Option Greeks are the various partial derivatives of the Black-Scholes Option Pricing Model, each with respect to a different variable. Greeks provide a way to measure the sensitivity of the option price to various factors, such as the underlying asset price, time, volatility, etc. The five key greeks commonly used by the traders are: <img class="aligncenter size-full wp-image-1258" title="greeks" src="https://financetrain.com/wp-content/uploads/2010/10/greeks.png" alt="" width="253" height="225" />
Delta
Delta measures the sensitivity of an option's value to a change in the price of the underlying asset. It indicates how much the value of an option will change with a one dollar change in the price of the underlying stock.
Gamma
Gamma is a measure of delta's sensitivity to changes in the price of the underlying asset. Gamma is the first derivative of delta with respect to the stock price. Gamma is also called curvature.
Theta
Theta is a measure of the sensitivity of an option to the time remaining until expiration. It is a measure of the time decay of the option. The value of theta is the dollar value that the option will lose each day with the passage of time.
Vega
Vega measures the sensitivity of the price of an option to changes in volatility. An increase in volatility increases the price of all options, and a decrease in volatility decreases the price of all options. Vega measures how much the value of the option will increase or decrease with change in volatility.
Rho
Rho is a measure of an option's sensitivity to changes in the risk free interest rate. Rho is the least used and least important greek. Unless the option has very long life, the changes in interest rates affect the premium only modestly.

Overview of Option Greeks

Delta is the change in option premium expected from a small change in the stock price.

It is an important by-product of the Black-Scholes model.

There are three common uses of delta.
<ul>
	<li>Measure of option sensitivity</li>
	<li>Hedge ratio</li>
	<li>Likelihood of becoming in-the-money</li>
</ul>
Measure of Option Sensitivity

For a call option, the delta is represented as:

<img class="aligncenter size-full wp-image-1202" title="delta of a call" src="https://financetrain.com/wp-content/uploads/2010/10/delta1.png" alt="" width="157" height="105" />

For a put option, the delta is represented as:

<img class="aligncenter size-full wp-image-1203" title="delta of a put" src="https://financetrain.com/wp-content/uploads/2010/10/delta2.png" alt="" width="150" height="90" />

Delta indicates how many shares of stock are required to replicate the returns of the option. For example, assume that a call option has a delta of 0.75. If the stock price rises by $1.00, the call option will increase by about 75 cents.

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Delta is always positive (between 0 and 1) for long calls and short puts and negative (between 0 and -1) for long puts and short calls. For a European option, the absolute values of the put and call deltas will sum to 1. In the Black-Scholes model, the call delta is exactly equal to N(d1).
<ul>
	<li>The delta of an at-the-money option declines linearly over time and approaches 0.50 at expiration</li>
</ul>
<ul>
	<li>The delta of an out-of-the-money option approaches zero as time passes</li>
</ul>
<ul>
	<li>The delta of an in-the-money option approaches 1.0 as time passes</li>
</ul>
Hedge Ratio

Delta is also used a measure of the hedge ratio.

Assume a short option position has a delta of 0.50. If an investor owns 100 shares of the stock, he can create a perfect hedge by writing 2 calls. This is because the underlying stock always has a delta of one.

Similarly, if a portfolio of options in a stock  is +2.50, the trader would be able to hedge the portfolio by selling short 250 shares of the underlying. This is called delta-hedging.

Likelihood of Becoming In-the-Money

Some traders also use delta is a measure of the probability that an option will be in the money at expiration.

For example, if an out-of the-money option has a delta 0.20, it indicates that the option has a 20% chance of being in the money at expiry, that is the stock price will be above the option striking price at expiration.

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Option Greeks: Delta

An option’s delta changes as the stock price changes. Gamma is a measure of the rate of change in the delta with respect to changes in the underlying price.

Gamma is the second derivative of the option premium with respect to the stock price. It is the first derivative of delta with respect to the stock price.

Gamma is represented as follows:

<img class="aligncenter size-full wp-image-1206" title="calculating_gamma" src="https://financetrain.com/wp-content/uploads/2010/10/calculating_gamma.png" alt="" width="262" height="227" />

Gamma is also called curvature. It is an important measure because it corrects the convexity of the option value.

Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passes.

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While establishing an effective delta-hedge for a portfolio, a trader may also want to neutralize the portfolio's gamma. This will ensure that the hedge will be effective over a wider range of underlying price movements.

Gamma will be larger for the at-the-money options, and gets lower as the options move in- and out-of-the-money options. Unlike delta, gamma is always positive for both calls and puts.

Options with gammas near zero have deltas that are not particularly sensitive to changes in the stock price

For a given striking price and expiration, the call gamma equals the put gamma.

Calculating Gamma

For calls and puts, gamma is calculated using the following formula:

<img class="aligncenter size-full wp-image-1207" title="calculating_gamma2" src="https://financetrain.com/wp-content/uploads/2010/10/calculating_gamma2.png" alt="" width="288" height="113" />

Option Greeks: Gamma

Theta is a measure of the sensitivity of an option to the time remaining until expiration. It is a measure of the time decay of the option. 

The value of theta is the dollar value that the option will lose each day with the passage of time. Theta value is calculated as a dollar value per year but is converted to a per day value by traders for their portfolios.

It is represented as follows:
<img class="size-full wp-image-1210 aligncenter" title="Theta Representation" src="https://financetrain.com/wp-content/uploads/2010/10/theta1.png" alt="" width="147" height="239" />
Theta is greater than zero because more time until expiration means more option value.

Theta is always negative for long calls and puts and positive for short (or written) calls and puts.

The passage of time hurts the option holder while it benefits the option writer. If you are long an option you are short theta, that is your portfolio will lose value with the passage of time (all other factors held constant).

Calculating Theta

For calls and puts, theta is calculated as follows:

<img class="aligncenter size-full wp-image-1211" title="calculating theta" src="https://financetrain.com/wp-content/uploads/2010/10/calculating-theta.png" alt="" width="438" height="206" />

The equations determine theta as a dollar value per year. A theta of -4.50, for example, means the option will lose $4.50 in value over a period of one year.

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Option Greeks: Theta

Vega is a measure of the sensitivity of the option price to the volatility of the underlying asset.

It is expressed as the amount of money an option will gain or lose, with a 1% increase or decrease in volatility.

The higher the volatility, the higher is the value of the option. For example, an option with a 0.25 vega will gain 0.25% in value for 1% increase in the volatility of the underlying asset.

It is represented as follows:

<img class="aligncenter size-full wp-image-1221" title="vega1" src="https://financetrain.com/wp-content/uploads/2010/10/vega1.png" alt="" width="209" height="211" />

All long options have positive vegas.

Vega is also called kappa, omega, tau, zeta, and sigma prime.

Vega can be an important Greek to monitor, especially in volatile markets since some of the option strategies can be very sensitive to changes in volatility. One such strategy is an option strangle, whose value is extremely dependent on changes to volatility.

Calculating Vega

Vega is calculated using the following formula:

<img class="aligncenter size-full wp-image-1222" title="vega2" src="https://financetrain.com/wp-content/uploads/2010/10/vega2.png" alt="" width="282" height="100" />

Option Greeks: Vega

Rho is a measure of an option's sensitivity to changes in the risk free interest rate.

It is expressed as the amount of money an option will lose or gain with a 1% change in interest rates.

For example, consider a call option with a rho of 0.05. This means that if the interest rates increase by 1%, the price of the option will increase by $0.05.

Calculating Rho

Rho is calculated using the following formula:

<img class="aligncenter size-full wp-image-1226" title="rho2" src="https://financetrain.com/wp-content/uploads/2010/10/rho2.png" alt="" width="271" height="151" />

Rho is the least used and least important greek.

Long calls and short puts have positive rho, that is, the option price will increase with an increase in interest rates and it will decrease with a decrease in interest rates.

Short calls and long puts have negative rho, that is, the option price will increase with a decrease in interest rates and it will decrease with a increase in interest rates.

For both call and put options, the more the time remaining for expiry, the higher is the impact of interest rates. Unless the option has very long life, the changes in interest rates affect the premium only modestly.

Deep out-of-the-money options have low rho compared to at-the-money and deep in-the-money options.

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Option Greeks: Rho

Delta neutral refers to a portfolio of underlying assets, where the value of the portfolio is unaffected by small changes in the value of the underlying assets.

Such a portfolio is created by taking positions in options such that the positive and negative deltas of various positions will offset each other thereby creating a portfolio whose value is insensitive to changes in the prices.

It is an important concept for institutional traders who establish large positions using various option strategies such as straddles, strangles, and ratio spreads.

We will take an example of a strangle to explain how delta neutrality can be achieved.
<h3>A Strangle Example</h3>
A stock currently trades at $50. The annual volatility of the stock is estimated to be 20%. The risk-free rate is 5%.

An options trader decides to write six-month strangles using $45 puts and $55 calls.

As a quick refresher, a strangle involves taking position in both a call and put with different strike prices but with the same maturity and underlying asset. This option strategy is profitable only if there are large movements in the price of the underlying asset.

The two options will have different deltas, so the trader will not write an equal number of puts and calls.

How many puts and calls should the trader use?

The delta for the call option is 0.3344

The delta for the put option is -0.1603

The ratio of the two deltas is -0.1603/0.3344 = -0.48. This means that delta neutrality is achieved by writing 0.48 calls for each put.

One approximate delta neutral combination is to write 25 puts and 12 calls.

Delta neutrality is useful for strategies in which a trader is neutral about the future dynamics for the market. So, the trader doesn’t assume either a bullish or a bearish position.

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Delta Neutrality

This video familiarizes traders with a set of Greek risk factors used to monitor a portfolio's profile. It explains the five key greeks:
<ol>
	<li>Delta: The relative change in an option's price for a given change in the underlying asset.</li>
	<li>Gamma: The rate of change of delta for a given change in the underlying asset's price.</li>
	<li>Theta: The relationship between time and the price of an option.</li>
	<li>Vega: The change in an option's price due to a change in its implied volatility.</li>
	<li>Rho: AN option's sensitivity to interest rates.</li>
</ol>
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Option Greeks

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