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The Greek letters derived from mathematical models represent various sensitivities of options behaviour. The variables that affect the Greeks are time, implied volatility, and the position of the underlying in relation to the strike prices.
Definition:
Delta is the change in the price of an option for a one point moves in the underlying.
Call options: 0 < Delta < 1
Put options: -1 < Delta < 0
In-the-money options: Delta approaches 1 (call: +1, put: -1)
At-the-money options: Delta is about 0.5 (call: +0.5, put: -0.5)
Out-of-the-money options: Delta approaches 0
Call deltas can be interpreted as the probability that the option will finish in the money. An at-the-money option, which has a delta of approximately 0.5, has roughly a 50/50 chance of ending up in-the-money.
Put deltas can be interpreted as -1 times the probability that the option will finish in the money.
Hedge ratio (futures equivalence):
Since 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 1 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 10 calls, each with a delta of 0.5 would require you to sell 5 futures contracts. (The delta of a futures contract is always 1).
Time to expiration:
As time passes, the delta of in-the-money options increases and the delta of out-of-the-money options decreases.
Volatility:
As volatility falls, the delta of in-the-money options increases and the delta of out-of-the-money options decreases.
Definition:
Gamma is the change in an option’s delta for a one-point change in the price of the underlying.
The gamma of a long option position (both calls and puts) is always positive. This means that the delta increases as the underlying price increases and that delta falls as the underlying price falls.
At-the-money options have the largest gamma. The further an option goes in-the-money or, out-of-the-money the smaller is gamma.
Time to expiration:
As time passes, the gamma of at-the-money options increases, the gamma of deep in-the-money and out-of-the-money options decreases.
Volatility:
As volatility falls, the gamma of at-the-money options increases, the gamma of deep in-the-money and out-of-the-money options decreases.
Definition:
Theta is defined as the change in the price of an option for a 1-day decrease in the time remaining to expiration.
At-the-money options have the greatest time value and the greatest rate of time decay (theta). The further an option goes in-the-money or out-of-the-money, the smaller is theta.
Time to expiration:
As time passes, the theta of at-the-money options increases, the theta of deep in-the-money and out-of-the-money options decreases.
Volatility:
As volatility falls, time value declines, theta declines.
Definition:
Vega is the change in the value of an option for a 1-percentage point increase in implied volatility. The vega of a long option position (both calls and puts) is always positive.
At-the-money options have the greatest vega. The further an option goes in-the-money or out-of-the-money, the smaller is vega.
Time to expiration:
As time passes, vega decreases. Time amplifies the effect of volatility changes. As a result, vega is greater for long-dated options than for short dated options.
Volatility:
As volatility falls, vega decreases for in-the-money and out-of-the-money options; vega is unchanged for at-the-money options.
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