Greeks (finance)

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of a derivative instrument such as an option to changes in one or more underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

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 relatively easy to calculate — a desirable property of financial models — and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging—such as [|delta], [|theta], and vega—are well-defined for measuring changes in the parameters spot price, time and volatility. Although [|rho] is a primary input into the Black–Scholes model, the overall impact on the value of a short-term option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common.
The most common of the Greeks are the first order derivatives: delta, [|vega], theta and rho; as well as [|gamma], a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive.
The players in the market make competitive trades involving many billions of underlying every day, so it is important to get the sums right. In practice they will use more sophisticated models which go beyond the simplifying assumptions used in the Black-Scholes model and hence in the Greeks.

Names

The use of Greek letter names is presumably by extension from the common finance terms alpha and beta, and the use of sigma and tau in the Black–Scholes option pricing model. Several names such as "vega" and "zomma" are invented, but sound similar to Greek letters. The names "color" and "charm" presumably derive from the use of these terms for exotic properties of quarks in particle physics.

First-order Greeks

Delta

Delta,, measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value of the option with respect to the underlying instrument's price.

Practical use

For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call and 0.0 and −1.0 for a long put ; depending on price, a call option behaves as if one owns 1 share of the underlying stock, or owns nothing, or something in between, and conversely for a put option. The difference between the delta of a call and the delta of a put at the same strike is equal to one. By put–call parity, long a call and short a put is equivalent to a forward F, which is linear in the spot S, with unit factor, so the derivative dF/dS is 1. See the formulas below.
These numbers are commonly presented as a percentage of the total number of shares represented by the option contract. This is convenient because the option will behave like the number of shares indicated by the delta. For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.25, it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements. The sign and percentage are often dropped – the sign is implicit in the option type and the percentage is understood. The most commonly quoted are 25 delta put, 50 delta put/50 delta call, and 25 delta call. 50 Delta put and 50 Delta call are not quite identical, due to spot and forward differing by the discount factor, but they are often conflated.
Delta is always positive for long calls and negative for long puts. The total delta of a complex portfolio of positions on the same underlying asset can be calculated by simply taking the sum of the deltas for each individual position – delta of a portfolio is linear in the constituents. Since the delta of underlying asset is always 1.0, the trader could delta-hedge his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. For example, if the delta of a portfolio of options in XYZ is +2.75, the trader would be able to delta-hedge the portfolio by selling short 2.75 shares of the underlying. This portfolio will then retain its total value regardless of which direction the price of XYZ moves..

As a proxy for probability

The Delta is close to, but not identical with, the percent moneyness of an option, i.e., the implied probability that the option will expire in-the-money. For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has approximately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls since the risk-free rate introduces some offset to the delta. The negative discounted probability of an option ending up in the money at expiry is called the dual delta, which is the first derivative of option price with respect to strike.

Relationship between call and put delta

Given a European call and put option for the same underlying, strike price and time to maturity, and with no dividend yield, the sum of the absolute values of the delta of each option will be 1 – more precisely, the delta of the call minus the delta of the put equals 1. This is due to put–call parity: a long call plus a short put replicates a forward, which has delta equal to 1.
If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta.
For example, if the delta of a call is 0.42 then one can compute the delta of the corresponding put at the same strike price by 0.42 − 1 = −0.58. To derive the delta of a call from a put, one can similarly take −0.58 and add 1 to get 0.42.

Vega

Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.
Vega is not the name of any Greek letter. The glyph used is a non-standard majuscule version of the Greek letter nu, written as. Presumably the name vega was adopted because the Greek letter nu looked like a Latin vee, and vega was derived from vee by analogy with how beta, eta, and theta are pronounced in American English.
The symbol kappa,, is sometimes used instead of vega
or capital lambda,
though these are rare).
Vega is typically expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1 percentage point. All options will gain value with rising volatility.
Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an at-the-money option straddle, for example, is extremely dependent on changes to volatility.
See Volatility risk.

Theta

Theta, , measures the sensitivity of the value of the derivative to the passage of time : the "time decay."
As time passes, with decreasing time to expiry and all else being equal, an option's extrinsic value decreases. Typically, this means an option loses value with time, which is conventionally referred to as long options typically having short theta. In fact, typically, the literal first derivative w.r.t. time of an option's value is a positive number. The change in option value is typically negative because the passage of time is a negative number. However, by convention, practitioners usually prefer to refer to theta exposure of a long option as negative, and so theta is usually reported as -1 times the first derivative, as above.
While extrinsic value is decreasing with time passing, sometimes a countervailing factor is discounting. For deep-in-the-money options of some types, as discount factors increase towards 1 with the passage of time, that is an element of increasing value in a long option. Sometimes deep-in-the-money options will gain more from increasing discount factors than they lose from decreasing extrinsic value, and reported theta will be a positive value for a long option instead of a more typical negative value.
By convention in options valuation formulas,, time to expiry, is defined in years. Practitioners commonly prefer to view theta in terms of change in number of days to expiry rather than number of years to expiry. Therefore, reported theta is usually divided by number of days in a year.

Rho

Rho,, measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk-free interest rate.
Except under extreme circumstances, the value of an option is less sensitive to changes in the risk-free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks.
Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk-free interest rate rises or falls by 1.0% per annum.