Chaos for generalized Black-Scholes equations

Abstract

The Nobel Prize winning Black-Scholes equation for stock options and the heat equation can both be written in the form \[ \frac{\partial u}{\partial t}=P_2(A)u, \] where $P_2(z)=\alpha z^2+ \beta z+\gamma$ is a quadratic polynomial with $\alpha > 0$. In fact, taking $A = x\frac{\partial}{\partial x}$ on functions on $[0,\infty) \times [0,\infty)$ the previous equality reduces to the Black-Scholes equation, while taking $A = \frac{\partial}{\partial x}$ for functions on $\mathbb{R} \times [0,\infty)$ it becomes the heat equation. Here, we ``connect'' the two previous problems by considering the generalized operator $A= x^a\frac{\partial}{\partial x}$ for functions on $[0,\infty) \times [0,\infty)$ with $0<a<1$, and our main result is that the corresponding degenerate parabolic equation is governed by a semigroup of operators which is chaotic on a class of Banach spaces. The relevant Banach spaces are weighted supremum norm spaces of continuous functions on $[0,\infty)$. This paper unifies, simplifies and significantly extends earlier results obtained for the Black-Scholes equation ($a=1$) in \cite{EGG} and the heat equation ($a=0$) in \cite{EGG1}.