Let $(Z_1,\ldots,Z_N)$ be an $N$-dimensional Brownian motion with correlation matrix $\rho$ and consider the multivariate Black-Scholes model \begin{align} dS_i(t) \ = \ (r-q_i)\, S_i(t) \, dt \, + \, \sigma_i \, S_i(t) \, dZ_j(t)\,. \end{align} where $r$ is the deterministic risk-free interest rate and $q_i$ are the dividend yields of asset $S_i$. The task is to calculate the price of a general geometric basket option with strike $K$, \begin{align} V(x,t) \ = \ e^{-r(T-t)} \; \mathbb E \Bigg[ \prod_{i=1}^N S_i(T)^{b_i} - K \ \Bigg| \ S(t) = x\Bigg] \end{align} where $b_i\in \mathbb R_{\geq 0}$, which---as the geometric mean of log-normal random variables is lognormal again---can be given by a formula of Black-Scholes type.

Can one please point me to a citable reference where this calculation is performed?


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