Efficiency testing of prediction markets: Martingale approach, likelihood ratio and Bayes factor analysis

This paper studies efficient market hypothesis in prediction markets and the results are illustrated for the in-play football betting market using the quoted odds for the English Premier League. Our analysis is based on the martingale property, where the last quoted probability should be the best predictor of the outcome and all previous quotes should be statistically insignificant. We use regression analysis to test for the significance of the previous quotes in both the time setup and the spatial setup based on stopping times, when the quoted probabilities reach certain bounds. The main contribution of this paper is to show how a potentially different distributional opinion based on the violation of the market efficiency can be monetized by optimal trading, where the agent maximizes logarithmic utility function. In particular, the trader can realize a trading profit that corresponds to the likelihood ratio in the situation of one market maker and one market taker, or the Bayes factor in the situation of two or more market takers.