Geometric stock index ($J_t={\prod }_{i=1}^n(S_t^i{)}^{1/n}$)

For positive stocks satisfying $d{S}_t^i/S_t^i={\sum }_j{\sigma }_{ij}d{W}_t^j+{\mu }_{i}dt$, let $q=n^{-1}1$ and $V=\sigma {\sigma }^T$. The Itô formula givesThus its volatility is ${\sigma }^{T}q$. This geometric average is generally not the value of the constant equal-weight self-financing portfolio: its instantaneous drift is $q^T\mu -\mathrm{tr}(V)/(2n)+q^{T}Vq/2$, whereas the fully invested equal-weight portfolio has drift $q^T\mu$.