The Greeks

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In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives, with each measuring a different aspect of the risk in an option position, and corresponding to the set of parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because most of the parameters are denoted by Greek letters.

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.

1 The Greeks
2 Black-Scholes
3 External links
4 See also

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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>\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>\frac{\partial^2 V}{\partial S^2}</math>.

The vega, which is not a Greek letter, measures sensitivity to implied volatility. The vega is the derivative of the option value with respect to the volatility of the underlying, <math>\frac{\partial V}{\partial \sigma}</math>; the term kappa, <math>\kappa</math>, is sometimes used instead of vega.

The theta measures sensitivity to the passage of time (see Option time value). The <math>\Theta</math> is the derivative of the option value with respect to the amount of time to expiry of the option, <math>\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>\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>\frac{\partial V}{\partial S}\times\frac{1}{V}</math>.
- The vega gamma or vulga 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.

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Black-Scholes

The Greeks under the Black-Scholes model are calculated as follows; where, <math> \phi </math> is the normal probability density function. Note that the gamma and vega formulas are the same for calls and puts.

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