Greeks (finance)

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"The Greeks" redirects here. For the ethnic group, see Greeks.

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 External links
5 See also

[edit] 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.

[edit] The Greeks

The delta measures sensitivity to price. The <math>\Delta</math>, of an instrument is the derivative of the value function with respect to the underlying price, <math>\Delta = \frac{\partial V}{\partial S}</math>.

The gamma measures second order sensitivity to price. 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>.

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, and some trading firms have also used the term tau, <math>\tau</math>.

The theta measures sensitivity to the passage of time (see Option time value). <math>\Theta</math> is minus 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>.
- 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 option value with respect to change in underlier price and underlier 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 measures the time decay of delta, <math>\frac{\partial \Delta}{\partial T}</math>. This can be important when hedging a position over a weekend.

[edit] 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.

	Calls	Puts
delta	<math> \Phi(d_1) \, </math>	<math> \Phi(d_1) - 1 \, </math>
gamma	<math> \frac{\phi(d_1)}{S\sigma\sqrt{T}} \, </math>
vega	<math> S \phi(d_1) \sqrt{T} \, </math>
theta	<math> - \frac{S \phi(d_1) \sigma}{2 \sqrt{T}} - rKe^{-rT}\Phi(d_2) \, </math>	<math> - \frac{S \phi(d_1) \sigma}{2 \sqrt{T}} + rKe^{-rT}\Phi(-d_2) \, </math>
rho	<math> KTe^{-rT}\Phi(d_2)\, </math>	<math> -KTe^{-rT}\Phi(-d_2)\, </math>