What are the Greeks?
Broadly, the Greeks measure the sensitivity of an option’s premium to changes in the underlying variables. They are necessary for determining how to properly hedge a portfolio and are therefore important for risk management.
In this presentation we’ll cover Greeks in theBlack-Scholesworld. This means we are assuming options are European on a non-dividend paying stock. It also assumes the underlying stock follows a geometric Brownian motion process.
This means the underlying variables are the follwing: the stock price, volatility, the risk-free rate, and time.
Note, each Greek (being a partial derivative of the Black-Scholes equation) assumes all other variables remain constant. The Black-Scholes equation for the premium of a European call option is shown on the next slide.
Black-Scholes Formula:
- `Call_0 = S_0N(d_1) - Xe^{-rT}N(d_2)`
- `Put_0 = N(-d_2)K\exp{-r(T-t)} - N(-d_1)S_0`
where
`d_1 = \frac{ln(\frac{S_0}{X}) + (r+\frac{\sigma^2}{2})T}{\sigma\sqrt(T)}`
`d_2 = d_1 - \sigma\sqrt(T)`
`S_0`: the value of the call option at time 0.
`N()`: the cumulative standard normal density function (NORMSDIST() in Excel)
`X`: the exercise or strike price.
`r`: the risk-free interest rate (annualized).
`T`: the time until option expiration in years.
`\sigma`: the annualized standard deviations of log returns.
- `e` and `ln` are the exponential and natural log functions respectively (EXP() and LN() in Excel).
Greeks
Let `P` refer to the equation for either a call or put option premium. Then thegreeksare defined as:
Delta (`\Delta = \frac{\partial P}{\partial S}`): Where `S`is the stock price.
Gamma (`\Gamma = \frac{\partial^2 P}{\partial S^2}`): Where `S` is the stock price.
Theta (`\Theta = \frac{\partial P}{\partial t}`): Where `t` is time.
Rho (`\rho = \frac{\partial P}{\partial r_f}`): Where `r_f` is the risk-free rate.
Vega (`v = \frac{\partial P}{\partial \sigma}`) (Not Greek): Where `\sigma` is volatility.
Delta: `\frac{\partial P}{\partial S}`
Delta is the rate of change on the option’s price with respect to changes in the price of the underlying asset (stock). For a call option the Delta is: `\Delta = N(d_1)`
where `N()` is the standard cumulative normal density function. The Delta for a put is: `\Delta = N(d_1) - 1`
Delta is very useful, because it is the number of shares to buy (or sell) to hedge out the risk of changes in the underlying stock’s price when short a call (or put) option.
In other words, if you have a portfolio short 1 call option and long Delta shares of stock, then your portfolio is riskless (over very short time periods). This is referred to asdelta hedging.
Similarly, a portfolio short one put and short Detla shares of stock is riskless.
Call Deltas range from 0 to 1, and put Deltas range from -1 to 0.
Interactive Apps
This presentation contains interactive apps for each Greek – on the following slide is the app for an option’s Delta.
Each app will allow you to graph the variation of a Greek, where you can choose the variable on the horizontal axis. You can also change the other inputs into the option pricing model and see how this affects the relationship.
Many of the relationships are greatly affected by themoneynessof the option, so first try changing the stock or strike price.
Gamma: `\frac{\partial^2 P}{\partial S^2}`
Gamma is the rate of change of the option’s Delta with respect to changes in the underlying stock. Gamma for abotha call and put is:
`\Gamma = \frac{N'(D_1)}{S_0 \sigma \sqrt(T)}`
- The higher the Gamma (in absolute value) the more often you’ll need to rebalance a delta-neutral portfolio.
Suppose the Gamma of a call option on a stock is 0.03.
- This means that a $1 increase in the stock’s price will increase the Delta of the option by 0.03.
To create a Gamma-neutral portfolio, you’ll have to trade in an option on the underlying stock – or some derivative which is not linearly related to the underlying stock.
