TABLE OF CONTENTS One of the most useful applications of the Black Scholes formula involves applying the partial derivatives of the formula to analyze and design portfolios containing derivatives securities. The presence of an option pricing formula permits partial derivatives to be solved directly, instead of having to rely on numerical techniques. Correct evaluation of the partial derivatives permits theoretical portfolios to be precisely constructed to have desirable properties.
Delta, theta, and gamma are names used to refer to the most commonly referenced partial derivatives. The partial derivatives are also referred to as Greeks, after the symbols used identify the derivatives. Applied to a call option, these three Greeks are defined as: [2]
While delta, theta, and gamma are typically the most commonly referenced partial derivatives, there are numerous other partial derivatives that could also be of value for certain types of situations. For example:
This is only a partial list. From put call parity, similar concepts can be derived for puts:
The other partial derivatives for puts follow appropriately.
To employ the derivative properties of Black Scholes to design portfolios of securities requires recognizing that the various securities that can be included in a given portfolio, such as stocks, commodities, futures, and options, all possess derivative properties. From linearity, this permits the delta, theta, gamma, and other Greeks of a portfolio to be calculated. More precisely, let V be the dollar value of the portfolio, V i represent...
