In this paper, we study the multi-asset Black-Scholes model in terms of the importance that the correlation parameter space (equivalent to an N dimensional hypercube) has in the solution of the pricing problem. We show that inside of this hypercube there is a surface, called the Kummer surface ΣK , where the determinant of the correlation matrix ρ is zero, so the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. Worse than that, in some regions outside this surface, the determinant of ρ becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside ΣK . On the Kummer surface instead, the rank of the ρ matrix is a variable number. By using the Wei-Norman theorem, we compute the propagator over the variable rank surface ΣK for the general N asset case. We also study in detail the three assets case and its implied geometry along the Kummer surface.

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