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 most of the parameters are denoted by Greek letters.

Use of the Greeks

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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

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• The delta measures sensitivity to price. The $\backslash Delta$, of an instrument is the derivative of the value function with respect to the underlying price, $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}$.

• The gamma measures second order sensitivity to price. The $\backslash Gamma$ is the second derivative of the value function with respect to the underlying price, $\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}$.

• 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, $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; \backslash sigma\}$; the term kappa, $\backslash kappa$, is sometimes used instead of vega.

• The theta measures sensitivity to the passage of time (see Option time value). $\backslash Theta$ is minus the derivative of the option value with respect to the amount of time to expiry of the option, $\backslash Theta\; =\; -\backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}$.

• The rho measures sensitivity to the applicable interest rate. The $\backslash rho$ is the derivative of the option value with respect to the risk free rate, $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; r\}$.

• Less commonly used:
  • The lambda, $\backslash lambda$ is the percentage change in option value per change in the underlying price, or $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}\backslash times\backslash frac\{1\}\{V\}$.
  • 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, $\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; \backslash sigma^2\}$.
  • The vanna measures cross-sensitivity of option value with respect to change in underlier price and underlier volatility, $\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S\; \backslash partial\; \backslash sigma\}$, 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, $\backslash frac\{\backslash partial\; \backslash Delta\}\{\backslash partial\; T\}$. This can be important when hedging a position over a weekend.

Black-Scholes

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

	Calls	Puts
delta	$N(d\_1)\; \backslash ,$	$N(d\_1)\; -\; 1\; \backslash ,$
gamma	$\backslash frac\{\backslash phi(d\_1)\}\{S\backslash sigma\backslash sqrt\{T\}\}\; \backslash ,$
vega	$S\; \backslash phi(d\_1)\; \backslash sqrt\{T\}\; \backslash ,$
theta	$-\; \backslash frac\{S\; \backslash phi(d\_1)\; \backslash sigma\}\{2\; \backslash sqrt\{T\}\}\; -\; rKe^\{-rT\}N(d\_2)\; \backslash ,$	$-\; \backslash frac\{S\; \backslash phi(d\_1)\; \backslash sigma\}\{2\; \backslash sqrt\{T\}\}\; +\; rKe^\{-rT\}N(-d\_2)\; \backslash ,$
rho	$KTe^\{-rT\}N(d\_2)\backslash ,$	$-KTe^\{-rT\}N(-d\_2)\backslash ,$

where

$d\_1\; =\; \backslash frac\{\backslash ln(S/K)\; +\; (r\; +\; \backslash sigma^2/2)T\}\{|\backslash sigma|\backslash sqrt\{T\}\}$

$d\_2\; =\; d\_1\; -\; |\backslash sigma|\backslash sqrt\{T\}.$

External links

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• Greeks for specific option models
  • options on non-dividend paying stocks, riskglossary.com
  • options on stock indexes, riskglossary.com
  • options on forwards (the Black model), riskglossary.com
  • foreign exchange options, riskglossary.com

• Discussion
  • The Greeks: riskglossary.com or optiontutor or investopedia.com or optiontradingtips.com
  • Delta: quantnotes.com or riskglossary.com
  • Gamma: quantnotes.com or riskglossary.com
  • Vega: riskglossary.com
  • Theta: quantnotes.com or riskglossary.com
  • Rho: riskglossary.com

• Calculation
  • Online Option Calculator, option-price.com
  • Free Option Pricing spreadsheet to calculate the Greeks, optiontradingtips.com

See also

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• Greek letters used in mathematics

Mathematical finance

Lettres grecques en mathématiques financières