Derivatives

CHAPTER 28The Greeks

Aims

• To show how ‘the Greeks’ (e.g. delta, gamma, rho, vega, and theta) provide useful ‘summary statistics’ for options, which can be used to provide an approximation to the change in the option price.
• To show how the Greeks are used to protect the value of an options portfolio from small changes (delta hedging) and large changes (delta-gamma hedging) in the price of the underlying asset.
• To demonstrate how the Greeks are used to protect the value of an options portfolio from large and small changes in the underlying asset's price, when the latter is also accompanied by (small) changes in volatility – this is ‘gamma-vega-delta’ hedging.
• To demonstrate how to calculate ‘the Greeks’ for the BOPM.

28.1 DIFFERENT GREEKS

The Black–Scholes formula (on a non-dividend paying stock) shows that the option premium varies with , all of which may change minute by minute. In this chapter we show how the change in price of the option can be represented in terms of a number of ‘summary statistics’ which are generally referred to as ‘the Greeks’. Knowing the numerical values for the various ‘Greeks’ also allows an options trader to set up effective hedge positions for her portfolio of options. As we shall see, the more sources of uncertainty the options trader wants to hedge (stocks, interest rates, volatility) and the smaller the desired hedging error, the more complex ...