Higher Order Greeks

Abstract

In the original Black-Scholes pricing framework, a continuously rebalanced delta hedge of an option against its underlying stock produces a perfectly risk free position. But in the real world, a basic delta hedge is still exposed to quite a lot of risk because rebalancing can only be done at discrete intervals, volatility is stochastically time-varying, and underlying assets do not follow true lognormal diffusions. Discrete rebalancing after a non-marginal price change makes gamma an important extra risk factor and with fluctuating implied volatility, vega becomes another one. This makes delta-gamma-vega hedging the next level of option risk management. But unhedged risk still remains. In this paper, Ederington and Guan investigate whether option risk factors measured by other second and third order derivatives of the option pricing function can capture this residual risk, and if so, which are the most important of these „higher order Greeks.” They find that, at least for S&P 500 index futures options, the additional risk factors can explain a substantial portion of the residual risk in a delta-gamma-vega hedge. A key finding is that an option's sensitivity to the higher order Greeks depends heavily on its moneyness, with several Greek letter risks that are unimportant for at the money options becoming much more relevant for out of the money contracts. They also show that a hedge strategy that neutralizes these additional factors can lower residual risk by an order of magnitude below that of a delta-gamma-vega hedge.