# Monthly Archives: January 2012

## Option Greeks ~ Delta and Gamma

To determine the theoretical price of an option, an option pricing model is used. Buyers and sellers on the marketplace determine the actual price constantly throughout the trading day. The most commonly used pricing model is the Black Scholes Model. From the Black Scholes Model we can derive some calculations used to determine how an option price will react to market variables, known as the Greeks because letters from the Greek alphabet are used to designate them. The Delta is the rate of change of option price with a corresponding one point move in the underlying instrument. A one hundred share position in the underlying instrument will always have a delta of 100. I like to express the delta in terms of the number of shares of the underlying that you hold. If you are long 100 shares of XYZ then you are a positive 100 delta position. If you are short 100 shares of XYZ then you have a negative -100 delta position. When you combine complex option and stock positions you’ll have to keep track of the deltas which are additive. At the money options will have a delta of about 50. So, if you own 100 XYZ and decide to sell an at the money call with a delta of 50, your net position delta will now be 50, 100-50 = 50. Which means that as long as the delta of the short call remains at 50 your combined position will behave like 50 shares of the underlying, not 100 shares, so the risk of ownership in the underlying is reduced by a 50 share equivalent. As call options move in the money, the delta will increase, as they move out of the money, the delta will decrease. The delta is also roughly equivalent to the probability of the option being in or out of the money at expiration. An at the money option will have a delta of about 50, which means that it will move half of what the underlying moves, but also has a 50/50 chance of being in the money at expiration. A long call with have a positive delta and a long put will have a negative delta. Conversely the short call will have negative delta and the short put will be a positive delta position. If you hold a 100 share position and are concerned about risk you could sell an at the money call and use the proceeds to buy an at the money put. If the short call has a -50 delta and the long put also has a -50 delta you now have a delta neutral position, 100-50-50=0. As long as the position stays delta neutral you do not have market risk. The deltas, however are not fixed but are variable so they change when the price of the underlying stock changes and adjustments have to be made if the investor wishes to maintain a delta neutral position. The rate at which the delta changes is known as the Gamma, the gamma is the rate of change of the delta with a corresponding one point underlying move.

Option income investors who use short straddles, short strangles, iron condors, iron butterflies, etc. will have delta neutral positions with short gamma, meaning that they don’t want the underlying to move, if the underlying moves the delta will change and the position will take on a directional bias. So they have to adjust to maintain a delta neutral position.

To determine the theoretical price of an option, an option pricing model is used. Buyers and sellers on the marketplace determine the actual price constantly throughout the trading day. The most commonly used pricing model is the Black Scholes Model. From the Black Scholes Model we can derive some calculations used to determine how an option price will react to market variables, known as the Greeks because letters from the Greek alphabet are used to designate them. The Delta is the rate of change of option price with a corresponding one point move in the underlying instrument. A one hundred share position in the underlying instrument will always have a delta of 100. I like to express the delta in terms of the number of shares of the underlying that you hold. If you are long 100 shares of XYZ then you are a positive 100 delta position. If you are short 100 shares of XYZ then you have a negative -100 delta position. When you combine complex option and stock positions you’ll have to keep track of the deltas which are additive. At the money options will have a delta of about 50. So, if you own 100 XYZ and decide to sell an at the money call with a delta of 50, your net position delta will now be 50, 100-50 = 50. Which means that as long as the delta of the short call remains at 50 your combined position will behave like 50 shares of the underlying, not 100 shares, so the risk of ownership in the underlying is reduced by a 50 share equivalent. As call options move in the money, the delta will increase, as they move out of the money, the delta will decrease. The delta is also roughly equivalent to the probability of the option being in or out of the money at expiration. An at the money option will have a delta of about 50, which means that it will move half of what the underlying moves, but also has a 50/50 chance of being in the money at expiration. A long call with have a positive delta and a long put will have a negative delta. Conversely the short call will have negative delta and the short put will be a positive delta position. If you hold a 100 share position and are concerned about risk you could sell an at the money call and use the proceeds to buy an at the money put. If the short call has a -50 delta and the long put also has a -50 delta you now have a delta neutral position, 100-50-50=0. As long as the position stays delta neutral you do not have market risk. The deltas, however are not fixed but are variable so they change when the price of the underlying stock changes and adjustments have to be made if the investor wishes to maintain a delta neutral position. The rate at which the delta changes is known as the Gamma, the gamma is the rate of change of the delta with a corresponding one point underlying move.

Option income investors who use short straddles, short strangles, iron condors, iron butterflies, etc. will have delta neutral positions with short gamma, meaning that they don’t want the underlying to move, if the underlying moves the delta will change and the position will take on a directional bias. So they have to adjust to maintain a delta neutral position.