152_v1.

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Date of creation: 2001
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Publication status: Published, Journal of Mathematical Economics, 2001, 35, 167-183
Handle: RePEc:hal:journl:halshs-00167152_v1

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Keywords: arbitrage; control problem;

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  1. Dokuchaev, Nikolai & Yu Zhou, Xun, 2001. "Optimal investment strategies with bounded risks, general utilities, and goal achieving," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 289-309, April. [Downloadable!] (restricted)
  2. Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October. [Downloadable!] (restricted)
  3. Elyès Jouini, 2003. "Market imperfections, equilibrium and arbitrage," Finance 0312005, EconWPA. [Downloadable!]
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  4. Nicole El Karoui & Monique Jeanblanc-Picqué, 1998. "Optimization of consumption with labor income," Finance and Stochastics, Springer, vol. 2(4), pages 409-440. [Downloadable!] (restricted)
  5. Elyès Jouini & Clotilde Napp, 1999. "Arbitrage and Investment Opportunities," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-034, New York University, Leonard N. Stern School of Business-. [Downloadable!]
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  6. Darrell Duffie & William Zame, 1988. "The Consumption-Based Capital Asset Pricing Model," Discussion Papers 88-10, University of Copenhagen. Department of Economics.
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  7. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June. [Downloadable!] (restricted)
  8. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April. [Downloadable!] (restricted)
  9. Hua He and Neil D. Pearson., 1989. "Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: The Infinite Dimensional Case," Research Program in Finance Working Papers RPF-191, University of California at Berkeley.
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  10. Elyes Jouini & Clotilde Napp, 1999. "Continuous Time Equilibrium Pricing of Nonredundant Assets," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-008, New York University, Leonard N. Stern School of Business-. [Downloadable!]
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  11. Elyès Jouini & Hédi Kallal, 1999. "Viability and Equilibrium in Securities Markets with Frictions," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-036, New York University, Leonard N. Stern School of Business-. [Downloadable!]
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  12. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-95, June. [Downloadable!] (restricted)
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  18. Carassus, Laurence & Jouini, Elyes, 2000. "A discrete stochastic model for investment with an application to the transaction costs case," Journal of Mathematical Economics, Elsevier, vol. 33(1), pages 57-80, February. [Downloadable!] (restricted)
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  22. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica, Econometric Society, vol. 53(2), pages 363-84, March. [Downloadable!] (restricted)
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  26. 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. [Downloadable!] (restricted)
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  1. 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. [Downloadable!] (restricted)
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