Free lunch and arbitrage possibilities in a financial market model with an insider

We consider financial market models based on Wiener space with two agents on different information levels: a regular agent whose information is contained in the natural filtration of the Wiener process W, and an insider who possesses some extra information from the beginning of the trading interval, given by a random variable L which contains information from the whole time interval. Our main concern are variables L describing the maximum of a pricing rule. Since for such L the conditional laws given by the smaller knowledge of the regular trader up to fixed times are not absolutely continuous with respect to the law of L, this class of examples cannot be treated by means of the enlargement of filtration techniques as applied so far. We therefore use elements of a Malliavin and Itô calculus for measure-valued random variables to give criteria for the preservation of the semimartingale property, the absolute continuity of the conditional laws of L with respect to its law, and the absence of arbitrage. The master example, given by supt∈[0,1] Wt, preserves the semimartingale property, but allows for free lunch with vanishing risk quite generally. We deduce conditions on drift and volatility of price processes, under which we can construct explicit arbitrage strategies.