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Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches

We consider simple models of financial markets with regular traders and insiders possessing some extra information hidden in a random variable that is accessible to the regular trader only at the end of the trading interval. The problems we focus on are the calculation of the additional utility of the insider and a study of his free lunch possibilities. The information drift—that is, the drift to eliminate in order to preserve the martingale property in the insider's filtration—turns out to be the crucial quantity needed to answer these questions. It is most elegantly described by the logarithmic Malliavin trace of the conditional laws of the insider information with respect to the filtration of the regular trader. Several examples are given to illustrate additional utility and free lunch possibilities. In particular, if the insider has advance knowledge of the maximal stock price process, given by a regular diffusion, arbitrage opportunities exist.