Greeks (finance)

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.

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

Black-Scholes

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	S e^{-q \tau} \phi(d_1) \sqrt{\tau} = K e^{-r \tau} \phi(d_2) \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}} \,

Greeks (finance)

Contents

Use of the Greeks