What are 'the Greeks'?

Why use Greeks?

Delta

Gamma

Vega

Theta

Rho

More information on Greeks

### What are 'the Greeks'?

This is a colloquial term given to the set of measures derived from the Black Scholes option pricing formula. Letters from the Greek alphabet are used to represent these derived measures:

- delta - a measure of an option’s sensitivity to changes in the price of the underlying asset
- gamma - a measure of delta’s sensitivity to changes in the price of the underlying asset
- vega - a measure of an option’s sensitivity to changes in the volatility of the underlying asset
- theta - a measure of an option’s sensitivity to time decay
- rho - a measure of an option’s sensitivity to changes in the risk free interest rate

These measures are not static, but are interdependent and change constantly. When you look at one measure it is on the basis that all other variables are held constant.

### Why use Greeks?

For the general investor and retail options trader, knowing the delta of your options position gives you an indication of how your option’s value will change with movements in the underlying stock price - all other variables remaining the same. Knowing your time decay (theta) gives you an indication of how much time value your option position is losing each day - all other variables remaining the same. Other measures are explained below.

The professional market uses the Greeks to measure exactly how much they need to hedge their portfolio. The Greeks also enable the measurement of how much risk the portfolio is exposed to, and where that risk lies (with movements in interest rates or volatility, for example).

### Delta

A by-product of the Black Scholes model is the calculation of delta: the degree to which an option price will move given a change in the underlying stock price, all else being equal. For example, an option with a delta of 0.5 will move half a cent for every one cent movement in the underlying stock.

A far out-of-the-money call will have a delta very close to zero; an at-the-money call a delta of 0.5; a deeply in-the-money call will have a delta close to 1.

Call deltas are positive; put deltas are negative, reflecting the fact that the put option price and the underlying stock price are inversely related. The put delta equals the call delta minus 1.

The delta is often called the neutral hedge ratio. For example if you have a portfolio of n shares of a stock then n divided by the delta gives you the number of calls you would need to write to create a neutral hedge - i.e. a portfolio which would be worth the same whether the stock price rose by a small amount or fell by a small amount. In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price should be exactly offset by a loss on the value of the calls written, and vice versa.

Note that as the delta changes with movements in the stock price and time to expiration the position would need to be continually adjusted to maintain the hedge. How quickly the delta changes with the stock price is given by gamma (see below).

The ASX Strategy Modelling Tool, which can be downloaded from this site, calculates and displays the delta for each individual option trade entered into the model.

For example, you may set up a covered call in the model using Black-Scholes European pricing (i.e. sell n calls and buy n underlying shares). If you then change the number of calls sold to be equal to the number of shares bought divided by the delta of the call options, you will have an example of a hedged position. Notice how the time line (i.e. the curved line showing the profit at the number of days to expiration) on the payoff diagram just touches (but doesn't pass through) the horizontal axis at one point only: the point equal to the current share price. A small share price move in either direction on this line will have no effect on profit. The profit/loss you make on the shares will be equal to the loss/profit you make on the options - you are 'delta hedged' .

The Strategy Modelling Tool also calculates the position delta for a range of stock prices and days to expiration - that is, the delta of the entire strategy consisting of multiple option trades and trades in the underlying stock. The position delta, sometimes called the Equivalent Stock Position (ESP), lets you see, for example, how a dollar rise in the underlying stock price will affect the overall profitability of the entire strategy. For example, if the ESP of a portfolio, or strategy, is -2,300 it means that the market exposure of the portfolio is equivalent to a portfolio short 2,300 shares. Thus a one dollar rise in the stock price will cause the profitability of the entire position to fall by $2,300.

The Strategy Modelling Tool also calculates the other position Greeks.

### Gamma

Gamma is a measure of the change in delta for a change in the underlying stock price.

If you are hedging a portfolio using the delta-hedge technique described above, then you will probably want to keep gamma as small as possible. The smaller gamma is, the less often you will have to adjust the hedge to maintain a delta neutral position. If gamma is too large a small change in stock price could break your hedge. Adjusting gamma, however, can be tricky and is generally done using options - unlike delta, it cannot be done by buying or selling the underlying asset, as the gamma of the underlying asset is, by definition, always zero so more or less of it will not affect the gamma of the total portfolio.

### Vega

Vega (sometimes known as kappa) is the change in option price given a one percentage point change in volatility. Vega is used for hedging volatility risk.

### Theta

Theta is the change in option price given a one day decrease in time to expiration. It is a measure of time decay. Theta is generally regarded as a descriptive statistic, used to gain an idea of how time decay is affecting your portfolio, rather than as the basis of a hedge. Hedging a portfolio against time decay, the effects of which are completely predictable, would be pointless.

### Rho

Rho is the change in option price given a one percentage point change in the risk-free interest rate.

### More information on Greeks

Like delta, all the Greeks vary with changes in each of the variables that are part of the Black Scholes pricing model. For example, the gamma varies as the volatility changes.

You can see graphically how each derived measure changes as the underlying Black-Scholes variables change. The Strategy Modelling Tool calculates the position Greeks, that is the Greeks both for individual option and stock trades, and for the entire strategy or portfolio (i.e. net position). This lets you see how the profitability of the strategy is affected by changes in stock price, time to expiration, volatility and so on.

All the Greeks can also be viewed graphically thereby highlighting how they change with changes in the value of the underlying asset and with time.