### E. Basic Option Greeks

**Option Intrinsic & Time Value**

In finance, the value of an option consists of two components, its intrinsic value and its time value. Time value is simply the difference between option value and intrinsic value. Time value is also known as extrinsic value, or instrumental value.

**Intrinsic Value**

The intrinsic value of an option is the value of exercising it now. If the option has a positive monetary value, it is referred to as being in-the-money, otherwise it is referred to as being out-of-the-money. If an option is out-of-the-money at expiration, its holder will simply abandon the option and it will expire worthless. For this reason we assume that the owner of the option will never choose to lose money by exercising, thus an option can never have a negative value.

Value of a call option: max[(S − K),0], or (S − K) +

Value of a put option: max[(K − S),0], or (K − S) +

As seen on the graph, the intrinsic value of a call option is positive when the underlying asset's spot price S exceeds the option's strike price K.

**Option Value**

Option value (i.e. price) is found via a formula such as Black-Scholes or using a numerical method such as the Binomial model. This price will reflect the "likelihood" of the option finishing "in-the-money". For an out-the-money option, the further in the future the expiration date - i.e. the longer the time to exercise - the higher the chance of this occurring, and thus the higher the option price; for an in-the-money option the chance in the money decreases; however the fact that the option cannot have negative value also works in the owner's favour. The sensitivity of the option value to the amount of time to expiry is known as the option's "theta"; see The Greeks. The option value will never be lower than its intrinsic value.

As seen on the graph, the full call option value (intrinsic and time value) is the red line.

**Time Value**

Time value is, as above, the difference between option value and intrinsic value, i.e.

Time Value = Option Value - Intrinsic Value.

More specifically, an option's time value reflects the probability that the option will gain in intrinsic value or become profitable to exercise before it expires. An important factor is the option's volatility. Volatile prices of the underlying instrument can stimulate option demand, enhancing the value. Numerically, this value depends on the time until the expiration date and the volatility of the underlying instrument's price. The time value of an option is not negative (because the option value is never lower than the intrinsic value), and converges towards zero with time. At expiration, where the option value is simply its intrinsic value, time value is zero. Prior to expiration, the change in time value with time is non-linear, being a function of the option price.

**Option Delta**

The delta of an option is the sensitivity of an option price relative to changes in the price of the underlying asset. It tells option traders how fast the price of the option will change as the underlying stock/future moves.

You can see how delta changes by spot change for call and put options at ATM (At-the-money), ITM (In-the-money) and OTM (Out-of-the-money) levels.

It can be formulated as;

Where V stands for Option Value and S stands for Spot Price of underlying.

As a rule of thumb ATM call options have delta of ~0.5 (or 50 in jargon) and ATM puts have ~-0.5 (-50 in jargon)

Delta shows the option traders what’s his exposure from these options in terms of underlying.

Also absolute value of delta roughly shows the probability of the option expiring in the money.

**Example**

Assume the trader holds 100 call options of XYZ stock with delta 0.60. By holding this many options, trader practically has a position of 100 x 0.60 = 60 XYZ stocks.

Which means if the price of XYZ goes up/down 1% he would make the same profit/loss from 100 options of XYZ with 0.60 delta and from 60 XYZ stocks.

**Relationship between call and put delta**

Given a call and put option for the same underlying, strike price and time to maturity, the sum of the absolute values of the delta of each option will be 1.00

If the value of delta for an option is known, one can compute the value of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from the known value. For example, if the delta of a call is 0.42 then one can compute the delta of the corresponding put at the same strike price by 0.42 - 1 = -0.58.

**Option Gamma**

The gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset. In other words, the Gamma shows the option delta's sensitivity to market price changes.

Gamma is important because it shows us how fast our position delta will change as the market price of the underlying asset changes.

Remember: One of the things the delta of an option tells us is effectively how many underlying contracts we are long/short. So, the Gamma is telling us how fast our "effective" underlying position will change.

In other words, Gamma shows how volatile an option is relative to movements in the underlying asset. So, by watching your gamma will let you know how large your delta (position risk) changes.

It can be formulated as;

Where V stands for Option Value and S stands for Spot Price of underlying.

The above graph shows Gamma vs Underlying price for 3 different strike prices. You can see that Gamma increases as the option moves from being in-the-money reaching its peak when the option is at-the-money. Then as the option moves out-of-the-money the Gamma then decreases.

Note: The Gamma value is the same for calls as for puts. If you are long a call or a put, the gamma will be a positive number. If you are short a call or a put, the gamma will be a negative number.

When you are "long gamma", your position will become "longer" as the price of the underlying asset increases and "shorter" as the underlying price decreases.

Conversely, if you sell options, and are therefore "short gamma", your position will become shorter as the underlying price increases and longer as the underlying decreases.

This is an important distinction to make between being long or short options - both calls and puts. That is, when you are long an option (long gamma) you want the market to move. As the underlying price increases, you become longer, which reinforces your newly long position.

If being "long gamma" means you want movements in the underlying asset, then being "short gamma" means that you do not want the price of the underlying asset to move.

A short gamma position will become shorter as the price of the underlying asset increases. As the market rallies, you are effectively selling more and more of the underlying asset as the delta becomes more negative.

**Option Theta**

Theta shows how much value the option price will lose for every day that passes.

An option contract has a finite life, defined by the expiration date. As the option approaches its maturity date, an option contract's expected value becomes more certain with each day.

This Time Value, also called Extrinsic Value, represents the uncertainty of an option.

Theta is the calcuation that shows how much of this time value is eroding as each trading day passes - assuming all other inputs remain unchanged. Because of this negative impact on an option price, the Theta will always be a negative number.

For example, say an option has a theoretical price of 3.50 and is showing a Theta value of -0.20. Tomorrow, if the underlying market opens unchanged (opens at the same price as the previous days close) then the theoretical value of the option will now be worth 3.30 (3.50 - 0.20).

It can be formulated as;

Where V stands for Option Value and T stands for time to expiration.

The above graph illustrates the effect on a OTM call option as it approaches maturity date. The increment as each day passes is what the Theta calculates.

You will notice that in the last remaining days of an option's life, it looses it's value quite rapidly.

This is one of the concepts traders use as a reason to short option contracts - to take advantage of this rapid rate of decay in an option's value as each trading day passes.

**Option Vega**

The Vega of an option indicates how much, theoretically at least, the price of the option will change as the volatility of the underlying asset changes.

Vega is quoted to show the theoretical price change for every 1 percentage point change in volatility. For example, if the theoretical price is 2.5 and the Vega is showing 0.25, then if the volatility moves from 20% to 21% the theoretical price will increase to 2.75.

Vega is most sensitive when the option is at-the-money and tapers off either side as the market trades above/below the strike.

It can be formulated as;

Where V stands for Option Value and σ stands for volatility.

The above graph plots the option Vega vs Underlying price for 3 different strike prices. Notice that the behaviour of an option Vega is similar to Gamma: increasing as the option moves from being in-the-money to at-the-money where it reaches its peak and then decreases as the option moves out-of-the-money.

**Note: **like the Gamma, Vega is the same value for calls and puts.

**Option Rho**

Rho is the change in option value that results from movements in interest rates.

The value is represented as the change in theoretical price of the option for a 1 percentage point movement in the underlying interest rate. For example, say you're pricing a call option with a theoretical value of 2.50 that is showing a Rho value of .25. If interest rates increase from 5% to 6%, then the price of the call option, theoretically at least will increase from 2.50 to 2.75.

It can be formulated as;

Where V stands for Option Value and r stands for interest rate.

Take a look at the above graphs, which plot the Rho of a call and a put option at 3 different points in time, across a range of strike prices, with a spot price of 100.

Unlike the other option greeks, Rho is larger for options that are in the money and decreases steadily as the option moves out of the money. Option Rho also increases with a greater amount of time to expiration.

These two factors are explained by the effect that interest rates have on the cost of carry of an option. ITM options, and options that have more time until expiration, will have higher premiums and therefore require more cash to hold the option until the expiration date.

Rho is generally the least important of all the option greeks. This is because option traders tend to focus on trading options that are close to expiration and out of the money.

The effect that interest rates have on option contracts is different for both call and put options.

There are two ways that interest rates affect option prices: the forward price and the premium discounting.

**Forward Price**

When calculating the price of an option, the first thing traders have to determine is what the price of he underlying asset will be at the expiration date. This is known as the forward price. Although an option trader can never know what this price will be, all other things being equal, it must be worth at least the cost of holding the asset until the expiration date. This value takes into account the cost of money i.e. what it would cost you in real terms to borrow and invest in the asset until the expiration date. This value, also referred to as the "cost of carry", uses the current interest rate to determine how much additional capital you will need to hold the asset until the expiration date.