Greeks (finance)

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“The Greeks” redirects here. For the ethnic group, see Greeks.

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.

1 Use of the Greeks
2 The Greeks
3 Black-Scholes
4 See also
5 External links

[edit] 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.

[edit] The Greeks

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

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

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

The vega, which is not a Greek letter ( $ν$ , 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, $κ$ , is sometimes used instead of vega, and some trading firms have also used the term tau, $τ$ .

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

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

Less commonly used:
- The lambda $λ$ is the percentage change in option value per change in the underlying price, or $\lambda = \frac{\partial V}{\partial S}\times\frac{1}{V}$ .
- 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, $\frac{\partial^2 V}{\partial \sigma^2}$ .
- The vanna measures cross-sensitivity of the option value with respect to change in the underlying price and the volatility, $\frac{\partial^2 V}{\partial S \partial \sigma}$ , 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, $\frac{\partial \Delta}{\partial T} = \frac{\partial^2 V}{\partial S \partial T}$ . 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, $\frac{\partial^3 V}{\partial S^2 \partial T}$ . It is the third derivative of the option value, twice to underlying asset price and once to time.

[edit] Black-Scholes

The Greeks under the Black-Scholes model are calculated as follows, where $φ$ (phi) is the standard normal probability density function and $Φ$ 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 Historic Volatility, $\sigma \,$ ...

	Calls	Puts
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	$Se^{-q \tau} \phi(d_1) \sqrt{\tau} \frac{d_1 d_2}{\sigma} = Vega \frac{d_1 d_2}{\sigma} \,$
vanna	$-e^{-q \tau} \phi(d_1) \frac{d_2}{\sigma} \, = \frac{Vega}{S}\left[1 - \frac{d_1}{\sigma\sqrt{\tau}} \right]\,$
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	$-e^{-q \tau} \frac{\phi(d_1)}{2S\tau \sigma \sqrt{\tau}} \left[2q\tau + 1 + \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}}d_1 \right] \,$
dual delta	$-e^{-r \tau} \Phi(d_2) \,$	$e^{-r \tau} \Phi(-d_2) \,$
dual gamma	$e^{-r \tau} \frac{\phi(d_2)}{K\sigma\sqrt{\tau}} \,$