IJPAM: Volume 76, No. 2 (2012)

THE BLACK-SCHOLES FORMULA AND THE GREEK
PARAMETERS FOR A NONLINEAR
BLACK-SCHOLES EQUATION

Joseph Eyang'an Esekon
Department of Mathematics and Applied Statistics
Maseno University
P.O. Box 333, Maseno, KENYA


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 $u_t + \tfrac{1}{2} \sigma^2 s^2 u_{ss} (1 + 2\rho s u_{ss}) = 0$ 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 $\tfrac{1}{\rho} f (s,t)$ and therefore they blow at $\rho = 0$.

Received: June 4, 2011

AMS Subject Classification: 35K10, 35K55

Key Words and Phrases: nonlinear black-scholes equation, black-scholes formula, illiquid markets, Greek parameters, transaction cost model

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 2