On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry

Abstract

In this paper, the multi-asset Black-Scholes model is studied in terms of the importance that the correlation parameter space (equivalent to an N dimensional hypercube) has in the solution of the pricing problem. It is shown that inside of this hypercube there is a surface, called the Kummer surface ∑ k , 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of the correlation matrix ρ is zero, so the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid.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 ∑ k . On the Kummer surface instead, the rank of the ρ matrix is a variable number. By using the Wei-Norman theorem, the propagator over the variable rank surface ∑ k for the general N asset case is computed. Finally, the three assets case and its implied geometry along the Kummer surface is also studied in detail.