### Abstract:

There has been a significant growth in research in the field of financial mathematics
since the derivation of the standard Black-Scholes-Merton Partial
Differential Equation by Black and Scholes, and Merton in 1973. The derivation
was done under the assumption that the market is liquid and frictionless
(no restrictions on trade and no transaction costs). The nonlinear equation
ut + 1
2σ2S2uSS(1 + 2ρSuSS) = 0 for modeling illiquid markets has only been
solved analytically using a positive gamma strategy. In order to price nonsingle-
signed-gamma assets, the solution to the nonlinear equation also needs
to be found via a negative gamma strategy for pricing any European styled
call option. Our main objective was to solve the equation analytically using
a negative gamma strategy, investigate volatility analytically and finally
compare and contrast the results from both the positive and negative gamma
strategies. The methodology involved transforming the equation into a nonlinear
porous medium-type equation. Assuming a traveling wave solution
yielded Ordinary Differential Equations (ODEs) which were solved to obtain
the solution to the Black-Scholes equation via a negative gamma strategy,
uss < 0. Volatilities arising from both positive and negative gamma strategies
were analysed showing an increasing trend with gamma resulting into a
concave shaped curve from positive gamma and convex shaped curve from
negative gamma amongst other results. In a real market situation, the solution
resulting from a negative gamma strategy may help in finding how
non-single-signed gamma assets can be valued hence contributing to the field
of Financial Mathematics.
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