GREEKS (FINANCE)


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.

Contents
Use of the Greeks
The Greeks
Black-Scholes
See also
External links
Discussion
Greeks for specific option models
Calculation

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 'Delta' of an instrument is the mathematical derivative of the value function with respect to the underlying cash flow, Delta = rac{partial V}{partial S}.

★ The 'gamma' measures the rate of change in the delta. The 'Gamma' is the second derivative of the value function with respect to the underlying price, Gamma = rac{partial^2 V}{partial S^2}.

★ The 'speed' measures third order sensitivity to price. The 'speed' is the third derivative of the value function with respect to the underlying price, rac{partial^3 V}{partial S^3}.

★ The 'vega', which is not a Greek letter (
u, ''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,
u= rac{partial V}{partial sigma}. The term 'kappa', kappa, is sometimes used instead of 'vega', and some trading firms have also used the term 'tau', au.

★ The 'theta' measures sensitivity to the passage of time (see Option time value). 'Theta' is the negative of the derivative of the option value with respect to the amount of time to expiry of the option, Theta = - rac{partial V}{partial T}.

★ The 'rho' measures sensitivity to the applicable interest rate. The '
ho' is the derivative of the option value with respect to the risk free rate,
ho = rac{partial V}{partial r}.

★ Less commonly used:

★
★ The 'lambda' 'lambda' is the percentage change in option value per change in the underlying price, or lambda = rac{partial V}{partial S} imes rac{1}{V}. 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, rac{partial^2 V}{partial sigma^2}.

★
★ The 'vanna' measures cross-sensitivity of the option value with respect to change in the underlying price and the volatility, rac{partial^2 V}{partial S partial sigma}, 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, rac{partial Delta}{partial T} = rac{partial^2 V}{partial S partial T}. 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, rac{partial^3 V}{partial S^2 partial T}. 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 phi (phi) is the standard normal probability density function and Phi 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, S , ,
Strike Price, K , ,
Risk-Free Rate, r , ,
Annual Dividend Yield, q , ,
Time to Maturity, au = T-t , , and
Historic Volatility, sigma , ...
{| border="1" cellspacing="0" cellpadding="10"
! !! Calls !! Puts
|-
! delta || e^{-q au} Phi(d_1) , || -e^{-q au} Phi(-d_1) ,
|-
! gamma ||colspan="2"| e^{-q au} rac{phi(d_1)}{Ssigmasqrt{ au}} ,
|-
! vega ||colspan="2"| Se^{-q au} phi(d_1) sqrt{ au} ,
|-
! theta || -e^{-q au} rac{S phi(d_1) sigma}{2 sqrt{ au}} - rKe^{-r au}Phi(d_2) + qSe^{-q au}Phi(d_1) , || -e^{-q au} rac{S phi(d_1) sigma}{2 sqrt{ au}} + rKe^{-r au}Phi(-d_2) - qSe^{-q au}Phi(-d_1) ,
|-
! rho || K au e^{-r au}Phi(d_2), || -K au e^{-r au}Phi(-d_2) ,
|-
! volga ||colspan="2"| Se^{-q au} phi(d_1) sqrt{ au} rac{d_1 d_2}{sigma} =
u rac{d_1 d_2}{sigma} ,
|-
! vanna ||colspan="2"| -e^{-q au} phi(d_1) rac{d_2}{sigma} , = rac{
u}{S}left[1 - rac{d_1}{sigmasqrt{ au}}
ight],
|-
! charm || -qe^{-q au} Phi(d_1) + e^{-q au} phi(d_1) rac{2(r-q) au - d_2 sigma sqrt{ au}}{2 au sigma sqrt{ au}} , || qe^{-q au} Phi(-d_1) - e^{-q au} phi(d_1) rac{2(r-q) au - d_2 sigma sqrt{ au}}{2 au sigma sqrt{ au}} ,
|-
! color ||colspan="2"| -e^{-q au} rac{phi(d_1)}{2S au sigma sqrt{ au}} left[2q au + 1 + rac{2(r-q) au - d_2 sigma sqrt{ au}}{2 au sigma sqrt{ au}}d_1
ight] ,
|-
! dual delta || -e^{-r au} Phi(d_2) , || e^{-r au} Phi(-d_2) ,
|-
! dual gamma ||colspan="2"| e^{-r au} rac{phi(d_2)}{Ksigmasqrt{ au}} ,
|}
where
: d_1 = rac{ln(S/K) + (r - q + sigma^2/2) au}{sigmasqrt{ au}}
: d_2 = rac{ln(S/K) + (r - q - sigma^2/2) au}{sigmasqrt{ au}} = d_1 - sigmasqrt{ au}
: phi(x) = rac{e^{- rac{x^2}{2}}}{sqrt{2 pi}}
: Phi(x) = int_{-infty}^x rac{e^{- rac{y^2}{2}}}{sqrt{2 pi}} ,dy = int_{-x}^{infty} rac{e^{- rac{y^2}{2}}}{sqrt{2 pi}} ,dy

See also



★ Alpha coefficient

★ Beta coefficient

★ Delta neutral

★ Greek letters used in mathematics

External links


Discussion


★ The Greeks: riskglossary.com or optiontutor or investopedia.com or investopedia.com or optiontradingtips.com or superderivatives.com

★ Surface Plots of Black-Scholes Greeks: Chris Murray

★ Delta: quantnotes.com or riskglossary.com

★ Gamma: quantnotes.com or riskglossary.com

★ Vega: riskglossary.com

★ Theta: quantnotes.com or riskglossary.com

★ Rho: riskglossary.com

★ Volga, Vanna, Speed, Charm, Color: Vanilla Options - Uwe Wystup or Vanilla Options - Uwe Wystup

★ Hedging Using the Greeks: Basic Fixed Income Derivative Hedging - Article on Financial-edu.com.
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
Calculation


★ Online realtime Option Calculator with all greeks, sitmo.com

★ Online Option Calculator, option-price.com

★ Option Pricing spreadsheet which calculates the Greeks, optiontradingtips.com

★ Complex Options Calculator, optionistics.com

★ Online real-time option prices and Greeks calculator when the underlying is normally distributed, by Razvan Pascalau, Univ. of Alabama

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