Insiders and their Free Lunches: the Role of Short Positions

Given a stock price process, we analyse the potential of arbitrage by insiders in a context of short-selling prohibitions. We introduce the notion of minimal supermartingale measure, and we analyse its properties in connection to the minimal martingale measure. In particular, we establish conditions when both fail to exist. These correspond to the case when the insider's information set includes some non null events that are perceived as having null probabilities by the uninformed market investors. These results may have different applications, such as in problems related to the local risk-minimisation for insiders whenever strategies are implemented without short selling.