Greek parameters of nonlinear Black-Scholes equation

Abstract

Derivatives are used in hedging European options against risks. The partial derivatives of the solution to either a variable or a parameter in the Black-Scholes model are called risk (Greek) parameters or simply the Greeks. Nonlinear versions of the standard Black-Scholes Partial Differential Equations have been introduced in financial mathematics in order to deal with illiquid markets. In this paper we derive the Greek parameters of a nonlinear Black-Scholes Partial Differential Equation whose nonlinearity is as a result of transaction costs for modeling illiquid markets. We compute the Greek parameters of a European call option price from the nonlinear equation ut + 1 2 σ 2S 2uSS(1 + 2ρSuSS) = 0. All these Greeks were of the form a + 1 ρ f(S, t). The methodology involved deriving the Greek parameters from the formula of the equation by differentiating the formula with respect to either a variable or a parameter. These Greeks may help a trader to hedge risks in a non-ideal market situation.