• Bocconi Students Options Club

The Greeks

What are “Greeks”

and what do we use them for?


In the realm of options, Greeks are used to study the interactions and links between parameters such as stock price, option price, time, volatility, etc. Some of the main Greeks are notably used to asses risk in light of specific factors, as well as giving other important information of the option’s behavior. These variables are named after Greek symbols (e.g. delta, gamma, theta, rho, vega…).

To each Greek is associated a value, that gives relevant information about the options’ behavior and movement. The most relevant Greeks (Delta, Vega, Theta, Gamma, and Rho) are found by calculating the partial derivative of the options pricing model (often the Black-Scholes model is used).

It is important to note that the value of Greeks change with time, and therefore an analysis of options with Greeks should be continuous, studying the variation in time.


Delta

Delta (Δ) is the rate of change between the price of the option and a $1 change in the underlying stock price. Indeed, Delta is a way to calculate the price sensitivity of the option in regards to the underlying asset. For call options, Delta varies between 0 and 1, while for put options, between 0 and negative 1.

In order to understand better, let’s look at an option with a delta of 0.70. This means that an increase in the underlying stock price by $1 would entail an increase in the options price by 70 cents.

In addition, knowing the delta of every asset allows the trader to implement a delta-neutral portfolio. A delta neutral portfolio averages out positive and negative deltas in the portfolio to obtain an overall delta of 0. Traders can use delta neutral strategies to profit from time decay or volatility of the options. In addition, a delta neutral strategy is also used to hedge positions and shield from short term stock movement. A delta neutral strategy is very risky (especially with high volatility stocks), and can incur losses when the price of the stock suddenly changes significantly in any direction.

Finally, an option’s delta can also be used to approximate the probability that an option will expire in the money. As an example, for an option with a delta of 0.60, a 60% chance of finishing in the money can be estimated.


Theta

Theta (Θ) represents the time sensitivity of an option, often called the option’s time decay. In other words, Theta indicates by how much the value of the option changes with each passing day. As an example, imagine an investor holds a call option with a theta of -0.40. This means that an option's price would decrease by 40 cents with every passing day, ceteris paribus.


The Theta value is always higher in at the money situations compared to in the money and out of the money. The time decay of an option is exponential; therefore, it will lose value faster when closer to its expiration date. Long calls and puts will have a negative Theta because as you get closer to your option maturity the odds of your option moving in the desired direction is lower as you don’t have much time.


For example, if your option is out of the money (set to expire worthless if the underlying stock doesn’t move) then logically a long option that expires in a few days won’t be worth much because the odds of the underlying stock moving strongly and your option becoming profitable are low while that same option expiring in three months will be worth more as there is a higher probability of the stock moving to make your option profitable.

On the other hand, short calls and puts will tend to have positive Theta with investors making money off of theta decay. Since Theta measures time decay, an asset whose value does not change because of time will have a Theta value equal to 0 (e.g. stocks).


Gamma

Gamma (Γ) studies the rate of change in delta given the price of the underlying asset. Gamma is also called second order price sensitivity. It indicates the change of delta given a $1 surge in the underlying’s asset price.

To give an example, let’s assume a trader purchases a call option on a certain stock. This option’s delta is 0.50 and the gamma is 0.30. Therefore, if the stock price varies by $1, the delta will vary in the same direction by 0.30.


Gamma studies the stability of an option's delta. A very high gamma value would entail a strong shift in delta after a change in stock price. Like Theta, Gamma increases as the expiration date approaches. In addition, Gamma is higher when the option is at the money, and lower when its strike price is very different from the stock price( whether the options is in or out of the money). Options that expire later will have lower Gamma values, since the sensitivity of delta to the underlying asset’s price.

As with the previously seen delta neutral portfolio strategy, traders can choose to hedge gamma, designing a delta gamma neutral strategy, which means that variation in the stock price will keep delta constant.


We’ve previously stated the advantages to keeping delta constant, but we’ve also seen that a delta neutral strategy can be risky when subject to volatility. The goal of gamma neutrality is to eliminate that risk, and to keeping volatility low. Indeed, since gamma represents the change in price sensitivity given a change in the underlying’s asset’s price, it can be considered as a measure of volatility. Keeping gamma close to zero can be a strategy to what is defined as “sealing profits” in. This means that in periods of high volatility (and therefore higher risk of losing all profits made to date), implementing a gamma neutral strategy would allow the position-holder to avoid losses by eliminating volatility.


Vega

Vega (v) measures the rate of change of an option's value given the underlying asset's implied volatility. Indeed, Vega is calculated measuring the change in the option’s price given a change of implied volatility of 1%. To give an example, if we have an option with a Vega of 0.70, and implied volatility changes by 1%, the option’s value will change by 70 cents.

There is a positive correlation between volatility of the underlying asset and its option’s value; therefore, an increase in volatility will increase an option’s value by Vega (and vice versa for a decrease in implied volatility).


Vega is higher for options further away from expiration (Vega decays with time), and it is higher when the option is at the money (as opposed to when the strike price is very different from.


Rho

Rho (p) indicates the option price sensitivity to the interest rate: therefore, it is the rate of change between the value of an option and a 1% variation in the interest rate.

As an example, let’s assume an option with a rho of 0.30 and a price of $2.50. If interest rates rise by 1%, the value of the call option would increase to $2.80, ceteris paribus. The opposite is also true in the case of put options. Rho is higher when the option is at the money and the option expires later.



Minor Greeks

We’ve discussed the most relevant and used Greeks in options trading. However, there are many other that are less known. These are some of the less known Greeks: epsilon, lambda, vera, color, vomma, speed, ultima, zomma.

These less known parameters are becoming increasingly more popular amongst traders, that want to find new strategies to take advantage of option behaviors and to understand their risks better.