Incomplete guide to Options Greeks

Finance

Greeks or Options Greeks are tools/statistics which measure sensitivity (responsiveness, calculus anyone?) of option price to changes in various parameters: underlying price, volatility, time, etc…

Terminology

Before digging into math/derivatives/formulas, let's remember the terminology once again, you can see mode details in Black-Scholes formula blog post.

  • O - option price of an Call/Put options

  • N - Cumulative Distribution Function (CDF)

  • P - Probability Density Function (PDF)

  • S - Spot price of the underlying asset

  • K - striKe price of option

  • t - time to expiraTion (in years)

  • σ - volatility - standard deviation of log returns (sigma)

  • σ² - variance - squared standard deviation of log returns

Intuition

What the heck are these Greeks?

Greeks are first/second/third/etc-order derivatives of Black-Scholes closed formula with respect to other greek(s) or different variable(s).

And here is an incomplete list of most important greeks used in options trading.

First-order Geeks

Given O, the option's value (price, premium, etc) the generic term for both C and P option prices. Formula for CALL options will be in green, for PUT in red and blue color when formula is same.

Delta - Δ

Option's sensitivity with respect to (w.r.t.) change in underlying price.

\[ \Delta = \frac{ΔO}{ΔS} \]

Formulas: \[ \textcolor{green}{ \Delta = N(d1) } \\ \textcolor{red}{ \Delta = -N(-d1) } \]

Vega / Kappa - V / Κ

Change in option's price w.r.t. change in volatility.

\[ V = \frac{ΔO}{Δσ} \]

Note. Vega is not a greek letter, so we will just use V.

Formula: \[ \textcolor{blue}{ V = S * P(d1) * \sqrt{t} } \]

Theta - Θ

Change in option's price w.r.t time to expiry.

\[ \Theta = \frac{ΔO}{Δt} \]

Formulas: \[ \textcolor{green}{ \Theta = -\frac{S*P(d1)*\sigma}{2\sqrt{t}}-r*K*e^{-rt} * N(d2) } \\ \textcolor{red}{ \Theta = -\frac{S*P(d1)*\sigma}{2\sqrt{t}}-r*K*e^{-rt} * N(-d2) } \]

Rho - Ρ

Changes in option's price w.r.t. risk-free interest rate.

\[ \rho = \frac{\Delta{O}}{\Delta{r}} \]

Formulas: \[ \textcolor{green}{ \rho = t*K*e^{-rt} * N(d2) } \\ \textcolor{red}{ \rho = -t*K*e^{-rt} * N(-d2) } \]

Second-order greeks

Second-order derivative of option's value with respect to some other variable(s).

Gamma - Γ

Change in delta w.r.t. the price of the underlying asset.

  • first-order derivative of delta w.r.t. underlying asset

\[ \Gamma = \frac{\partial{Δ}}{\partial{S}} \]

  • second-order derivative of option value w.r.t. underlying asset

\[ \Gamma = \frac{\partial^2{O}}{\partial{S^2}} \]

Formula: \[ \textcolor{blue}{ \Gamma = \frac{P(d1)}{S * \sigma * \sqrt{t}} } \]

Vomma / Volga

Sensitivity of vega to a change in the volatility.

  • first-order derivative of vega w.r.t. volatility

\[ volga = \frac{\partial{V}}{\partial{σ}} \]

  • second-order derivative of option's value w.r.t volatility

\[ volga = \frac{\partial^2{O}}{\partial{σ^2}} \]

Formula: \[ \textcolor{blue}{ volga = P(d1) * \sqrt{t} * \frac{d1 * d2}{\sigma} } \]

Vanna

This is volatility's cross Greek and is the sensitivity of delta/vega w.r.t. volatility/spot.

  • first-order derivative of delta w.r.t. volatility

\[ vanna = \frac{\partial{Δ}}{\partial{σ}} \]

  • first-order derivative of vega w.r.t. underlying spot

\[ vanna = \frac{\partial{V}}{\partial{S}} \]

  • second-order derivative of option value w.r.t. both underlying spot and volatility

\[ vanna = \frac{\partial^2{O}}{\partial{\sigma} \partial{S}} \]

Formula: \[ \textcolor{blue}{ vanna = P(d1) * \sqrt{t} * (1 - d1) } \]

Charm / Delta decay

Change in delta w.r.t. time to expiry.

  • first-order derivative of delta w.r.t time

\[ charm = \frac{\partial{Δ}}{\partial{t}} \]

  • first-order derivative of theta w.r.t. underlying spot

\[ charm = \frac{\partial{\theta}}{\partial{S}} \]

  • second-order derivative of option value w.r.t. both underlying spot and time

\[ charm = \frac{\partial^2{O}}{\partial{S} \partial{t}} \]

Vera / Rhova

Change in rho w.r.t. volatility.

  • first-order derivative of rho w.r.t. volatility

\[ vera = \frac{\partial{\rho}}{\partial{σ}} \]

  • first-order derivative of vega w.r.t. interest rate

\[ vera = \frac{\partial{V}}{\partial{r}} \]

  • second-order derivative of option price w.r.t. interest rate and volatility

\[ vera = \frac{\partial^2{O}}{\partial{r} \partial{\sigma}} \]

Veta / Vega decay

Change in vega w.r.t. time.

  • first-order derivative of vega w.r.t. time to expiry

\[ veta = \frac{\partial{V}}{\partial{t}} \]

  • first-order derivative of theta w.r.t. volatility

\[ veta = \frac{\partial{\theta}}{\partial{\sigma}} \]

  • second-order derivative of option's value w.r.t. volatility and time

\[ veta = \frac{\partial^2{O}}{\partial{\sigma} \partial{t}} \]

Third-order Greeks

Third-order derivative of option's price with respect to some other variable(s).

Color / Gamma decay

Gamma w.r.t. time.

\[ color = \frac{\partial{Γ}}{\partial{t}} \]

Speed

Gamma w.r.t. underlying spot.

\[ speed = \frac{\partial{Γ}}{\partial{S}} \]

Ultima

Vomma w.r.t. volatility.

\[ ultima = \frac{\partial{vomma}}{\partial{σ}} \]

Zomma

Gamma w.r.t. volatility.

\[ zomma = \frac{\partial{Γ}}{\partial{σ}} \]

TO BE CONTINUED

comments powered by Disqus