Arbitrage and Pricing in Financial Markets with Interval Data

Abstract

Financial data are often affected by uncertainty: imprecision, incompleteness etc.. Uncertain data may be represented by intervals. Intervals may be useful for representing uncertainty in financial data or, by converse it may be useful to construct intervals from scalar financial data, for analyzing the uncertainty in the solution of real financial problems. Considering this different form of input data, a review of some financial models and definitions has been necessary. The notion of arbitrage is crucial in the modern theory of finance. It is the cornerstone of the option pricing theory. Roughly speaking a market is arbitrage-free if there is no way of making risk less profits. How to extend this definition when the returns are intervals? In the present work the definition of a system of returns which does not allow arbitrage opportunities is given for the case of interval returns. It is proved that, given a two-period economy T = (t0, t1) and n securities, the system of returns at time t0 which does not allow interval arbitrage opportunities, is an interval vector. Furthermore using the IntervalCAPM (Interval Capital Asset Pricing Model) methodology, in the present work the region of the plane, risk vs. expected return, where surely there are arbitrage opportunities is described. Some numerical results are presented: the interval beta and the interval alpha of the asset ABBOT (Abbot Laboratories), which belongs to the SP500 (Standard and Poor's 500 Composite) index, is estimated using the IntervalCAPM approach. The used algorithm has been implemented in MATLAB. The solutions obtained are always well interpretable.