On statistical indistinguishability of the complete and incomplete markets

TL;DR

This paper demonstrates that in continuous time diffusion stock market models, the distinction between complete and incomplete markets is statistically indistinguishable due to their non-robust nature, as small deviations can switch their classification.

Contribution

It shows that market incompleteness is non-robust and that incomplete markets can be statistically indistinguishable from complete markets.

Findings

Small deviations can convert incomplete markets into complete ones.

Complete and incomplete markets have arbitrarily close stock price paths.

Market statistics cannot reliably distinguish between complete and incomplete markets.

Abstract

The possibility of statistical evaluation of the market completeness and incompleteness is investigated for continuous time diffusion stock market models. It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into a incomplete one. The paper shows that market incompleteness is also non-robust: small deviations can convert an incomplete model into a complete one. More precisely, it is shown that, for any incomplete market from a wide class of models, there exists a complete market model with arbitrarily close paths of the stock prices and the market parameters. This leads to a counterintuitive conclusion that the incomplete markets are indistinguishable from the complete markets in the terms of the market statistics.