The Greeks

The greeks are crucial to understanding options. This page provides an overview of the different option greeks.

Option Greeks

This article provides a light introduction to option Greeks, which capture how the price of an option changes with respect to changes in other variables such as volatility, the underlying price or the time to expiration of the underlying asset. These various rates of change are termed option Greeks, or just Greeks, and are represented with Greek letters. From a practical perspective, Greeks are important tools that practitioners use as guidance when gauging the risks of options positions.

Let V be the price of the option (either a call, C, or a put, P), S the price of the underlying asset, and σ the volatility of the underlying asset. The volatility of an asset is a measure of how much its price fluctuates over time.

Delta

$$\Delta = \frac{\partial V}{\partial S}$$

Delta is by far the most important Greek in pricing an option. It's the change in the option price (V) per unit change in the underlying asset It is typically given as a numerical percentage. If you have a 50-delta (50d) option on Apple, and Apple goes up $3, then the price of your option goes up $1.50. Similarly, the delta of an option could also be negative. If you had a -30d option on Apple, and Apple goes up $2, then the price of your option will go down by $0.60.

Vega

$$\nu = \frac{\partial V}{\partial \sigma}$$

Vega represents the change in the option price(V) per unit change in the volatility of the underlying(σ). For example, if SPY has a vega of 2, and SPY has a vol of 10 annually, which then increases to 11 annually, the price of the option goes up to $2.

Traders can have a long vega position or a short vega position. A long vega position corresponds to a long option position and is profitable if volatility rises, all else being equal. When volatility rises, the prices of options tend to rise. Conversely, the prices of options tend to fall when volatility falls. Buying either puts or calls results in exposure to long vega, and writing them results in exposure to short vega.

In times of calm or no real market events, it makes sense to have a short exposure to vega; in contrast, expecting upcoming market volatility will make you want to have a long vega position.

Theta

$$\Theta = \frac{\partial V}{\partial \tau}$$

Theta quantifies the risk that time poses to a long option holder, as the amount of time remaining (𝜏) before the option can be exercised will affect the price of the option. In the case of a long, out-of-the-money call, as time passes, the option has a smaller probability of finishing in-the-money, so its value goes down. This is what’s known as theta: the rate of change of the price of the option with 𝜏 (the time remaining, typically in days, before the option expires).

For example, an NVDA call option with a strike price of $300 and price of $5, when the underlying is trading at $250, with one day to expiration, has a daily theta of -$5, since the option will be worth $0 tomorrow with high probability. Recall that the intrinsic value of an option is the value it would have if it were exercised immediately. There is no intrinsic value to the option because it is out-of-the-money, so the entire value of the option is time value, which is given by theta.

Rho

$$\rho = \frac{\partial V}{\partial r}$$

Rho measures the price sensitivity of an option to the risk-free interest rate. By convention, it is measured in the change of an option’s price per 1% change in the risk-free interest rate. For example, if the risk-free interest rate increases by .5%, and your option has a rho of .1, then the price of your option goes up by $.05. When the risk-free interest rate is relatively stable, rho is smaller compared to the other Greeks, so rho has a comparatively smaller effect on the price of an option.