On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry

TL;DR

This paper analyzes the multi-asset Black-Scholes model focusing on the correlation parameter space, revealing geometric structures like the Kummer surface where the correlation matrix becomes degenerate, affecting the model's validity and solution.

Contribution

It introduces the geometric and algebraic structure of the correlation space, especially the Kummer surface, and computes the propagator for the multi-asset Black-Scholes model using the Wei-Norman theorem.

Findings

The correlation matrix determinant vanishes on the Kummer surface.

Outside the surface, the propagator becomes complex and divergent.

The Wei-Norman theorem enables propagator computation on the variable rank surface.

Abstract

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 $\Sigma_K$, where the determinant of the correlation matrix $\rho$ is zero, so the usual formula for the propagator of the $N$ asset Black-Scholes equation is no longer valid.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 $\rho$ becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside $\Sigma_K$. On the Kummer surface instead, the rank of the $\rho$ matrix is a variable number. By using the Wei-Norman theorem, we compute the propagator over the variable rank surface $\Sigma_K$ for the general $N$ asset case. We also study in detail the three assets case and its implied geometry along the Kummer surface.