Discrete geometric average under the stock-numeraire measure ($\log A\sim N(m,v)$)

For sampling times $t_i$ and $\bar{t}=n^{-1}{\sum }_i{t}_i$, the geometric mean $A=({\prod }_i{S}_{t_i}{)}^{1/n}$ has $\log A$ normally distributed under the stock-numeraire measure in the Black-Scholes model. Its mean is $\log S_0+(\rho +{\sigma }^2/2)\bar{t}$, and its variance is $v={\sigma }^2{n}^{-2}{\sum }_{i,j}\min (t_i,t_j)$. For descending times the sum is ${\sum }_i(2i-1)t_i$. The drift shift comes from exponential tilting, and must be included when the delivered payment contains a factor $S_T$.

A delivery $S_T\max (r,c/A)$ has value $S_0[r\Phi (d)+c{e}^{-m+v/2}\Phi (-d+\sqrt{v})]$, where $m,v$ are the mean and variance of $\log A$ under the stock-numeraire measure in the Black-Scholes model, and $d=(m-\log (c/r))/\sqrt{v}$. Split the normal integral at $\log (c/r)$ and complete the square in $e^{-y}$ times its normal probability density.