# π₯Greeks

Delta, Ξ

Ξ **is the amount an option price is expected to move based on a $1 change in the underlying stock**

Call options have positive delta, between 0 and 1. This means if the price of the underlying asset increases and no other variables change, the price of the call option will increase.

*Example: A call has a delta of 0.5 and the underlying crypto increases in price by $1, the price of the call option will go up by around $0.5.*

Puts have a negative delta, between 0 and -1. This means if the underlying assets increases in value and no other variables change, then the price of the option will go down.

In general, options which are in-the-money will move more than out-of-the-money options, and options which are closer to expiry will react more than longer-term options to the same price changes in an underlying asset.

As options approach expiry, the delta for in-the-money calls approaches 1, whilst the delta for out of the money calls approaches 0 and doesn't react to changes in the stock. This is because if the options are held to expiry, they'll either be exercised and become the underlying asset, or they expire worthless.

## Gamma, πͺ

**πͺ represents the rate of change between an options's Delta and a $1 change in the underlying asset's price**

This 'acceleration' is best shown in this figure:

A $50 strike call option has a 50 delta (0.50) with the spot price of the underlying equal to $50. If the spot price increases by $1, the option will increase in value by $0.50, and the delta changes too. As the delta increases to 0.6 in the figure above. The 0.1 change in delta is approximately equal to the gamma of the option.

## Theta, π―

π― **refers to the rate of decline in the value of an option due to the passage of time**

Theta is typically seen as the decline in value of an option, for a 1 day change in time to expiry.

This all means that an options loses value as it moves closer to its expiry, as long as nothing else changes. the figure above shows the price (premium) of an at-the-money call option as it gets closer to expiry.

At-the-money options suffer more significant dollar losses over-time than in or out of the money options with the same underlying asset and expiry. This is due to at-the-money options having the most time value built into the premium, thus there being more to lose.

## Vega, v

**v is the amount an option's value chances for a one-point change in implied volatility (IV) of the asset**

For example if Bob buys a call with an IV of 50% and $4.00 of vega. If the IV increases to 51%, Aliceβs option value will increase by $4.00. Vega risk is extremely important to assess and understand when trading options, the next section illustrates why.

Usually, as IV increases, the value of an option increases. As this increase would suggest an increased range of potential movement for the underlying asset.

The longer the time remaining to option expiration, the higher the vega. This is because it encodes the 'time value' of the option which is sensitive to changes in volatility.

## Rho, π

**π is the amount an option's value will change based on a one percentage point change in interest rates**

If a call option is priced at $100 and has a π of 0.50 and the risk free interest rate was to increase from 0% to 1%, then the price of the call would increase to $100.50, all else equal.

This is because interest rates can have an effect on the value of an option as they affect the cost of carrying this position over time. A long call gives the right to purchase the underlying asset, usually the cost of this call is cheaper than the exercisable value of the underlying. The difference of these numbers could be deposited into an interest bearing account. This is what makes call options more valuable in a high interest rate environment.