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Option Greeks

Delta, Gamma, Rho, Theta, and Vega

Option Greeks are measures of the sensitivity of an option's price to changes in underlying parameters.

1. Delta (Δ):

  • Definition: The change in the option's price for a $1 change in the underlying asset's price.
  • Calculation (Stock Option, No Dividend):
    • Call Option: Δ = N(d1)
    • Put Option: Δ = N(d1) - 1 Where N(d1) is the cumulative standard normal distribution function of d1. d1 = [ln(S/K) + (r + (σ^2)/2)T] / (σ√T)
  • Calculation (Stock Option, with Dividend):
    • Call Option: Δ = e^(-qT) * N(d1)
    • Put Option: Δ = e^(-qT) * [N(d1) -1] Where q is the continuous dividend yield.
  • Calculation (Currency Option):
    • Call Option: Δ = e^(-rfT) * N(d1)
    • Put Option: Δ = -e^(-rfT) * N(-d1) Where rf is the foreign risk-free rate. d1 = [ln(S/K) + (rd - rf + (σ^2)/2)T] / (σ√T) Where rd is the domestic risk-free rate, S is the spot exchange rate (domestic/foreign), and K is the strike price.
  • Interpretation:
    • Call Delta: A call option's delta is between 0 and 1.
    • Put Delta: A put option's delta is between -1 and 0.
    • Currency Option Delta: Depends on the specific currency and rates but represents sensitivity to changes in the exchange rate.

2. Gamma (Γ):

  • Definition: The rate of change of the option's delta for a $1 change in the underlying asset's price. It measures the curvature of the option price with respect to the underlying asset price.
  • Calculation (Stock and Currency Options):
    • Γ = N'(d1) / (Sσ√T) Where N'(d1) is the probability density function of the standard normal distribution evaluated at d1.
  • Interpretation: Gamma is always positive for both calls and puts (except for extreme in-the-money or out-of-the-money options where it approaches zero). High gamma means delta is very sensitive to price changes.

3. Rho (ρ):

  • Definition: The change in the option's price for a 1% change in the risk-free interest rate.
  • Calculation (Stock Option, No Dividend):
    • Call Option: ρ = KTe^(-rT)N(d2) / 100
    • Put Option: ρ = -KTe^(-rT)N(-d2) / 100 Where d2 = d1 - σ√T
  • Calculation (Currency Option):
    • Call Option: ρ = KTe^(-rdT)N(d2) / 100
    • Put Option: ρ = -KTe^(-rdT)N(-d2) / 100
  • Interpretation:
    • Calls: Positive Rho (higher interest rates increase call option value).
    • Puts: Negative Rho (higher interest rates decrease put option value).

4. Theta (Θ):

  • Definition: The change in the option's price for a one-day decrease in time to expiration. Often referred to as "time decay."
  • Calculation (Stock Option, No Dividend):
    • Call Option: Θ = [-SσN'(d1) / (2√T) - rKe^(-rT)N(d2)] / 365
    • Put Option: Θ = [-SσN'(d1) / (2√T) + rKe^(-rT)N(-d2)] / 365
  • Calculation (Currency Option):
    • Call Option: Θ = [-SσN'(d1) / (2√T) + rfKe^(-rdT)N(d2) - rdSe^(-rfT)N(d1)] / 365
    • Put Option: Θ = [-SσN'(d1) / (2√T) - rfKe^(-rdT)N(-d2) + rdSe^(-rfT)N(-d1)] / 365
  • Interpretation: Theta is usually negative (options lose value as time passes), but can be positive for deep in-the-money puts.

5. Vega (ν):

  • Definition: The change in the option's price for a 1% change in the volatility of the underlying asset.
  • Calculation (Stock and Currency Options):
    • ν = S√T N'(d1) / 100
  • Interpretation: Vega is always positive for both calls and puts. Options are more valuable when volatility is higher.

Important Notes:

  • These formulas are based on the Black-Scholes model assumptions.
  • Real-world option prices may deviate from these theoretical values.
  • Dividends and foreign interest rates significantly impact option prices and Greeks.
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