Option Pricing - Delta

The delta is a measure of how the option price changes for a 1-point price change in the underlying future. The delta of an option is simply the slope of the line which depicts the price of the option as a function of the price of the future. The delta changes as the price of the future changes, as can be observed from the graph. The delta for a call will always be positive and range from 0 to 1. The delta of a call is positive because an increase in the price of the future will result in an increase in the price of the call, but a drop in the price of the future will result in a decrease in the price of the call.

When the future is at 100, the December 100 call will move approximately 1/2 point for a 1 point move in the future. The delta is calculated:

Delta = change in price of the option / change .in price of the future

        = 1/2

 

The delta is the slope of the option price line and measures how the price of the option changes for a specific change in price of the future. The delta of a call is positive because the slope of the line is positive.

The delta for a put will always be negative and range from -1 to 0. The delta of a put is negative because an increase in the price of the future will cause the price of the put to drop, whereas a decrease in the price of the future will cause the put value to increase. The delta is -1/2 at the 100 strike in this example because a 2-point decline in the future will cause the put to increase by approximately 1. If we use calculus, the delta is the first derivative in the Black Scholes equation.

The delta of an option indicates how many options approximate one future. For example a call which has a delta of 1/2 should move about 1/2 point for a 1-point move in the future. Therefore 2 calls will approximately equal the movement of 1 future when the delta is 1/2. From the previous option graphs we can see that the delta of a call or put is usually 1/2 when the future is at the strike price. The trader must remember the delta changes as the future price changes which introduces the next idea-the gamma.

The gamma is the amount the delta changes with respect to a change in price of the underlying security.

The delta constantly changes as the price of the security changes. The trader must also monitor the gamma to determine how the delta changes so that the position does not become too long or short. The delta changes the most when the future is at the strike price and changes less the further the underlying is from the strike price. Therefore the gamma is greatest at the strike price. The gamma provides the trader with an indication of how long or short a position may become as the underlying future price changes. The delta and gamma are dynamic and usually measured by a 1-point move on the future.

Any position that is long options will have a positive gamma irrespective of whether the options are calls or puts. For example, assume you are long the January 100 straddle, which means being long 1 January 100 call and long 1 January 100 put and the market is currently trading at 100. The net delta of this position is 0 because the call delta is 1/2 and the put delta is -1/2 (recall, if the future is trading at the strike price the delta of a call is approximately 1/2 and the delta of a put is approximately -1/2). If the market rallies, the delta of the position will become more positive and increase from 0 and rise to as high as 1. If the market drops, the delta of the position will become more negative and decrease from a to as low as -1. This position will always become longer as the market rises and shorter as the market drops.

Any position that is short options will have a negative gamma irrespective of whether the options are calls or puts. Assume the opposite strategy and you are now short 1 February 100 call and short 1 February 100 put and the market is trading at 100. The net delta of the position is 0 as before. If the market increases, the delta will decrease and become negative so the position will become increasingly short up to -1. If the market drops, the delta will increase and become more positive so the position will become longer up to 1. This position will always become shorter as the market rallies and longer as the market drops.

Most novice traders are easily enticed by the benefits of a positive gamma which automatically allows the trader to get longer or shorter as the market goes up or down. A positive gamma is somewhat similar to pyramiding a position as it goes in your favor. A negative gamma is similar to pyramiding a position as it goes against you. However, the gamma must be evaluated with another important factor-the theta.