A table of the most commonly used option Greeks, their symbols and how they are derived.
In the table above, the symbol ∂ is pronounced “delta” and refers to an incremental change in a given variable. This should not be confused with the option Greek called delta described below.
The derivatives shown in the table above are expressed using the following variables:
V = the option value S = the underlying asset’s price r = the risk-free interest rate t = time 𝜎 = the option’s implied volatility
Delta The delta, symbol Δ, of an option measures the rate of change of its theoretical value with respect to a change in the price of the underlying stock by $1. Delta is the first derivative of an option’s value or V with respect to the price of the underlying asset S. This is the equation used to compute delta:
Δ = ∂ V / ∂ S
The delta of a vanilla option can range between 0.0 and 1.0 for a long call or a short put position, and between 0.0 and −1.0 for a long put or short call position. Its value will depend on how close the option’s strike price is to the current price of the underlying asset. If those prices are equal in the case of an at-the-money (ATM) option, then the delta is usually 0.5 or 50%.
For example, if you own a portfolio with 100 call options each on 100 shares of ABC stock that have a 0.5 or 50% delta, then the value of the option contract will theoretically rise in value by .50 if the stock price increases by $1.
Also, the total delta of a portfolio of various option positions with the same underlying asset is computed by summing up the deltas of each individual position. This practice is used by almost all option portfolio risk managers.
The absolute value of the delta can also be used by traders as a rough estimate for the probability of an option ending up in-the-money (ITM) at expiration. For example, an option with a 0.15 or 15% delta could be thought of as having approximately a 15% chance of being exercised.
Furthermore, due to a rule known as put-call parity, if you know the delta value of an option, you can compute the delta of the option with the opposite right that has the same strike price. You do this by subtracting the call option’s delta from 1 to get the delta of the put with the same strike price or by subtracting the put option’s delta from 1 to get the corresponding call option’s delta. So for example, if you have a delta of .4 on a call option, then delta on the corresponding put option will be .6. The sum of the deltas between the corresponding put and call options can not exceed 1.
Gamma The gamma, symbol γ, of an option measures the rate of change in the option’s delta with respect to an incremental change in the price of the underlying asset. Gamma is the second derivative of the value function with respect to the underlying asset’s price S and the only second-order derivative commonly used by options traders.
This is the equation used to compute gamma: γ = − ∂2 V / ∂2 S
In general, long option positions have positive gamma, while short option positions have negative gamma. The gamma of an option is the highest when the option nears being at-the-money, and it falls as an option goes further in-the-money or out-of-the-money (OTM).
Theta The theta, symbol Θ, of an option measures the sensitivity of its value to the passage of time, which is usually expressed in one-day increments. Theta is the first derivative of the option value function with respect to time t.
Theta is also sometimes called the "time decay" of an option. This is the equation used to compute theta:
Θ = − ∂ V / ∂ t
For vanilla options, the theta of a long option position is generally less than or equal to zero, while the theta of a short option position is almost always greater than or equal to zero.
Options traders with a long option position are said to be “short theta”, and their portfolio will decay in value as time passes if other factors are held steady. Conversely, traders with a short option position are “long theta”, and they will experience a rise in their portfolio’s value over time if other factors remain constant.
Vega Vega, symbol 𝜈, measures an option position’s sensitivity to movements in implied volatility. Vega is the first derivative of an option’s value with respect to the implied volatility or 𝜎 of the underlying asset.
Those actually familiar with the Greek language might note that there is no Greek letter called vega. To get around this, the symbol “𝜈” for the Greek letter nu is typically used to represent vega.
Vega is also sometimes called the "volatility sensitivity" of an option. This is the equation used to compute vega:
𝜈 = ∂ V / ∂ 𝜎
Most option traders will express vega as the change in an option’s value as implied volatility goes up or down by 1 percent. Vanilla options will generally gain in value as implied volatility increases.
Option portfolio traders tend to watch their vega closely to make sure they are either hedged or positioned correctly according to their implied volatility view. Also, those trading very volatility-sensitive option strategies like at-the-money straddles might want to keep an eye on their vega exposure, especially if they intend to trade out of the straddle before expiration.
Rho Rho, symbol ρ, measures an option’s sensitivity to the risk-free interest rate. Rho is therefore the derivative of an option’s value with respect to the risk-free interest rate pertaining to the option’s remaining time until expiration.
Rho can be called the "interest rate sensitivity" of an option. This is the equation used to compute rho:
ρ = ∂ V / ∂ r
In most cases, rho reflects a relatively minor risk for options traders because an option’s price is typically less sensitive to interest rate shifts than to changes in other pricing parameters. Also, options traders typically express rho as the change in an option’s value as the risk-free interest rate goes up or down by 1 percent per annum.
While stock options only have the rho Greek, currency options have another Greek called phi that relates to the foreign currency interest rate, while the rho for currency options relates to the domestic currency interest rate.
Implied Volatility While not a Greek letter, Implied Volatility either denoted short hand as IV or less commonly with the greek symbol, 𝜎, is the estimated range a security’s price will either go up or down within 68% of the time (one standard deviation) in a one-year time period. Option traders will use Implied Volatility to analyze how much a security may move. It’s important to monitor implied volatility depending on your option strategy
How to Apply the Greeks in Webull Now that you have learned about the option and stock market Greeks and how to use them, you may want to apply that knowledge to your portfolio and/or an option strategy using Webull.
Within the Webull mobile app, website, and desktop, you can view the options chain. Within the options chain, you can review the implied volatility, delta, gamma and theta Greeks for each option. Note that you can also view potentially useful information like the current stock price and each option’s bid/ask prices, breakeven, percent change and trading volume.
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