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

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.