= Geometric stock index
{title2=$J_t=\prod_{i=1}^n(S_t^i)^{1/n}$}
For positive <stocks> satisfying $dS_t^i/S_t^i=\sum_j\sigma_{ij}dW_t^j+\mu_i dt$, let $q=n^{-1}\mathbf1$ and $V=\sigma\sigma^T$. The <Itô formula> gives
$$
\log(J_t/J_0)=q^T\sigma W_t+\left(q^T\mu-\frac1{2n}\operatorname{tr}V\right)t.
$$
Thus its volatility is $\sigma^Tq$. This geometric average is generally not the value of the constant equal-weight <self-financing portfolio>: its instantaneous drift is $q^T\mu-\operatorname{tr}(V)/(2n)+q^TVq/2$, whereas the fully invested equal-weight portfolio has drift $q^T\mu$.
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