Abstract. We study a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of transaction costs that lead to market illiquidity. After reducing the equation into a nonlinear parabolic porous medium type equation, we find that the assumption of a traveling wave profile to the porous medium type equation reduces it further to ordinary differential equations. Solutions to all these transformed equations together with the use of localizing boundary conditions facilitate a twice continuously differentiable solution to the nonlinear Black-Scholes equation. We also find that the option is always more volatile compared to the stock. All the risk parameters except Gamma are negative throughout time .

Received: July 16, 2012

AMS Subject Classification: 35K10, 35K55

Key Words and Phrases: porous medium equation, analytic solution, illiquid markets, transaction cost

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