Greeks (finance)

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

In mathematical finance, the Greeks are the quantities representing the sensitivities of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the sensitivities are often denoted by Greek letters.

Use of the Greeks

The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example Delta hedging.

The Greeks in the Black-Scholes model are easy to calculate, a desirable property of financial models.

Delta

Delta is the first derivative of the value, $V$, of a portfolio of derivative securities on a single underlying instrument, $S$, with respect to the underlying instrument's price, $\Delta ={\frac {\partial V}{\partial S}}$. Since delta measures sensitivity to a small change in the price of the underlying, it may be used to construct an instantaneously riskless portfolio consisting only of cash, a position in the underlying instrument and an offsetting position in any derivative securities on it.

Gamma

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, $\Gamma ={\frac {\partial ^{2}V}{\partial S^{2}}}$. Gamma is important because it corrects for the convexity of delta.

Vega

Vega, which is not a Greek letter ($\nu \,$, 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, $\nu ={\frac {\partial V}{\partial \sigma }}$. The term kappa, $\kappa$, is sometimes used (by academics) instead of vega, as is tau, $\tau$, though this is rare.

Theta

Theta, or "time decay," measures sensitivity to the passage of time (see Option time value). $\Theta$, $\Theta =-{\frac {\partial V}{\partial T}}$. The value of an option is made up of two parts: the intrinsic value (finance) and the time value. The intrinsic value is the amount of money you would gain if you exercised the option immediately, so a call with strike $50 on a stock with price $60 would have intrinsic value of $10, whereas the corresponding put would have zero intrinsic value. The time value is the worth of having the option of waiting longer when deciding to exercise. Even a deeply out of the money put will be worth something as there is some chance the stock price will fall below the strike. However, as time approaches maturity, there is less chance of this happening, so the time value of an option is decreasing with time. Thus if you are long an option you are short theta: your portfolio will lose value with the passage of time (unless there is enough volatility to offset this).

Traders with a high theta position are known to describe this as "bleeding theta".

Rho

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

Higher-order and Cross-derivatives

• Lambda $\lambda$ is the percentage change in option value per change in the underlying price, or $\lambda ={\frac {\partial V}{\partial S}}\times {\frac {1}{V}}$. It is the logarithmic derivative.
• 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, ${\frac {\partial ^{2}V}{\partial \sigma ^{2}}}$.
• Vanna measures cross-sensitivity of the option value with respect to change in the underlying price and the volatility, ${\frac {\partial ^{2}V}{\partial S\partial \sigma }}$, which can also be interpreted as the sensitivity of delta to a unit change in volatility.
• Delta decay, or charm, measures the time decay of delta, ${\frac {\partial \Delta }{\partial T}}={\frac {\partial ^{2}V}{\partial S\partial T}}$. This can be important when hedging a position over a weekend.
• Gamma decay or colour measures the sensitivity of the charm, or delta decay to the underlying asset price, ${\frac {\partial ^{3}V}{\partial S^{2}\partial T}}$. It is the third derivative of the option value, twice to underlying asset price and once to time.
• Speed measures third order sensitivity to price. The speed is the third derivative of the value function with respect to the underlying price, ${\frac {\partial ^{3}V}{\partial S^{3}}}$.

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, $\tau =T-t\,$, and Volatility $\sigma \,$...

	Calls	Puts
value	$e^{-q\tau }S\Phi (d_{1})-e^{-r\tau }K\Phi (d_{2})\,$	$e^{-r\tau }K\Phi (-d_{2})-e^{-q\tau }S\Phi (-d_{1})\,$
delta	$e^{-q\tau }\Phi (d_{1})\,$	$-e^{-q\tau }\Phi (-d_{1})\,$
gamma	$e^{-q\tau }{\frac {\phi (d_{1})}{S\sigma {\sqrt {\tau }}}}\,$
vega	$Se^{-q\tau }\phi (d_{1}){\sqrt {\tau }}\,$
theta	$-e^{-q\tau }{\frac {S\phi '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\Phi (d_{2})+qSe^{-q\tau }\Phi (d_{1})\,$	$-e^{-q\tau }{\frac {S\phi '(d_{1})\sigma }{2{\sqrt {\tau }}}}+rKe^{-r\tau }\Phi (-d_{2})-qSe^{-q\tau }\Phi (-d_{1})\,$
rho	$K\tau e^{-r\tau }\Phi (d_{2})\,$	$-K\tau e^{-r\tau }\Phi (-d_{2})\,$
volga	$Se^{-q\tau }\phi (d_{1}){\sqrt {\tau }}{\frac {d_{1}d_{2}}{\sigma }}=\nu {\frac {d_{1}d_{2}}{\sigma }}\,$
vanna	$-e^{-q\tau }\phi (d_{1}){\frac {d_{2}}{\sigma }}\,={\frac {\nu }{S}}\left[1-{\frac {d_{1}}{\sigma {\sqrt {\tau }}}}\right]\,$
charm	$-qe^{-q\tau }\Phi (d_{1})+e^{-q\tau }\phi (d_{1}){\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{2\tau \sigma {\sqrt {\tau }}}}\,$	$qe^{-q\tau }\Phi (-d_{1})+e^{-q\tau }\phi (d_{1}){\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{2\tau \sigma {\sqrt {\tau }}}}\,$
color	$-e^{-q\tau }{\frac {\phi (d_{1})}{2S\tau \sigma {\sqrt {\tau }}}}\left[2q\tau +1+{\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{\sigma {\sqrt {\tau }}}}d_{1}\right]\,$
speed	$-e^{-q\tau }{\frac {\phi (d_{1})}{S^{2}\sigma {\sqrt {\tau }}}}\left({\frac {d_{1}}{\sigma {\sqrt {\tau }}}}+1\right)\,$
dual delta	$-e^{-r\tau }\Phi (d_{2})\,$	$e^{-r\tau }\Phi (-d_{2})\,$
dual gamma	$e^{-r\tau }{\frac {\phi (d_{2})}{K\sigma {\sqrt {\tau }}}}\,$

where

$d_{1}={\frac {\ln(S/K)+(r-q+\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}$

$d_{2}={\frac {\ln(S/K)+(r-q-\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}=d_{1}-\sigma {\sqrt {\tau }}$

$\phi (x)={\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}$

$\Phi (x)=\int _{-\infty }^{x}{\frac {e^{-{\frac {y^{2}}{2}}}}{\sqrt {2\pi }}}\,dy=\int _{-x}^{\infty }{\frac {e^{-{\frac {y^{2}}{2}}}}{\sqrt {2\pi }}}\,dy$

See also

• Alpha coefficient
• Beta coefficient
• Delta neutral
• Greek letters used in mathematics

External links

• Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula, Nassim Taleb and Espen Haug
• Surface Plots of Black-Scholes Greeks: Chris Murray
• Delta: quantnotes.com,
• Delta, Gamma, GammaP, Gamma symmetry, Vanna, Speed, Charm, Saddle Gamma: Vanilla Options - Espen Haug,
• Volga, Vanna, Speed, Charm, Color: Vanilla Options - Uwe Wystup, Vanilla Options - Uwe Wystup
• Online real-time option prices and Greeks calculator when the underlying is normally distributed, by Razvan Pascalau, Univ. of Alabama