DOI: 10.12732/ijam.v37i4.5
ANALYTIC SOLUTION OF A
NONLINEAR BLACK–SCHOLES
EQUATION VIA LONG AND SHORT
GAMMA POSITIONS
Joseph Eyang’an Esekon
Department of Pure and Applied Sciences
Kirinyaga University
P.O. Box 143 – 10300, Kerugoya, KENYA
Abstract. We study a nonlinear Black–Scholes equation whose nonlinearity is due to feedback effects. The market involved here is illiquid as a result of transaction costs. An analytic solution to the equation via long and short gamma positions is currently unknown. After transforming the equation into a parabolic nonlinear porous medium-type equation, we find that the assumption of a traveling wave profile to the later equation reduces it to Ordinary Differential Equations (ODEs). This together with the use of long and short gamma positions facilitate a twice continuously differentiable solution. The solution can be used to price a call option. Both positive and negative gamma exposures can lead to the value of a short call.
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to cite this paper?
DOI: 10.12732/ijam.v37i4.5
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 4
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