On statistical indistinguishability of complete and incomplete market models

Purpose This paper aims to investigate possibility of statistical detection of market completeness for continuous time diffusion stock market models. Design/methodology/approach The paper uses theory of forecasting to find criteria of predictability of market parameters such as volatilities and the appreciation rates. Findings It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into an incomplete one. The paper shows that market incompleteness is also non-robust: 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. Originality/value The paper results lead to a counterintuitive conclusion that the incomplete markets are indistinguishable in the terms of the market statistics.

Good Deal Bounds

In this chapter we study an incomplete market, but we do not look for a unique martingale measure. Instead we try to find “reasonable” bounds on arbitrage free prices. The terms “reasonable” is formalized in terms of a price rule with bounded Sharpe ratio–so-called good deal bounds. We study a factor model and show that the good deal bounds can be obtained by solving a control problem where the likelihood process acts as a state variable, and the Girsanov kernel is the control variable. The theory is then applied to concrete examples.