Analysis of The Greek Parameters of a Nonlinear Black-Scholes Partial Differential Equation

Abstract

Derivatives are used in hedging European options against risks. The partial derivatives of the solution with respect to either a variable or a parameter in the Black-Scholee model are called risk parameters or simply the Greeks. Nonlinear versions of the standard Black- Scholes Partial Differential Equation have been introduced in financial mathematics in order to deal with illiquid markets. Market liquidity is relevant in the risk management of derivatives since in an illiquid market the implementation of a dynamic hedging strategy affects the price process of the underlying. Different hedging strategies and suitable pricing adjustments are needed. We studied the Greek parameters of a nonlinear Black- Scholes Partial Differential Equation whose nonlinearity is as a result of transaction costs for modelling illiquid markets. The objective of this study was to compute the Greek parameters of a European call option in illiquid markets whose illiquidity is arising from transaction costs. This is in relation to Cetin et al. model in which transaction costs have been incorporated (with zero interest rate). These Greeks were compared with those derived from the formula of Bakstein and Howison (2003) equation (with positive interest 1 rate). All these Greeks were of the form a+ - j(S, t). The methodology involved deriving p 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. Greeks show how to protect one's position against adverse movements in critical market variables such as the stock price, time and interest rate.