# Option Greeks

## Overview of Option Greeks

Option greeks measure the options sensitivity to various risk components inherent to the price of an option. These measure include the speed of the underlying securities price movement, interest rate movement, time decay of an option, and volatility.

Delta and Gamma measure the options sensitivity to the speed of price changes in the underlying security, Rho measures the options interest rate sensitivity, Theta measures the change in the options price due to a change in the time left till expiration on the option, and Vega measures the change in the options price due to changes in the options historical volatility.

## Delta Basics

Delta measures the rate of change in the option price over the rate of change in the price of the underlying security. Therefore, we can say that delta measures the speed of the option price movement relative to a single point move in the underlying security. Long calls and naked puts have positive delta while short calls and long puts have negative delta.

Delta = (rate of change in option price) / (rate of change in underlying security price)

If you own a call option at $2 when the stock was $30 and now the stock moves up to $31 and the option does as well to $2.50. In our formula above, the delta in this situation would be ($2.5 - $2) / ($31 - $30) = .5

Conversely, let's say you owned a put at $2.00 and the stock moved higher by one point which resulted in the put option decreasing in value by $.5. The delta in this case would be ($1.5 - $2) / ($31 - $30) = -.5

The closer the delta is to 1 or -1, the greater response in the price of the option when the price of the underlying changes.

The examples above assumed that nothing else changed; however, in reality, changes in vega, theta, and rho can impact delta.

### Delta Ranges for Calls and Puts

If an option is in the money, the delta for a call will approach 1 while the delta of a put will approach -1. At the money (ATM) options will be near .5 and -.5 while out of the money options will have a delta value approaching 0 as they move further out of the money. Also remember, as the option comes closer to expiration, especially within 30 days, the delta curve becomes steeper; basically, the option becomes more sensitive to price movement in the underlying.

### Delta Neutral Trading

Delta neutral trading, also known as "hedge" trading is a method of trading where the total position delta is 0. The idea is to hedge your position by slowing your position speed down. Delta neutral trading is used by many traders to make profitable adjustments on their trade as the price of the security moves up and down. For example, a popular strategy is to make adjustments to your total position to bring it back to delta neutral after the underlying security has moved 20% in either direction. This can be done by making adjustments to the profitable side of your trade.

Let's create a simple example of a delta neutral trade. Assume we bought 100 shares of Microsoft which would represent +100 deltas. We would need to either buy 2 at the money puts OR sell 2 at the money calls OR buy 1 at the money put and sell 1 at the money call. Either scenario would get you to delta neutral.

Remember, delta neutral does NOT mean that you have set up a risk free position, it means that you have slowed down the speed of the percentage changes of your position.

## Gamma Basics

The delta of a stock relies on the price of the stock in relation to the strike price of the option. Therefore, when the stock price changes, so does the delta. This is where gamma becomes relevant. Gamma is an estimation of the change in delta for a 1 point move in a stock and can be thought of as the second derivative of delta. A large gamma value indicates that delta will shift strongly as the underlying security moves up and down.

A long call and long put will have positive gamma while the short counterparts will always be negative. A positive gamma refers to the idea that the delta of a long will become higher, or closer to 1, as the underlying security moves higher. The long put gamma will also move closer to -1 as the underlying security continues to move lower. The opposite can be said for short calls and short puts.

Gamma reaches its highest value when a stock is trading at the money or near the money. This value goes lower and lower as the security moves further out of the money or further in the money. This makes logical sense as the option price has the highest probability of moving from being OTM to ITM or ITM to OTM.

## Theta Basics

Theta represents the measure for time decay of an option. Remember, an option price consists of intrinsic value and time premium. Theta measures the decay in time premium as every day passes until options expiration. Therefore, we can say that the theta for a long call or put will be negative while the opposite can be said for the short call and put. This is true because when you are long an option, you will lose money in that option every day all else being equal due to the time premium decaying. However, the time decay in a short option will increase your profits.

Theta does not adjust evenly as time goes on. The closer and closer the option is to expiration, the greater the time decay. Theta will accelerate at a higher rate especially when the option has less than 30 days to go. This also makes logical sense since the option has less time to get or stay in a profitable situation. Additionally, an options theta will be highest when the stock is at the money. Since the stock has basically no intrinsic value, the time value component is the majority of the premium and will fluctuate strongly as expiration approaches.

### Relationship between Theta and Gamma

There is a direct correlation between theta and gamma. When an options gamma is high, the theta moves higher as well. When we say higher, it means theta becomes more negative which negatively impacts the time premium for a long option holder. Some options traders will actually play the high theta by selling shorter term options and buying that same strike option with a greater term to maturity at the same time. They are banking on the fact that the longer dated option will have slower time decay than the shorter dated option.

## Vega Basics

Moving on to the volatility component of an option; we measure the options price sensitivity to volatility using Vega. Vega may also be referred to as kappa by some. Volatility can be calculated by measuring the standard deviation of the last 30 days of closing prices in the underlying security, commonly known as historical volatility. For example, if a $50 stock has a 30% volatility figure, you can say that the expectation is for the stock trade between $35 and $65. Historical volatility is used to determine the fair value of the option; however, options rarely trade in the open market at fair value. "Implied" volatility then estimates the volatility using the market premium of the options. We do not want to go into too much detail on this but just know that there are two measurements for volatility and that one is derived from past market data and one is derived from current options premiums themselves.

We can go one step further to say that vega is a measurement of the change in the option price for each 1% adjustment in volatility. Using our example, if the vega for this option was 20, we can say that the option would move higher by $1.00 if the stock's volatility

Higher volatility, or vega, results in higher option prices. This is true because higher volatility gives the option a better chance to expire in the money.

Options exhibit the highest vega when the underlying is at the money and gradually declines as the stock moves ITM or OTM.

## Rho Basics

Our last greek, Rho, measures theoretical option price changes due interest rate shifts. While this measure of option price sensitivity is the least used, it has more relevant context when applied to higher priced stocks. Why? Remember that a call option commands a large amount of stock with a relatively small amount of investment. Most times the value of the underlying that the option commands is worth in excess of 10 times the value of the option itself. If you would have to buy the stock, you would need quite a bit more money and the interest expense related that amount is built into a call option premium. As you can see, as interest rates increase, a call option will increase in value and a put option will decrease in value. It is for this reason that calls have a positive Rho when interest rates rise. Conversely, if interest rates fall, put premiums will increase while call premiums will decrease.