Greeks (finance)

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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

The Greeks are vital tools in risk management. With the exception of the theta, each of the quantities represent a specific measure of risk in owning an option. Thus a desirable property of a model of a financial market is the ability to compute 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 delta, $\Delta$, of an instrument is the derivative of the value function with respect to the underlying price, ${\frac {\partial V}{\partial S}}$; delta measures sensitivity to price.

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

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

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

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

• Less commonly used:
  • The lambda, $\lambda$ is the percentage change in option value per change in the underlying price, or ${\frac {\partial V}{\partial S}}\times {\frac {1}{V}}$.
  • The vega gamma or vulga, is the second derivative of the option value with respect to the volatility of the underlying, ${\frac {\partial ^{2}V}{\partial \sigma ^{2}}}$; vega measures second order sensitivity to implied volatility.

Black-Scholes

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

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

External links

• 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
  • Delta: quantnotes.com or riskglossary.com
  • Gamma: quantnotes.com or riskglossary.com
  • Vega: riskglossary.com
  • Theta: quantnotes.com or riskglossary.com
  • Rho: riskglossary.com