The analysis of gamma strategies for a nonlinear black-scholes equation

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. v