Information geometry provides a correspondence between differential geometry and statistics through the Fisher information matrix. In particular, given two models from the same parametric family of distributions, one can define the distance between these models as the length of the geodesic connecting them in a Riemannian manifold whose metric is given by the model’s Fisher information matrix. One limitation that has hindered the adoption of this similarity measure in practical applications is that the Fisher distance is typically difficult to compute in a robust manner. We review such complications and provide a general form for the distance function for one parameter model. We next focus on higher dimensional extreme value models including the generalized Pareto and generalized extreme value distributions that will be used in financial risk applications. Specifically, we first develop a technique to identify the nearest neighbors of a target security in the sense that their best fit model distributions have minimal Fisher distance to the target. Second, we develop a hierarchical clustering technique that utilizes the Fisher distance. Specifically, we compare generalized extreme value distributions fit to block maxima of a set of equity loss distributions and group together securities whose worst single day yearly loss distributions exhibit similarities.

We propose a theoretical framework for the detection and identification of triangular arbitrage opportunities between currency exchange rates in the spot foreign exchange market. We obtain sufficient conditions for the exclusion of triangular arbitrage opportunities in the setting of non-trivial transaction costs in terms of the currency rates of the market under consideration. Then we propose an efficient computational approach which can detect triangular arbitrage opportunities in real time. Finally, we consider numerical studies that utilize spot currency exchange rate quotes to substantiate and present applications of the theoretical findings as well as to demonstrate the efficiency of the proposed computational arbitrage detection and identification methods.

We propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options with an underlying asset that evolves according to a one-dimensional Markov process, which includes diffusion and jump-diffusion processes. The prices and Greeks of a discretely monitored double barrier option are explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice—for example, time-dependent barriers and nonuniform monitoring time intervals. We provide numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature in a unified framework. TOPICS:Derivatives, options Key Findings ▪ Markov chain approximations to stochastic processes are used to develop a simple closed-form pricing formula for discretely monitored barrier options. ▪ This article explores prices and Greeks computations within the constant elasticity of variance model and Merton’s and Kou’s jump-diffusions models. ▪ Numerical examples are provided to demonstrate the robustness and accuracy of the pricing technique both to re-create specialized results in related barrier option literature and to extend to multiple novel settings.

The time dependent notion of equity market centrality can uncover the influence of the pairwise and risk evolution of securities with respect to system stability