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Equilibrium Pricing Bound on Option Prices

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  • Jouini, Elyès
  • Chazal, Marie
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    Abstract

    We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kind of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. We obtain a weak sufficient condition for strong duality and existence for the dual problem to hold, for options defined by general payoff functions. Explicit bounds are provided for the call option. Finally, we provide a numerical example.

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    Bibliographic Info

    Paper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/30.

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    Date of creation: Jun 2008
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    Publication status: Published in Mathematics and Financial Economics, 2008, Vol. 1, no. 3-4. pp. 251-281.Length: 30 pages
    Handle: RePEc:dau:papers:123456789/30

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    Related research

    Keywords: Conic duality; Semi-infinite programming; Option bounds; Equilibrium prices;

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    References

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    1. Elyès Jouini & Clotilde Napp, 2003. "A class of models satisfying a dynamical version of the CAPM," Post-Print, HAL halshs-00167159, HAL.
    2. Peter Ritchken & Shyanjaw Kuo, 1989. "On Stochastic Dominance and Decreasing Absolute Risk Averse Option Pricing Bounds," Management Science, INFORMS, INFORMS, vol. 35(1), pages 51-59, January.
    3. Bruce D. Grundy, . "Option Prices and the Underlying Asset's Return Distribution (Reprint 012)," Rodney L. White Center for Financial Research Working Papers, Wharton School Rodney L. White Center for Financial Research 11-91, Wharton School Rodney L. White Center for Financial Research.
    4. Napp, Clotilde & Jouini, Elyès, 2003. "A Class of Models satisfying a Dynamical Version of the CAPM," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/56, Paris Dauphine University.
    5. Levy, Haim, 1985. " Upper and Lower Bounds of Put and Call Option Value: Stochastic Dominance Approach," Journal of Finance, American Finance Association, American Finance Association, vol. 40(4), pages 1197-1217, September.
    6. Ritchken, Peter H, 1985. " On Option Pricing Bounds," Journal of Finance, American Finance Association, American Finance Association, vol. 40(4), pages 1219-33, September.
    7. Jouini, Elyès, 2003. "Convergence of the equilibrium prices in a family of financial models," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/359, Paris Dauphine University.
    8. Perrakis, Stylianos & Ryan, Peter J, 1984. " Option Pricing Bounds in Discrete Time," Journal of Finance, American Finance Association, American Finance Association, vol. 39(2), pages 519-25, June.
    9. Grundy, R.D., 1991. "Option Prices and the Underlying Asset's Return Distribution," Weiss Center Working Papers, Wharton School - Weiss Center for International Financial Research 11-91, Wharton School - Weiss Center for International Financial Research.
    10. Constantinides, George M. & Perrakis, Stylianos, 2002. "Stochastic dominance bounds on derivatives prices in a multiperiod economy with proportional transaction costs," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 26(7-8), pages 1323-1352, July.
    11. Lo, Andrew W., 1987. "Semi-parametric upper bounds for option prices and expected payoffs," Journal of Financial Economics, Elsevier, Elsevier, vol. 19(2), pages 373-387, December.
    12. Basso, A. & Pianca, P., 2001. "Option pricing bounds with standard risk aversion preferences," European Journal of Operational Research, Elsevier, Elsevier, vol. 134(2), pages 249-260, October.
    13. Elyès Jouini, 2003. "Convergence of the equilibrium prices in a family of financial models," Post-Print, HAL halshs-00167153, HAL.
    14. Grundy, Bruce D, 1991. " Option Prices and the Underlying Asset's Return Distribution," Journal of Finance, American Finance Association, American Finance Association, vol. 46(3), pages 1045-69, July.
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