Dynamic hedging of equity call options

Abstract

The theory of option pricing assumes generally that options can be replicated through dynamic hedging in the underlying stock. First, we outline the assumptions behind the popular models, such as regarding the distribution of stock returns, and the probability of the terminal stock value reaching certain levels. Then, we define the common "Greeks" of call options, that is the sensitivity of option values to changes in particular variables. Figures show the sensitivity of those Greeks to stock price levels, and time to expiration. Then, we attempt to show that delta and complex hedges, using options with more than one exercise price, are the "solutions" for simultaneous equations establishing delta and gamma (and eventually vega) neutrality, subject to a budget constraint. Finally, we examine the relative profitability and effectiveness (in terms of variance reduction) of delta hedging strategies for three trade positions (in, at and out-of-the-money).