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A discrete stochastic model for investment with an application to the transaction costs case

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  • Carassus, Laurence
  • Jouini, Elyes

Abstract

This work consists of two parts. In the first one, we study a model where the assets are investment opportunities, which are completely described by their cash-flows. Those cash-flows follow some binomial processes and have the following property called stationarity: it is possible to initiate them at any time and in any state of the world at the same condition. In such a model, we prove that the absence of arbitrage condition implies the existence of a discount rate and a particular probability measure such that the expected value of the net present value of each investment is non-positive if there are short-sales constraints and equal to zero otherwise. This extends the works of Cantor–Lippman who studied a deterministic setup. In the second part, we apply this result to a financial model in the spirit of Cox–Ross–Rubinstein Cox, but where there are transaction costs on the assets. This model appears to be stationary. At the equilibrium, the Cox–Ross–Rubinstein's price of a European option is always included between its buying and its selling price. Moreover, if there is transaction cost only on the underlying asset, the option price will be equal to the Cox–Ross–Rubinstein's price. Those results give more information than the results of Jouini–Kallal Jouini, which where working in a finite horizon model.
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  • 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.
    • Handle: RePEc:eee:mateco:v:33:y:2000:i:1:p:57-80
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    References listed on IDEAS

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    1. 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.
    2. Elyès Jouini, 2001. "Arbitrage and investment opportunities," Finance and Stochastics, Springer, vol. 5(3), pages 305-325.
    3. repec:dau:papers:123456789/5630 is not listed on IDEAS
    4. Ilan Adler & David Gale, 1997. "Arbitrage and Growth Rate for Riskless Investments in a Stationary Economy," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 73-81, January.
    5. Laurence Carassus & Elyès Jouini, 1998. "Investment and Arbitrage Opportunities with Short Sales Constraints," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 169-178, July.
    6. Cantor, David G & Lippman, Steven A, 1995. "Optimal Investment Selection with a Multitude of Projects," Econometrica, Econometric Society, vol. 63(5), pages 1231-1240, September.
    7. repec:crs:wpaper:9513 is not listed on IDEAS
    8. repec:dau:papers:123456789/5604 is not listed on IDEAS
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    10. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    11. Cantor, David G & Lippman, Steven A, 1983. "Investment Selection with Imperfect Capital Markets," Econometrica, Econometric Society, vol. 51(4), pages 1121-1144, July.
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    Cited by:

    1. Bruno Bouchard & Elyès Jouini, 2010. "Transaction Costs in Financial Models," Post-Print halshs-00703138, HAL.
    2. Jouini, Elyes, 2001. "Arbitrage and control problems in finance: A presentation," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 167-183, April.
    3. repec:dau:papers:123456789/5590 is not listed on IDEAS

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