Detection of arbitrage opportunities in multi-asset derivatives markets

We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet–Hoeffding bounds in the presence of additional information on functionals of a copula by [18], we can extend the results of [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

The Role of Item Distributions on Reliability Estimation: The Case of Cronbach’s Coefficient Alpha

Simulations concerning the distributional assumptions of coefficient alpha are contradictory. To provide a more principled theoretical framework, this article relies on the Fréchet–Hoeffding bounds, in order to showcase that the distribution of the items play a role on the estimation of correlations and covariances. More specifically, these bounds restrict the theoretical correlation range [−1, 1] such that certain correlation structures may be unfeasible. The direct implication of this result is that coefficient alpha is bounded above depending on the shape of the distributions. A general form of the Fréchet–Hoeffding bounds is derived for discrete random variables. R code and a user-friendly shiny web application are also provided so that researchers can calculate the bounds on their data.

Performance of Test Supermartingale Confidence Intervals for the Success Probability of Bernoulli Trials

Given a composite null hypothesis H0, test supermartingales are non-negative supermartingales with respect to H0 with an initial value of 1. Large values of test supermartingales provide evidence against H0. As a result, test supermartingales are an effective tool for rejecting H0, particularly when the p-values obtained are very small and serve as certificates against the null hypothesis. Examples include the rejection of local realism as an explanation of Bell test experiments in the foundations of physics and the certification of entanglement in quantum information science. Test supermartingales have the advantage of being adaptable during an experiment and allowing for arbitrary stopping rules. By inversion of acceptance regions, they can also be used to determine confidence sets. We used an example to compare the performance of test supermartingales for computing p-values and confidence intervals to Chernoff-Hoeffding bounds and the “exact” p-value. The example is the problem of inferring the probability of success in a sequence of Bernoulli trials. There is a cost in using a technique that has no restriction on stopping rules, and, for a particular test supermartingale, our study quantifies this cost.

Multivariate Distributions with Fixed Marginals and Correlations

Consider the problem of drawing random variates (X1, …, Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj. Any achievable correlation between Xi and Xj is a convex combination of these bounds. We call the value λ(Xi, Xj) ∈ [0, 1] of this convex combination the convexity parameter of (Xi, Xj) with λ(Xi, Xj) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F1, …, Fn of (X1, …, Xn), we show that λ(Xi, Xj) = λij if and only if there exist symmetric Bernoulli random variables (B1, …, Bn) (that is {0, 1} random variables with mean ½) such that λ(Bi, Bj) = λij. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

Multivariate Distributions with Fixed Marginals and Correlations

Consider the problem of drawing random variates (X 1, …, X n ) from a distribution where the marginal of each X i is specified, as well as the correlation between every pair X i and X j . For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between X i and X j . Any achievable correlation between X i and X j is a convex combination of these bounds. We call the value λ(X i , X j ) ∈ [0, 1] of this convex combination the convexity parameter of (X i , X j ) with λ(X i , X j ) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F 1, …, F n of (X 1, …, X n ), we show that λ(X i , X j ) = λ ij if and only if there exist symmetric Bernoulli random variables (B 1, …, B n ) (that is {0, 1} random variables with mean ½) such that λ(B i , B j ) = λ ij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.