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Option Greeks

Financial measures that determine the sensitivity of an option’s price to its underlying parameters

What are Option Greeks?

Option Greeks are financial measures of the sensitivity of an option’s price to its underlying determining parameters, such as volatility or the price of the underlying asset. The Greeks are utilized in the analysis of an options portfolio and on sensitivity analysis of an option or portfolio of options. The measures are essential to making informed decisions in options trading.

Delta, Gamma, Vega, Theta, and Rho are the key option Greeks. However, there are many other option Greeks that can be derived from those mentioned above.

 

NameDependent VariableIndependent Variable
DeltaOption priceValue of underlying asset
GammaDeltaValue of underlying asset
VegaOption priceVolatility
ThetaOption priceTime to maturity
RhoOption priceInterest rate

 

Delta

Delta (Δ) is a measure of the sensitivity of an option’s price changes relative to the changes in the underlying asset’s price. In other words, if the price of the underlying asset increases by $1, the price of the option will change by Δ amount. Mathematically, the delta is found by:

 

Option Greeks - Formula for Delta

 

Where:

  • ∂ – the first derivative
  • V – the option’s price (theoretical value)
  • S – the underlying asset’s price

 

Keep in mind that we take the first derivative of the option and underlying asset prices because the derivative is a measure of the rate change of the variable at a given period.

The delta is usually calculated as a decimal number from -1 to 1. Call options can have a delta from 0 to 1, while puts have a delta from -1 to 0. The closer the option’s delta to 1 or -1, the deeper in-the-money is the option.

The delta of an option’s portfolio is the weighted average of the deltas of all options in the portfolio.

Delta is also known as a hedge ratio. If a trader knows the delta of the option, he can hedge his position by buying or shorting the number of underlying assets multiplied by delta.

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Gamma

Gamma (Γ) is a measure of the delta’s change relative to the changes in the price of the underlying asset. If the price of the underlying asset increases by $1, the option’s delta will change by the gamma amount. The main application of gamma is the assessment of the option’s delta.

 

Formula for Gamma

 

Long options have a positive gamma. An option has a maximum gamma when it is at-the-money (option strike price equals the price of the underlying asset). However, gamma decreases when an option is deep-in-the-money or out-the-money.

 

Vega

Vega (ν) measures the sensitivity of an option price relative to the volatility of the underlying asset. If the volatility of the underlying asses increases by 1%, the option price will change by the vega amount.

 

Formula for Vega

 

Where:

  • ∂ – the first derivative
  • V – the option’s price (theoretical value)
  • σ – the volatility of the underlying asset

 

The vega is expressed as a money amount rather than a decimal number. An increase in vega generally causes an increase in the option value (both calls and puts).

 

Theta

Theta (θ) is a measure of the sensitivity of the option price relative to the option’s time to maturity. If the option’s time to maturity decreases by one day, the option’s price will change by the theta amount. Theta is also referred to as time decay.

 

Formula for Theta

 

Where:

  • ∂ – the first derivative
  • V – the option’s price (theoretical value)
  • τ – the option’s time to maturity

In most cases, theta is negative for options. However, it may be positive for some European options. Theta shows the most negative amount when the option is at-the-money.

 

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Rho

Rho (ρ) measures the sensitivity of the option price relative to the interest rate. If the interest rate increases by 1%, the option price will change by the rho amount. The rho is considered the least significant among other Greeks because option prices are generally less sensitive to interest rate changes than to the changes in other parameters.

 

Formula for Rho

 

Where:

  • ∂ – the first derivative
  • V – the option’s price (theoretical value)
  • r – interest rate

 

Generally, call options have a positive rho, while the rho for put options is negative.

 

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