Abstract:
This study presents a nonlinear Black-Scholes equation whose nonlinearity is
due to feedback effects. The market involved 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, 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. Both positive
and negative gamma exposure can lead to an out-of-the-money option.
