A time before which insiders would not undertake risk

Abstract

A continuous-path semimartingale market model with wealth processes discounted by a riskless asset is considered. The numéraire portfolio is the unique strictly positive wealth process that, when used as a benchmark to denominate all other wealth, makes all wealth processes local martingales. It is assumed that the numéraire portfolio exists and that its wealth increases to infinity as time goes to infinity. Under this setting, an initial enlargement of the filtration is performed, by including the overall minimum of the numéraire portfolio. It is established that all nonnegative wealth processes, when stopped at the time of the overall minimum of the numéraire portfolio, become local martingales in the enlarged filtration. This implies that risk-averse insider traders would refrain from investing in the risky assets before that time. A partial converse to the previous result is also established in the case of complete markets, showing that the time of the overall minimum of the numéraire portfolio is in a certain sense unique in rendering undesirable the act of undertaking risky positions before it. The aforementioned results shed light to the importance of the numéraire portfolio as an indicator of overall market performance.