## Options: the Greeks

##### Published: May 2011

Corporate Treasurers notoriously dislike the use of options. They don’t like paying premium upfront; they don’t like managing the accounting issues and they don’t like the complexity of option valuation. To understand option prices you need to know a little bit of Greek.

### Delta

Occasionally referred to as the hedge ratio, the delta of an option is a measure of the effect a change in the price of an underlying asset has on the price of a put or call option taken out on that asset. It is used to ascertain the rate at which the price of a derivative will change as the price of the underlying asset fluctuates.

The delta is conventionally assigned a decimal figure between one and minus one to indicate the percentage move in the option price for a one unit move in the underlying, but traders generally use the percentage itself. Whereas changes in vanilla call option prices are indicated by positive deltas, vanilla put option deltas are assigned negative values.

### Call options

To take an example, a vanilla call option with a delta of 0.6 (or 60) means that for every €1 increase in the price of an underlying asset, the price of the call option will increase commensurately by €0.60. Conversely, when the price of the underlying stock decreases the value of the call option also decreases by the amount of the derivative’s delta.

### Put options

The deltas for vanilla put options are negative values because as the underlying security increases in price, the price of the option decreases. For example, a delta of -0.6 for a put option means that the derivate decreases in price by €0.60 for every €1 the underlying asset rises in value.

Like call options, put derivatives respond to changes in price of the underlying stock price and their deltas respond to these changes.

For example, when the price of the underlying stock increases, the price of the put option decreases by its delta value. Conversely, when the underlying stock falls in price, the price of the put option moves in the opposite direction by the size of its delta.

If a call option is described as in-the-money (ITM), its delta is above 0.5 and the underlying stock is trading above the strike price of the option. Conversely, an option that is out-of-the-money (OTM) means that the derivative has a delta of less than 0.5 and the strike price of the option is above the price at which the underlying stock is currently trading.

A call option trading at-the-money (ATM) will have a delta of 0.5 or somewhere close to that figure. The strike price of the option and the price of the underlying asset are equal or are nearing the same price.

When a call option’s strike price is far below the market price of the underlying asset, the option is referred to as deep-in-the-money and its delta approaches one. This means that the price moves of the option match those of the underlying asset. At the other end of the scale, far out-of-the-money options have small deltas.

### Hedge ratio

Since option delta is a measure of how sensitive an option’s price is to changes in the underlying, it is useful as a hedge ratio. A futures option with a delta of 0.5 means that the option price increases 0.5 for every one point increase in the futures price. For small changes in the futures price therefore, the option behaves like one-half of a futures contract.

Constructing a delta hedge for a long position in ten calls, each with a delta of 0.5 would require you to sell five futures contracts. (The delta of a futures contract is always one.)

### Gamma

An option’s gamma is used to gauge the sensitivity of delta to changes in the underlying stock. It measures the responsiveness of the derivative’s delta to a one-point shift in the underlying asset. When the option is nearing its strike price, the gamma will be larger than it is for out-of-the-money deltas. Unlike delta, gamma is positive for both call and put options.

### Theta

An option’s theta is a measure of how much value the option loses every day before it reaches maturity. The more time there is for the underlying to hit the strike price of an option, the more valuable, all other things being equal, the option is. And clearly at expiration the time-value of any option is zero. So options are always losing part of the time-value component of their overall value, which is why theta values are always negative and the theta of an option that is further away from its date of expiration is always smaller than that of one which is close to maturation.