Control of the Black-Scholes equation

Abstract

There are hundreds of papers dealing with the Black-Scholes equation. Yet, no one seems to have ever used the scaling invariance coming from the heat equation. And there appears to be very few studies on the "control aspect" of the equation. In previous works, J.Y. Chemin and Cl. David obtained new results for the mass critical nonlinear Schr\"odinger equation, with a simple principle: one uses the fact that for this nonlinear equation, solutions of scales which are different enough almost do not interact. The nonlinearity of the original equation just required to determine a condition about the size of the scale which depends continuously on the data. The idea is to apply the building technique of the map described above, starting from the point that the Black-Scholes equation can be transformed into a linear heat equation, which has a natural scaling invariance, and that the linearity of the transformed equation enables one to obtain exact solutions, and, thus, yield interesting results. The aim of this paper is to present a new way to change either the Delta, or other greeks, by a given amount ; this technique appears thus as a new way to control the equation, without resorting to control functions. Indeed, one can change the greeks directly using the general solution of the Black-Scholes equation, nevertheless, our technique appears as an alternative and simple one to implement.