The Null Volatility Limit of the Chaotic Black-Scholes Equation

Abstract

In [4, 5] it is proved that the Black-Scholes semigroup is chaotic in various Banach spaces of continuous functions. Here we will see why in these spaces the null volatility case is not governed by a chaotic semigroup, while if we consider the generalized Black-Scholes equation in which there are two interest rates \(r_1\) and \(r_2\) with \(r_1 > r_2\), then the corresponding Black-Scholes semigroup converges strongly to a chaotic semigroup when the volatility \(\sigma \rightarrow 0\). It is then shown that, keeping the volatility fixed and positive, the coefficients in the lower order terms in the generalized Black-Scholes equation can be replaced by any real constants, and one still obtains chaotic semigroups. Finally, the heat equation on the real line with arbitrary coefficients in the lower order terms is shown to be chaotic.