#### Gamma

Gamma, measures the rate of change of Delta. When call options are deep out of the money, they generally have a small Delta. This is because changes in the underlying bring about only tiny changes in the price of the option. But as the call option gets closer to the money, resulting from a continued rise in the price of the underlying, the Delta gets larger.

Gamma is the change in an option’s delta for unit change in the value of the underlying asset. The gamma of a long option position (both calls and puts) is always positive. 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.

- If you are long gamma you expect the underlying to make large moves. Traders with long positions expect positive gamma
- If you are short gamma you expect the underlying to remain relatively inactive. Traders with short positions expect negative gamma
- Gamma is a useful indication of the risk associated with a futures position. A large gamma number, whether positive or negative indicates a high degree of risk and a low gamma number indicates a low degree of risk.

#### Delta

Delta is the amount by which the option changes compared to the underlying asset. It is a measure of the probability that an option will expire in the money. Call deltas can be interpreted as the probability that the option will finish in the money. Put deltas can be interpreted as -1 times 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”. For example, if an at-the-money wheat call option has a Delta of .5, and if wheat makes a 10-cent move higher, the premium on the option will increase approximately by 5 cents (.5 x 10 = 5), or $250 (each cent in premium is worth $50). The interpretation of Delta values is as under:

- Call options: 0 to 1
- Put options: -1 to 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
- Long Calls have a positive delta -You want the market to go up
- Short Calls have a negative delta -You want the market to go down
- Long Puts have a negative delta -You want the market to go down
- Short Puts have a positive delta -You want the market to go up

Delta 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.

As time passes, the delta of in-the-money options increases and the delta of out-of-the-money options decreases.

#### Theta

Theta is defined as the change in the price of an option for a 1-day decrease in the time left for 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. As volatility falls, the time value declines and hence theta also declines.

- Theta is the rate at which an option loses its value as each day passes.
- The inherent assumption is that the options are a “wasting asset.”
- Long options have negative theta
- Short options have positive theta

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.

Theta has the exact opposite characteristics of gamma. Thus the size of a gamma position correlates to the size of the theta position. A large positive gamma position goes in hand with a large negative theta position, while a large negative gamma position goes hand in hand with a large positive theta position. What this means is that every option position is a tradeoff between market movement and time decay.

Theta is not used much by traders, but it is an important conceptual dimension. Theta measures the rate of decline of time-premium resulting from the passage of time. In other words, an option premium that is not intrinsic value will decline at an increasing rate as expiration nears.

#### Vega

Vega is the change in the value of an option for a 1-percentage point increase in implied volatility of the underlying asset price. Implied volatility is measured as the annualized standard deviation of an asset’s daily price changes. 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 the Vega. 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.

- As volatility falls, Vega decreases for in-the-money and out-of-the-money options; Vega is unchanged for at-the-money options.
- Vega is the option’s change in theoretical value with a change in volatility.
- Most options have a positive Vega because they gain value with rising volatility and lose with falling volatility

Vega of most options decline as time to expiration grows shorter. Vega tells us approximately how much an option price will increase or decrease given an increase or decrease in the level of implied volatility. Option sellers’ benefit from a fall in implied volatility, and it’s just the reverse for option buyers. Vega can increase or decrease even without price changes of the underlying because implied volatility is the level of expected volatility

If you would like to discuss these methods of option buying and selling, or assembling an option portfolio, please feel free to call us at 800-263-3004.