The black-Scholes formula and the Greek parameters for a nonlinear Black-Scholes equation

Abstract

We study the Greek (risk) parameters of a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of transaction costs. These parameters are derived from the Black-Scholes formula of the nonlinear Black-Scholes equation ut + 1 2 2s2uss(1 + 2 suss) = 0 by differentiating the formula with respect to either a variable or a parameter in the equation. The Black-Scholes formula and all the Greek parameters are of the form 1 f(s, t) and therefore they blow at = 0.