Analytic Solutions Of Call And Put Options Of Nonlinear Black-Scholes Equation With Transaction Costs And Price Slippage

Abstract

A nonlinear Black-Scholes Partial Dierential Equation whose nonlinearity

is as a result of transaction costs and a price slippage impact

that lead to market illiquidity with feedback eects was studied.

Most of the solutions obtained in option pricing especially

using nonlinear equations are numerical which gives approximate

option values. To get exact option values, analytic solutions for

these equations have to be obtained. Analytic solutions to the nonlinear

Black-Scholes Partial Dierential Equation for pricing call

and put options to expiry time are currently unknown. The main

purpose of this study was to obtain analytic solutions of European

call and put options of a nonlinear Black-Scholes Partial Dierential

Equation with transaction costs and a price slippage impact. The

methodology involved reduction of the equation into a second-order

nonlinear Partial Dierential Equation. By assumption of a traveling

wave prole the equation was further reduced to Ordinary Differential

Equations. Solutions to all the transformed equations gave

rise to an analytic solution to the nonlinear Black-Scholes equation

for a call option. Using the put-call parity relation the put option's

value was obtained. The solutions obtained will be used to price

put and call options in the presence of transaction costs and a price

slippage impact. The solutions may also help in tting the Black-

Scholes option pricing model in the modern option pricing industry

since it incorporates real world factors hence signicantly contributing

to the eld of mathematical nance. We, therefore, recommend

to hedgers and speculators in derivatives markets to make use of option

pricing formulae obtained in this research for accurate option

pricing so that they can maximize their prots. In conclusion, further

research needs to be done to study the exposure from writing

a covered call and the exposure from writing a naked put.