As we now know, the price of an option depends on many factors: the price of the asset, its implied volatility, and the time to expiration, amongst other things. But how does the price of an option change as these variables change? If volatility goes up 2%, how much does my call go up? If the asset increases by $10, how much does my put go down?

The Greeks offer an answer to these questions by measuring the sensitivity of an option’s price to changes in the input parameters. There are many different Greeks, but we will focus on just five of them.

Delta is the option’s price sensitivity to price changes in the underlying asset. If a call has a 0.50 (or ‘50' delta) and the asset moves up $1.00, the option will increase $0.50 in value.

Another way to think about delta is a rough estimate of the probability of the asset finishing in the money at expiration. Our 50 delta call implies the asset has a roughly 50% chance of trading higher than our strike price on expiration. Measuring delta is a way of calculating the equivalent position in the underlying asset. If you are long 1 delta, you are effectively long one share of the underlying asset.

Vega is the option’s price sensitivity to changes in the implied volatility of the asset. For example, if Alice buys a call with an IV of 40% and $2.00 of vega. If the IV increases to 41%, Alice’s option value will increase by $2.00. Vega risk is extremely important to assess and understand when trading options, the next section illustrates why.

I bought a call and the asset price increased, why am I losing money?

More often than not, the culprit is vega. A common misconception is that if you buy a call and the price of the asset goes up, you will make money (vice versa with a put). There are often times where traders buy a call for an asset with a relatively high IV, and are taken aback when they lose money if the asset price grinds up slowly. For example, if a trader buys an option with 100% IV (implying ~5% per move per day) and the asset moves up 0.50% for 3 days in a row, they could end up losing money if the market decides that the asset's IV should really be 80%. This would result in a reduction in the option's value of approximately 20 * optionVega. If this drop is larger than the effect of the spot price increase, the trader will lose money.

Gamma is the rate of change of an option's delta for a $1 move in the underlying asset. To illustrate, imagine a $2000 strike call option has a 50 delta (0.50) with the spot price of the underlying equal to $2000. If the spot price increases by $1, the option will increase in value by $0.50, and the delta will change too. Imagine the delta increases to 52 (0.52). The 0.02 change in delta is approximately equal to the gamma of the option.

Gamma is the Greek which clearly identifies the power of options. Other products like leveraged perpetuals and futures can offer enhanced first order exposure to an asset, but it is the second derivative properties of options which give them their non-linearity or convexity. Options with gamma increase in value at an increasing rate if the underlying moves in the desired direction.

The gamma of a given option changes with respect to time. For an out-of-the-money option, gamma is maximized when the option is far away from expiration. For an at-the-money option, gamma is maximized near expiration.

The theta of an option measures how much value it will lose as a result of the passage of time. The theta is typically quoted as the dollar amount which an option will lose if time is fast-forwarded one day. For example, if the theta of an option is $5.00, the option will be worth $5.00 less in one day's time, assuming all else equal.

Rho is the rate at which the price of an option changes relative to the risk-free interest rate. Imagine a call option is priced at $100 and has a rho of 0.50. If the risk free interest rate was to increase from 0 to 1%, then the price of the call would increase to $100.50.

​