Author(s)

Mauricio Contreras¹, Alejandro Llanquihuén², Marcelo Villena¹

¹Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibánez, Santiago, Chile.
²Facultad de Ciencias Exactas, Universidad Andrés Bello, Santiago, Chil.

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of the correlation matrix ρ is zero, so the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid.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.

Multi-Asset Black-Scholes Equation, Wei-Norman Theorem, Correlation Matrix Eigenvalues, Kummer Surface, Propagators

Contreras, M. , Llanquihuén, A. and Villena, M. (2016) On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry. Journal of Mathematical Finance, 6, 562-579. doi: 10.4236/jmf.2016.64043.