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.
Weak convergence of partial sums and multilinear forms in independent random variables and linear processes to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and conceptually simple method for obtaining such forms of convergence. The method relies on the fact that the econometric quantities of interest involve discrete time martingales or semimartingales and shows how in the limit these quantities become continuous martingales and semimartingales. The limit theory itself uses very general convergence results for semimartingales that were obtained in work by Jacod and Shiryaev (2003). The theory that is developed here is applicable in a wide range of econometric models and many examples are given.
One notable outcome of the new approach is that it provides a unified treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity, as both these cases are subsumed within the martingale convergence approach and different rates of convergence are accommodated in a natural way. The approach is also useful in developing asymptotics for certain nonlinear functions of integrated processes, which are now receiving attention in econometric applications, and some new results in this area are presented. The paper is partly of pedagogical interest and the conceptual simplicity of the methods is appealing. Since this is the first time the methods have been used in econometrics, the exposition is presented in some detail with illustrations of new derivations of some well-known existing results, as well as some new asymptotic results and the unification of the limit theory for autoregression.