Charm-Adjusted Delta and Delta Gamma Hedging

Abstract

Hedging an option is easy in the basic Black-Scholes world. The only stochastic variable is the stock price, and by holding a short position in the stock equal to minus the partial derivative of the call price with respect to the stock, a momentarily riskless hedge of a call option is achieved. With no transactions costs or impediments to trading, the hedge can be rebalanced continuously through the entire life of the option. But delta hedging like this isn’t possible in the real world, where it only makes sense to rebalance a hedge periodically, and the change in the delta as the stock price moves—the option’s gamma—becomes a source of unhedged risk. Delta–gamma hedges can manage both delta risk and the change in the delta, but volatility also changes over time, necessitating another Greek letter (or quasi-Greek in this case)—and vega was invented. As consideration extended to further higher-order derivatives, it put such a terrific strain on the Greek alphabet that an extension of the partial derivative naming convention was required. For example, “charm” is the derivative of delta with respect to time. And as this article shows, in an option hedge, charm becomes as important a risk factor as gamma close to maturity. The delta of an in-the-money (out-of-the-money) call converges rapidly to +1.0 (0.0) right at maturity, so a delta-hedged position can quickly become quite unhedged in the last few days. When charm is large, a more accurate hedge can be achieved using a modified hedge ratio, different from the standard delta.