When using options as a hedge we use specific terms to define each risk associated with how the risk management variable impact the price of the option. For some of the numbers they can be remembered easily using the first letter of the word. For others, you will find them quite easily memorized even without a memory trick. **DELTA** – Calculates how much the option will change for a $1 change in the underlying asset **GAMMA** – Calculates how much Delta will change as the underlying asset changes **THETA** – TIME: how much the option price will decay as time passes **VEGA** – VOLATILITY: how much the option price will change for a 1% change in volatility **RHO** – RATES: how much the option price will change for a 1% change in rates **PSI:** – DIVIDEND PAYMENT: measures a change in dividends paid during life of option For each Greek this section provides a definition of the risk measurement, followed by two graphs. Each Greek is impacted by the price of the underlying and the time remaining to expiration. Notice the matrix below differs from the matrix in the first section. In this section there are three strike prices: the 60 call & put; the 65 call & put and the 70 call & put.

An in the money call option would be, for example the $60 call. An option to buy the stock at $60/ share with the stock at $65 has $5 of intrinsic value. An in the money put option would be, for example the $70 put. An option to sell the stock at $70/share with the stock at $65 has $5 of intrinsic value. Remember: IN-the-money options have INtrinsic value. N.B.: Notice the theoretical value for the $60 call is $6.49. Less $5 of intrinsic value leaves $1.49 of time value.

An at-the-money call & put option would be, for example the $65 call and put. The right to buy or sell the stock at $65 with the stock at $65 Remember: At-the-money means the strike price and underlying asset price is the same N.B.: The 65 call at $3.44 is all time value

An out of the money call option would be, for example the $70 call. An option to buy the stock at $70/ share with the stock at $65. The premium is 100% time value. The stock would need to rise $5 or 7.14% to be in-the-money. An out of the money put option would be, for example the $60 put. An option to sell the stock at $60/share with the stock at $65 has only time value. The stock would need to fall $5 or 8.33% to be in the money.

Reading the Matrix below from left to right starts with the inputs to the Black-Scholes pricing model. To the right of the words CALL and PUT are the outputs: Option price, Delta, Gamma, Vega, Theta & Rho.

When studying each Greek all other variables are presumed unchanged to learn each risk management figure with respect to the variable we use it for. In our Membership program The Greeks are discussed in much greater detail.

– Delta defines how much the option price will change for a one unit change in the underlying asset. – Calls have positive delta’s; puts have negative delta’s. This is useful so you always know the impact on price. In other words, if the underlying asset rises:

- The call’s value will rise (a positive asset change * a positive delta)
- The put value will fall (a positive asset change * a negative delta)

**The second definition for delta: **it represents the probability the option will expire IN THE MONEY. Consider this while viewing the graph below. **The graph below shows the $65 call delta (X axis) as the asset price changes (Y axis).** With the asset at $65, you will note the delta is approximately .50. In other words, there’s a 50% probability of the call expiring in the money.

The graph below shows the put delta as price changes. You’ll note the $65 put delta with the asset at $65 is approximately -.50. In other words, there’s a 50% probability the put expires in the money.

Gamma tells us how much **delta** will change as price changes and time passes. Gamma measures, indirectly, how the option price will move as price and time change. **The gamma on the $65 call is +.05** Gamma is always shown as a positive number, until a long or short position is opened. Gamma on a long position is a positive number: Are a positive number * a long position = a positive gamma Gamma on a short position is a negative number: Are a positive number * a short position = a negative gamma A positive number * a positive number = a positive number

The change in the price of a put with respect to gamma is also greatest At-The-Money and becomes lower as the asset decreases in price.

Options risk over time is important to understand. Time’s impact on the price of an option is vital to understand how option risk changes as time passes. To understand each risk management number, We’ll define the risk over time, provide an example and show a graph of gamma over time. Notice below the graph of gamma over time. The slope gets steeper the closer the option gets to expiration. Inside of 14 days, the slope really ramps up. Bear in mind the graph below presue the aasset is unchanged at $65 per share for the entire 60 days.

- Theta is the rate of change of the value of the option with respect to the passage of time
- Also known as Time decay; it is a negative number shown most often one days decay.
- Theta and Gamma are related, as you learned more advanced concepts in market making you will better understand why this relationship is so important.
- At-The-Money options have the greatest Gamma & Theta, regardless of time to expiration
- Out-of-the-money options gamma and theta have relatively low gamma and theta, regardless of time to expiration.
- Longer dated ATM option = low theta (& low gamma)
- Shorter dated ATM option = high theta (& high gamma)

The theta of put is identical to the call. As with Gamma we saw no reason to repeat those graphs. It’s most important to understand the concepts. Theta over time increases (as a negative number) as expiration gets closer. As with GAMMA, theta also slope becomes larger within 14 days to expiration.

- Vega (n) is the rate of change of the value of an option (or derivatives portfolio) with respect to a 1% change in volatility
- As premiums rise, implied volatility is higher. Our option increases .105 for a 1% increase in volatility.
- Vega is highest when option is at-the-money. Vega is greater for longer term options; Vega is smaller for shorter term options

Vega is always a positive number for a long position; and a negative number for a short position. The graph below shows that vega decreases as time to expiration draws close.

As your interest in more advance topics grows you’ll see how vega is used to trade earnings with little volatility risk. The knowledge center doesn’t cover Rho & PSI. This information is covered in the advanced material to separate the concepts of your primary Greeks to managing the changes in rates and dividends.