Modeling the Volatility and Expected Value of a Diversified World Index

This paper considers a diversified world stock index in a continuous financial market with the growth optimal portfolio (GOP) as the reference unit or benchmark. Diversified broadly based portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstrated that a key financial quantity is the drift of the discounted GOP, which can be expressed explicitly using a certain quadratic variation term. Using real market approximations for the discounted GOP it is shown that its drift does not vary greatly in the long term. For a diversified world index this leads to a natural model where the discounted index is a time transformed squared Bessel process of dimension four. The inverse of the squared GOP volatility then follows a square root process of dimension four.