Greeks (finance)

In mathematical finance, the Greeks are the quantities (known in calculus as partial derivatives; first-order or higher) 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 (as are some other finance measures). Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

Use of the Greeks

{| class="wikitable floatright" style="width:350px;"

{| border="0" cellspacing="1" cellpadding="1" style="width:100%;"

! rowspan=2 style="vertical-align:bottom; text-align:right;" | Underlying parameter

! colspan=3 | Option parameter

! style="vertical-align:bottom;" | Spot price S

! style="vertical-align:bottom;" | Volatility <math>\sigma</math>

! style="vertical-align:bottom;" | Passage of time

! style="text-align:right;" | Value (V) 

| style="background:#8ebded; text-align:center;" | <math>\Delta</math> Delta

| style="background:#8ebded; text-align:center;" | <math>\mathcal{V}</math> Vega

| style="background:#8ebded; text-align:center;" | <math>\Theta</math> Theta

! style="text-align:right;" | Delta (<math>\Delta</math>)

| style="background:#90ee90; text-align:center;" | <math>\Gamma</math> Gamma

| style="background:#90ee90; text-align:center;" | Vanna

| style="background:#90ee90; text-align:center;" | Charm

! style="text-align:right;" | Vega (<math>\mathcal{V}</math>)

| style="background:#90ee90; text-align:center;" | Vanna

| style="background:#90ee90; text-align:center;" | Vomma

| style="background:#90ee90; text-align:center;" | Veta

! style="text-align:right;" | Theta (<math>\Theta</math>)

| style="background:#90ee90; text-align:center;" | Charm

| style="background:#90ee90; text-align:center;" | Veta

| style="background:#90ee90; text-align:center;" |

! style="text-align:right;" | Gamma(<math>\Gamma</math>)

| style="background:#eded8e; text-align:center;" | Speed

| style="background:#eded8e; text-align:center;" | Zomma

| style="background:#eded8e; text-align:center;" | Color

! style="text-align:right;" | Vomma

| style="background:#eded8e; text-align:center;" |

| style="background:#eded8e; text-align:center;" | Ultima

| style="background:#eded8e; text-align:center;" |

! style="text-align:right;" | Charm

| style="background:#eded8e; text-align:center;" |

| style="background:#eded8e; text-align:center;" | Parmicharma

| style="padding:0.6em;" | Definition of Greeks as the sensitivity of an option's price and risk (in the first row) to the underlying parameter (in the first column). First-order Greeks are in blue, second-order Greeks are in green, and third-order Greeks are in yellow. Vanna, charm and veta appear twice, since partial cross derivatives are equal by Schwarz's theorem. Rho, lambda, epsilon, and vera are left out as they are not as important as the rest. Three places in the table are not occupied, because the respective quantities have not yet been defined in the financial literature.

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 (a relatively simple idealised model of certain financial markets) 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 (the partial derivative with respect to the risk-free interest rate) 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 $, £ or €) 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 (the standard deviation of logarithmic returns) and tau (time to expiry) in the Black–Scholes option pricing model. Several names such as "vega" (whose symbol is similar to the lower-case Greek letter nu; the use of that name might have led to confusion) 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, <math>\Delta</math>, 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 <math>V</math> of the option with respect to the underlying instrument's price <math>S</math>.

: <math>\Delta = \frac{\partial V}{\partial S}</math>

Practical use

For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (or a short put) and 0.0 and −1.0 for a long put (or a short call); depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), 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(s). This is convenient because the option will (instantaneously) 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 (= 25%), it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements (100 option contracts covers 10,000 shares). The sign and percentage are often dropped – the sign is implicit in the option type (negative for put, positive for call) 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 (unless they are zero). 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 (expressed as shares of the underlying) 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. (Albeit for only small movements of the underlying, a short amount of time and not-withstanding changes in other market conditions such as volatility and the rate of return for a risk-free investment).

As a proxy for probability

The (absolute value of) 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 (if the market moves under Brownian motion in the risk-neutral measure). 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 (positive) minus the delta of the put (negative) equals 1. This is due to put–call parity: a long call plus a short put (a call minus a 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.

: <math>\Delta(\text{call}) - \Delta(\text{put}) = 1, \text{ therefore: } \Delta(\text{call}) = \Delta(\text{put}) + 1 \text{ and } \Delta(\text{put}) = \Delta(\text{call}) - 1. </math>

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

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 (both calls and puts) 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, <math>\Omega</math>, or elasticity <math>\varepsilon</math> (also known as psi, <math>\psi</math>), is the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk. The dividend yield impact is in practice determined using a 10% increase in those yields. Obviously, this sensitivity can only be applied to derivative instruments of equity products.

: <math>\varepsilon = \psi = \frac{\partial V}{\partial q}</math>

Numerically, all first-order sensitivities can be interpreted as spreads in expected returns. Information geometry offers another (trigonometric) interpretation.thumb|alt=A graph showing the relationship between long option Delta, underlying price, and Gamma|Long option delta, underlying price, and gamma. Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM). Gamma is important because it corrects for the convexity of value.

When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements.

Vanna

Vanna, and DdeltaDvol, measures the instantaneous rate of change of delta over the passage of time.

: <math>\text{Charm} = -\frac{\partial \Delta}{\partial \tau} = \frac{\partial \Theta}{\partial S} = -\frac{\partial^2 V}{\partial \tau \, \partial S}</math>

Charm has also been called DdeltaDtime. vega convexity, vega decay or DvegaDtime (sometimes rhova) which uses quoted call option prices to estimate the risk-neutral probabilities implied by such prices.

: <math>\varpi =\frac{\partial^2 V}{\partial K^2}</math>

For call options, it can be approximated using infinitesimal portfolios of butterfly strategies.

Third-order Greeks

Speed

Speed or DgammaDtime Parmicharma is a third-order derivative of the option value, twice to time and once to underlying asset price. In order to better maintain a delta-hedge portfolio as time passes, the trader may hedge charm in addition to their current delta position. It is also commonly known as cega.

Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying.

Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying.