"The Greeks" measure the sensitivity of the value of a derivative or a portfolio to changes in parameter value(s) while holding the other parameters fixed. They are partial derivatives of the price with respect to the parameter values
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"The Greeks" measure the sensitivity of the value of a derivative or a portfolio to changes in parameter value(s) while holding the other parameters fixed. They are partial derivatives of the price with respect to the parameter values. One Greek, "gamma" (as well as others not listed here) is a partial derivative of another Greek, "delta" in this case.
The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are not speculating and following a delta-neutral hedging approach as defined by Black–Scholes.
The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.[10]
Note that from the formulas, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and put options. This can be seen directly from put–call parity, since the difference of a put and a call is a forward, which is linear in S and independent of σ (so a forward has zero gamma and zero vega).
In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).
(Vega is of course not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.)
The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are not speculating and following a delta-neutral hedging approach as defined by Black–Scholes.
The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.[10]
| Calls | Puts | ||
|---|---|---|---|
| Delta |  |  |  |
| Gamma |  |  | |
| Vega |  |  | |
| Theta |  |  |  |
| Rho |  |  |  |
In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).
(Vega is of course not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.)
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