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Greeks (finance)

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In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives. Each "Greek" measures a different aspect of the risk in an option position, and corresponds to a parameter on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the parameters are often denoted by Greek letters.

1 Use of the Greeks
2 The Greeks
3 Black-Scholes
4 See also
5 External links

Use of the Greeks

The Greeks are vital tools in risk management. Each Greek (with the exception of theta - see below) represents a specific measure of risk in owning an option, and option portfolios can be adjusted accordingly ("hedged") to achieve a desired exposure; see for example Delta hedging. As a result, a desirable property of a model of a financial market is that it allows for easy computation of the Greeks. The Greeks in the Black-Scholes model are very easy to calculate and this is one reason for the model's continued popularity in the market.

The Greeks

The delta measures the sensitivity to changes in the price of the underlying asset. The <math>\Delta</math> of an instrument is the mathematical derivative of the value function with respect to the underlyer's price, <math>\Delta = \frac{\partial V}{\partial S}</math>.
The gamma measures the rate of change in the delta. The <math>\Gamma</math> is the second derivative of the value function with respect to the underlying price, <math>\Gamma = \frac{\partial^2 V}{\partial S^2}</math>. Gamma is important because it indicates how a portfolio will react to relatively large shifts in price.
The vega, which is not a Greek letter (<math>\nu</math>, nu is used instead), measures sensitivity to volatility. The vega is the derivative of the option value with respect to the volatility of the underlying, <math>\nu=\frac{\partial V}{\partial \sigma}</math>. The term kappa, <math>\kappa</math>, is sometimes used instead of vega, some math finance training materials sometimes mistakenly use the term tau, <math>\tau</math>.
The speed measures third order sensitivity to price. The speed is the third derivative of the value function with respect to the underlying price, <math>\frac{\partial^3 V}{\partial S^3}</math>.

The theta measures sensitivity to the passage of time (see Option time value). <math>\Theta</math> is the negative of the derivative of the option value with respect to the amount of time to expiry of the option, <math>\Theta = -\frac{\partial V}{\partial T}</math>.
The rho measures sensitivity to the applicable interest rate. The <math>\rho</math> is the derivative of the option value with respect to the risk free rate, <math>\rho = \frac{\partial V}{\partial r}</math>.
Less commonly used:
- The lambda <math>\lambda</math> is the percentage change in option value per change in the underlying price, or <math>\lambda = \frac{\partial V}{\partial S}\times\frac{1}{V}</math>. It is the logarithmic derivative.
- The vega gamma or volga measures second order sensitivity to implied volatility. This is the second derivative of the option value with respect to the volatility of the underlying, <math>\frac{\partial^2 V}{\partial \sigma^2}</math>.
- The vanna measures cross-sensitivity of the option value with respect to change in the underlying price and the volatility, <math>\frac{\partial^2 V}{\partial S \partial \sigma}</math>, which can also be interpreted as the sensitivity of delta to a unit change in volatility.
- The delta decay, or charm, measures the time decay of delta, <math>\frac{\partial \Delta}{\partial T} = \frac{\partial^2 V}{\partial S \partial T}</math>. This can be important when hedging a position over a weekend.
- The color measures the sensitivity of the charm, or delta decay to the underlying asset price, <math>\frac{\partial^3 V}{\partial S^2 \partial T}</math>. It is the third derivative of the option value, twice to underlying asset price and once to time.

Black-Scholes

The Greeks under the Black-Scholes model are calculated as follows, where <math>\phi</math> (phi) is the standard normal probability density function and <math>\Phi</math> is the standard normal cumulative distribution function. Note that the gamma and vega formulas are the same for calls and puts. For a given: Stock Price, <math> S \, </math>, Strike Price, <math> K \, </math>, Risk-Free Rate, <math> r \, </math>, Annual Dividend Yield, <math> q \, </math>, Time to Maturity, <math> \tau = T-t \, </math>, and Historic Volatility, <math> \sigma \, </math>...

	Calls	Puts
delta	<math> e^{-q \tau} \Phi(d_1) \, </math>	<math> -e^{-q \tau} \Phi(-d_1) \, </math>
gamma	<math> e^{-q \tau} \frac{\phi(d_1)}{S\sigma\sqrt{\tau}} \, </math>
vega	<math> Se^{-q \tau} \phi(d_1) \sqrt{\tau} \, </math>
theta	<math> -e^{-q \tau} \frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} - rKe^{-r \tau}\Phi(d_2) + qSe^{-q \tau}\Phi(d_1) \, </math>	<math> -e^{-q \tau} \frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} + rKe^{-r \tau}\Phi(-d_2) - qSe^{-q \tau}\Phi(-d_1) \, </math>
rho	<math> K \tau e^{-r \tau}\Phi(d_2)\, </math>	<math> -K \tau e^{-r \tau}\Phi(-d_2) \, </math>
volga	<math> Se^{-q \tau} \phi(d_1) \sqrt{\tau} \frac{d_1 d_2}{\sigma} = \nu \frac{d_1 d_2}{\sigma} \, </math>
vanna	<math> -e^{-q \tau} \phi(d_1) \frac{d_2}{\sigma} \, = \frac{\nu}{S}\left[1 - \frac{d_1}{\sigma\sqrt{\tau}} \right]\, </math>
charm	<math> -qe^{-q \tau} \Phi(d_1) + e^{-q \tau} \phi(d_1) \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}} \, </math>	<math> qe^{-q \tau} \Phi(-d_1) - e^{-q \tau} \phi(d_1) \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}} \, </math>
color	<math> -e^{-q \tau} \frac{\phi(d_1)}{2S\tau \sigma \sqrt{\tau}} \left[2q\tau + 1 + \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}}d_1 \right] \, </math>
dual delta	<math> -e^{-r \tau} \Phi(d_2) \, </math>	<math> e^{-r \tau} \Phi(-d_2) \, </math>
dual gamma	<math> e^{-r \tau} \frac{\phi(d_2)}{K\sigma\sqrt{\tau}} \, </math>

where

<math> d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)\tau}{\sigma\sqrt{\tau}} </math>

<math> d_2 = \frac{\ln(S/K) + (r - q - \sigma^2/2)\tau}{\sigma\sqrt{\tau}} = d_1 - \sigma\sqrt{\tau} </math>

<math> \Phi(x) = \int_{-\infty}^x \frac{e^{- \frac{y^2}{2}}}{\sqrt{2 \pi}} \,dy = \int_{-x}^{\infty} \frac{e^{- \frac{y^2}{2}}}{\sqrt{2 \pi}} \,dy</math>