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Sign Tests for Dependent Observations and Bounds for Path-Dependent Options

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  • Donald Brown
  • Rustam Ibragimov
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    Abstract

    The present paper introduces new sign tests for testing for conditionally symmetric martingale-difference assumptions as well as for testing that conditional distributions of two (arbitrary) martingale-difference sequences are the same. Our analysis is based on the results that demonstrate that randomization over zero values of three-valued random variables in a conditionally symmetric martingale-difference sequence produces a stream of i.i.d. symmetric Bernoulli random variables and thus reduces the problem of estimating the critical values of the tests to computing the quantiles or moments of Binomial or normal distributions. The same is the case for randomization over ties in sign tests for equality of conditional distributions of two martingale-difference sequences. The paper also provides sharp bounds on the expected payoffs and fair prices of European call options and a wide range of path-dependent contingent claims in the trinomial financial market model in which, as is well-known, calculation of derivative prices on the base of no-arbitrage arguments is impossible. These applications show, in particular, that the expected payoff of a European call option in the trinomial model with log-returns forming a martingale-difference sequence is bounded from above by the expected payoff of a call option written on a stock with i.i.d. symmetric two-valued log-returns and, thus, reduce the problem of derivative pricing in the trinomial model with dependence to the i.i.d. binomial case. Furthermore, we show that the expected payoff of a European call option in the multiperiod trinomial option pricing model is dominated by the expected payoff of a call option in the two-period model with a log-normal asset price. These results thus allow one to reduce the problem of pricing options in the trinomial model to the case of two periods and the standard assumption of normal log-returns. We also obtain bounds on the possible fair prices of call options in the (incomplete) trinomial model in terms of the parameters of the asset's distribution. Sharp bounds completely similar to those for European call options also hold for many other contingent claims in the trinomial option pricing model, including those with an arbitrary convex increasing function as well as path-dependent ones, in particular, Asian options written on averages of the underlying asset's prices.

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    Bibliographic Info

    Paper provided by Yale School of Management in its series Yale School of Management Working Papers with number amz2581.

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    Date of creation: 01 Jun 2005
    Date of revision: 01 Jul 2005
    Handle: RePEc:ysm:somwrk:amz2581

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    Web page: http://icf.som.yale.edu/
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    Related research

    Keywords: Sign tests; dependence; martingale-difference; Bernoulli random variables; conservative tests; exact tests; option bounds; trinomial model; binomial model; semiparametric estimates; fair prices; expected payoffs; path-dependent contingent claims; effcient market hypothesis;

    References

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    2. Dufour, J.M. & Hallin, M., 1992. "Improved Eaton Bounds for Linear Combinations of Bounded Random Variables with Statistical Applications," Cahiers de recherche 9224, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
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    Cited by:
    1. Anat Bracha & Jeremy Gray & Rustam Ibragimov & Boaz Nadler & Dmitry Shapiro & Glena Ames & Donald J. Brown, 2005. "Randomized Sign Test for Dependent Observations on Discrete Choice under Risk," Cowles Foundation Discussion Papers 1526, Cowles Foundation for Research in Economics, Yale University.

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