Option Greeks

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What Are Option Greeks?


The mathematical characteristics of the Black-Scholes model are named after the greek letters used to represent them in equations. These are known as the Option Greeks. The 5 Option Greeks measure the sensitivity of the price of stock options in relation to 4 different factors; Changes in the underlying stock price, interest rate, volatility, time decay.

Option Greeks allow option traders to objectively calculate changes in the value of the option contracts in their portfolio with changes in the factors that affects the value of stock options. The ability to mathematically calculate these changes gives option traders the ability to hedge their portfolio or to construct positions with specific risk/reward profiles. This alone makes knowing the Option Greeks priceless in options trading.

For the amateur trader, knowing the delta (Greek Symbol δ) of your options position is the most important as it gives you an indication of how your option's value will change with movements in the underlying stock price - all other variables remaining the same. Knowing your time decay (theta θ) gives you an indication of how much time value your options trading position is losing each day - all other variables remaining the same.

Professionals use the Option Greeks to measure exactly how much they need to hedge their portfolio and to surgically remove specific risk factors from their portfolio. The Option Greeks also enable the measurement of how much risk the portfolio is exposed to, and where that risk lies (with movements in interest rates or volatility, for example). Having a comprehensive knowledge of options greeks is essential to long term success in options trading.

The 5 Option Greeks are:

Delta (Greek Symbol δ) - a measure of an option's sensitivity to changes in the price of the underlying asset

Gamma (Greek Symbol γ) - a measure of delta's sensitivity to changes in the price of the underlying asset

Vega - a measure of an option's sensitivity to changes in the volatility of the underlying asset

Theta (Greek Symbol θ) - a measure of an option's sensitivity to time decay

Rho (Greek Symbol ρ) - a measure of an option's sensitivity to changes in the risk free interest rate


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Option Greeks - Delta (δ)

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Option Delta - Introduction


Delta value is the most well known and the most important of the option greeks. It is the degree to which an option price will move given a change in the underlying stock price, all else being equal. For example, an option with a delta of 0.5 will move half a cent for every one cent movement in the underlying stock. Which means, stock options with a higher delta will increase / decrease in value more with the same move on the underlying stock versus stock options with a lower delta value.


Why Is Option Delta Important?


Knowing the delta value of your options is important for option traders who do not hold stock options until expiration. In fact, few options traders hold speculative positions to expiration in options trading. If you are speculating a quick $1 surge in the underlying stock within a few days and bought call options in order to prepare for the move, the delta of your call options will tell you exactly how much money you will make with that $1 surge. The option delta therefore helps you plan for how much call options to buy if you are planning to capture a definite cash value in profits and helps you calculate the options leverage involved.


Option delta is also important for option traders who uses complex position trading option strategies. If an option trader is planning to profit from the time decay of his short term stock options, then that option trader needs to make sure that the overall delta value of his position is near to zero so that changes in the underlying stock price do not affect the overall value of his position. This is known as Delta Neutral in options trading.


Characteristics of Option Delta And Reading Delta Values


A far out of the money stock option will have a delta very close to zero; an at the money stock option a delta of 0.5; a deeply in the money stock option will have a delta close to 1.

delta of short term call options

The picture above are real delta values for MSFT's call options with 29 days left to expiration. Notice that the delta value increases nearer to 1 as the option becomes more In-The-Money and decreases nearer to 0 as the option becomes more and more Out-of-The-Money. Call options with delta of 1 means that it will move up $1 as the underlying stock go up by $1, perfectly shadowing every move of the underlying stock. In a way, the delta of a stock option also tells you the probability that the option will expire In-The-Money. That is why far Out-of-The-Money options have a delta of zero, reflecting that there is almost no chance of that option expiring In-The-Money.

delta of long term call options

The picture above are real delta values for the same MSFT call options but this time with 183 days to expiration. Notice that as the expiration date gets further away, the delta values for call options of the same strike price drops as well. The April27.50Call option in the previous picture has a delta value of 0.779 but the Oct27.50Call option in this picture only has a delta value of 0.697. This shows that in terms of profitability, nearer term options are more profitable as they are cheaper and has a higher delta but also carries more risk as it allows less time for the underlying stock to move in your favor. Options trading is all about balancing risk and reward.

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Call deltas are positive; put deltas are negative, reflecting the fact that the put option price and the underlying stock price are inversely related.

The delta is often called the neutral hedge ratio. For example if you have a portfolio of shares, divide the amount by the delta gives you the number of calls you would need to write to create a neutral hedge - i.e. a portfolio which would be worth the same whether the stock price rose by a small amount or fell by a small amount. In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price should be exactly offset by a loss on the value of the calls written, and vice versa. That gave rise to the important concept of "Delta Neutral" hedging or positions. Learn all about Delta Neutral Hedging now!


Does Option Delta Stay The Same Till Expiration?


Sadly, option delta changes all the time. Option delta changes as the price of the underlying stock changes, bring that option more and more in the money or more and more out of the money. This effect is governed by the option gamma. Even if the underlying stock remain stagnant, option delta for In The Money options increases as expiration nears and option delta for Out Of The Money options decreases as expiration nears.


Option Delta Formula


The formula for calculation of option delta is:

option delta formula

Where...
C = Value of the Call Option
St = Current value of underlying asset
N(d1) = Rate of change of the option price with respect to the price of the underlying asset
T = Option life as a percentage of year
ln = Natural log of
Rf = Risk free rate of return


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Option Greeks - Gamma (γ)

Gamma - Introduction


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 changes in relation to the market price of the underlying asset, however, it is not normally needed for calculation for most option trading strategies. Gamma is particularly important for delta neutral traders who wants to predict how to reset their delta neutral positions as the price of the underlying stock changes.

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Reading Gamma Values



gamma of short term call options

The picture above depicts the real gamma value of MSFT's call options with 29 days to expiration while the bottom picture depicts the gamma value of the same call options with 183 days to expiration. You would notice that as expiration date gets further away, the gamma value becomes smaller. This makes stock options with longer expiration less sensitive to delta changes as the underlying stock value changes.

gamma of long term call options


Option Gamma Formula


The formula for calculation of option gamma is:

option gamma formula

Where...
d1 = Please refer to Delta Calculation above
S = Current value of underlying asset
T = Option life as a percentage of year


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Option Greeks - Vega

Vega - Introduction


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 implied volatility. For example, if the theoretical price is 2.5 and the Vega is showing 0.25, then if the implied volatility moves from 20% to 21% the theoretical price will increase to 2.75.


Reading Vega Value


Vega is most sensitive when the option is at-the-money and tapers off either side as the market trades above/below the strike. Some option trading strategies that are particularly vega sensitive are Long Straddle (where a profit can be made when volatility increases without a move in the underlying asset) and a Short Straddle (where a profit can be made when volatility decreases without a move in the underlying asset). As you can see from the below picture of MSFT Call Option's real vega values, it reduces drastically as it goes in the money and out of the money.

vega of short term call options

The below picture depicts the same MSFT call options with 183 days to expiration and you can see that the vega value is much higher than the above picture with only 29 days to expiration. This shows that stock options with longer expiration changes in value as volatility changes than nearer term stock options.

vega of long term call options

Vega is also the greek that most affect option prices second to Delta.

To completely understand this, you will need to understand how stock options are priced and how volatility is factored into it. Learn About Implied Volatility Here.


Option Vega Formula


The formula for calculation of option vega is:

option vega formula

Where...
d1 = Please refer to Delta Calculation above
S = Current value of underlying asset
T = Option life as a percentage of year
C = Value of Call Option


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Option Greeks - Theta (θ)

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Theta - Introduction


Theta measures how fast the premium of a stock option decay with time. By Time Decay, we mean the depreciation of the premium value of a stock option contract. To completely understand what the premium of a stock option is, you need to understand how stock options are priced.

The theta value indicates how much value a stock option's price will diminish per day with all other factors being constant. If a stock option has a theta value of -0.012, it means that it will lose 1.2 cents a day. Such a stock option contract will lose 2.4 cents over a weekend. (Yes, the effect of theta value and time decay is active even when markets are closed!)

The nearer the expiration date, the higher the theta and the farther away the expiration date, the lower the theta. Some option trading strategies that are particularly theta sensitive are Calendar Call Spread and Calendar Put Spread where traders need to maintain a net positive theta in order to ensure a profit.


Reading Theta Value



theta of short term call options

Compare the theta values for MSFT Call Options with 29 days left to expiration above and the same call options with 183 days left to expiration in the picture below and you will notice that stock options with a longer expiration date has a lower theta value and therefore a lower rate of time premium decay than stock options with a shorter expiration date. Hence it is not wise to buy short term stock options with a high premium value. Notice also that theta value drops as the stock option gets further in the money and out of the money as there are very little premium value left in deep in the money and out of the money options.

theta of long term call options


Characteristics Of Theta Value


You might have noticed something perculiar about the theta of Out of The Money (OTM) options when comparing the two pictures above and that is, theta value for OTM options are higher with longer expiration and lower with nearer expiration.

Indeed, theta behaves differently for ITM/ATM options and OTM options:

ITM/ATM Options Theta


Further Expiration : Low Theta
Nearer EXpiration : High Theta

OTM Options Theta


Further Expiration : High Theta
Nearer EXpiration : Low Theta

As you can see from the below illustrations, ITM and ATM options decays fastest during the last 30 days to expiration whereas OTM options decays the least during the final 30 days, which is also due to the fact that OTM options near to expiration has too little premium value left to decay on anyways.

theta decay of ITM / ATM options

theta decay of OTM options


Option Theta Formula


The formula for calculation of option theta is:

option theta formula

Where...
d1 = Please refer to Delta Calculation above
T = Option life as a percentage of year
C = Value of Call Option
St = Current price of underlying asset
X = Strike Price
Rf = Risk free rate of return
N(d2) = Probability of option being in the money


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Option Greeks - Rho (ρ)

Rho - Introduction


Rho measures the sensitivity of an option or options portfolio to a change in interest rate.

For example, if an option or options portfolio has a rho of 0.017, then for every percentage-point increase in interest rates, the value of the option increases $0.017. However, it is not normally needed for calculation for most option trading strategies.


Reading Rho Value


Notice from the below real rho values for MSFT Call Options that rho values are usually pretty low and therefore a percentage increase or decrease in interest rates don't really make much of a difference to a stock option.

rho of short term call options

Notice also from the below picture that longer term stock options have a higher rho value than nearer term ones, however, even higher ones like these hardly come close to even 1. That means that a percentage change in interest rates only make a slight $0.105 change in the option value even if the rho values are as high as 0.105 below.

rho of long term call options


Rho Value Formula
The formula for calculation of option rho is:

option rho formula

Where...
d1 = Please refer to Delta Calculation above
T = Option life as a percentage of year
C = Value of Call Option
X = Strike Price
N(d2) = Probability of option being in the money


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