A Benchmark Approach to Investing and Pricing

This paper introduces a general market modeling framework, the benchmark approach, which assumes the existence of the numeraire portfolio. This is the strictly positive portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales, that is intuitively speaking downward trending or trendless. It can be shown to equal the Kelly portfolio which maximizes expected logarithmic utility. In several ways the Kelly or numeraire portfolio is the "best" performing portfolio and can not be out performed systematically by any other nonnegative portfolio. Its use in pricing as numeraire leads directly to the real world pricing formula, which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shown to be approximated by well diversified portfolios.