# Greeks (finance)

Greeks (finance)

In mathematical finance, the Greeks are the quantities representing the sensitivities of the price of derivatives such as options to a change in 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 often denoted by Greek letters. Collectively these have also been called the risk sensitivities,[1] risk measures[2] or hedge parameters.[3]

## Use of the Greeks

Spot
Price (S)
Volatility
(σ)
Time to
Expiry (τ)
Risk-Free
Rate (r)
Value (V)  Δ Delta ν Vega Θ Theta ρ Rho
Delta (Δ Γ Gamma Vanna Charm
Gamma (Γ Speed Zomma Color
Vomma  Ultima Totto
 The table shows the relationships between the more common sensitivities to (derivatives of) the four primary inputs into the Black–Scholes model (namely, the spot price of the underlying security, the volatility of that price, the time remaining until the option expires, and the rate of return of a risk-free investment) and to the option's value, delta, gamma, vega and vomma. Greeks which are first-order derivatives are in blue, second-order derivatives are in green, and third-order derivatives are in yellow. Note that vanna appears, intentionally, in two places in the table, as the two mixed derivatives are mathematically equivalent.

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 delta, theta, and vega are well-defined for measuring changes in Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an 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.

## First-order Greeks

### Delta Δ

 $\Delta = \frac{\partial V}{\partial S}$

Delta,[4] Δ, measures the rate of change of option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value V of the option with respect to the underlying instrument's price S.

#### Practical use

For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (and/or short put) and 0.0 and −1.0 for a long put (and/or 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.

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 25 shares of XYZ as the price changes for small price movements.

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 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

Some option traders also use the absolute value of delta as the probability that the option will expire in-the-money (if the market moves under Brownian motion).[5] For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has appropriately 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 puts and calls have a delta of approximately[6] 0.5 and −0.5 respectively (however, this approximation rapidly goes out the window when looking at a term of just a few years, with the ATM call commonly having a delta over 0.60 or 0.70), or each will have a 50% chance of expiring in-the-money. The correct, exact calculation for the probability of an option finishing in the money is its 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 1.00 – 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 compute the value of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from the known value. 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. While in deriving delta of a call from put will not follow this approach e.g. – delta of a put is −0.58 and if we follow the same approach then delta of a call with same strike should be −1.58. so delta should be = opposite sign ( abs(delta) − 1).

### Vega ν

 $\nu=\frac{\partial V}{\partial \sigma}$

Vega[7] 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. However, the glyph used is the Greek letter nu. 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 English.

The symbol kappa, κ, is sometimes used (by academics) instead of vega (as is tau (τ), though this is rare).

#### Practical use

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%.

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 option straddle, for example, is extremely dependent on changes to volatility.

### Theta Θ

 $\Theta = -\frac{\partial V}{\partial \tau}$

Theta,[7] Θ, measures the sensitivity of the value of the derivative to the passage of time (see Option time value): the "time decay."

#### Practical use

The mathematical result of the formula for theta (see below) is expressed in value per year. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount of money per share of the underlying that the option loses in one day. Theta is almost always negative for long calls and puts and positive for short (or written) calls and puts. An exception is a deep in-the-money European put. The total theta for a portfolio of options can be determined by summing the thetas for each individual position.

The value of an option can be analysed into two parts: the intrinsic value 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 value of having the option of waiting longer before 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 before the expiry date. 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 (all other factors held constant).

### Rho ρ

 $\rho = \frac{\partial V}{\partial r}$

Rho,[7] ρ, measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term). 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.

#### Practical use

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 (100 basis points).

## Higher-order Greeks

### Charm

 $\text{Charm} =- \frac{\partial \Delta}{\partial \tau} = \frac{\partial \Theta}{\partial S} = -\frac{\partial^2 V}{\partial S \, \partial \tau}$

Charm[7] or delta decay, measures the instantaneous rate of change of delta over the passage of time. Charm has also been called DdeltaDtime.[8] Charm can be an important Greek to measure/monitor when delta-hedging a position over a weekend. Charm is a second-order derivative of the option value, once to price and once to the passage of time. It is also then the derivative of theta with respect to the underlying's price.

#### Practical use

The mathematical result of the formula for charm (see below) is expressed in delta/year. It is often useful to divide this by the number of days per year to arrive at the delta decay per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate.

### Color

 $\text{Color} = \frac{\partial \Gamma}{\partial \tau} = \frac{\partial^3 V}{\partial S^2 \, \partial \tau}$

Color,[9] gamma decay or DgammaDtime[10] measures the rate of change of gamma over the passage of time. Color is a third-order derivative of the option value, twice to underlying asset price and once to time. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes.

#### Practical use

The mathematical result of the formula for color (see below) is expressed in gamma/year. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate.

 $\frac{\partial \nu}{\partial \tau} = \frac{\partial^2 V}{\partial \sigma \, \partial \tau}$

DvegaDtime,[11] measures the rate of change in the vega with respect to the passage of time. DvegaDtime is the second derivative of the value function; once to volatility and once to time.

#### Practical use

It is common practice to divide the mathematical result of DvegaDtime by 100 times the number of days per year to reduce the value to the percentage change in vega per one day.

### Gamma Γ

 $\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$

Gamma,[7] Γ, measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price. All long options have positive gamma and all short options have negative 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. However, in neutralizing the gamma of a portfolio, alpha (the return in excess of the risk-free rate) is reduced.

### Lambda

 $\lambda = \frac{\partial V}{\partial S}\times\frac{S}{V}$

Lambda, λ, omega, Ω, or elasticity[7] is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.

### Speed

 $\text{Speed} = \frac{\partial\Gamma}{\partial S} = \frac{\partial^3 V}{\partial S^3}$

Speed[7] measures the rate of change in Gamma with respect to changes in the underlying price. This is also sometimes referred to as the gamma of the gamma[12] or DgammaDspot.[13] Speed is the third derivative of the value function with respect to the underlying spot price. Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio.

### Ultima

 $\text{Ultima}= \frac{\partial \text{vomma}}{\partial \sigma} = \frac{\partial^3 V}{\partial \sigma^3}$

Ultima[7] measures the sensitivity of the option vomma with respect to change in volatility. Ultima has also been referred to as DvommaDvol.[7] Ultima is a third-order derivative of the option value to volatility.

### Vanna

 $\text{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \nu}{\partial S} = \frac{\partial^2 V}{\partial S \partial \sigma}$

Vanna,[7] also referred to as DvegaDspot and DdeltaDvol,[8] is a second order derivative of the option value, once to the underlying spot price and once to volatility. It is mathematically equivalent to DdeltaDvol,[8] the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price.

### Vomma

 $\text{Vomma} = \frac{\partial \nu}{\partial \sigma} = \frac{\partial^2 V}{\partial \sigma^2}$

Vomma,[14] Volga,[14] Vega Convexity,[14] Vega gamma or dTau/dVol measures second order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes. With positive vomma, a position will become long vega as implied volatility increases and short vega as it decreases, which can be scalped in a way analogous to long gamma. And an initially vega-neutral, long-vomma position can be constructed from ratios of options at different strikes. Vomma is positive for options away from the money, and initially increases with distance from the money (but drops off as vega drops off). (Specifically, vomma is positive where the usual d1 and d2 terms are of the same sign, which is true when d2 < 0 or d1 > 0.)

### Zomma

 $\text{Zomma} = \frac{\partial \Gamma}{\partial \sigma} = \frac{\partial \text{vanna}}{\partial S} = \frac{\partial^3 V}{\partial S^2 \, \partial \sigma}$

Zomma[7] measures the rate of change of gamma with respect to changes in volatility. Zomma has also been referred to as DgammaDvol.[13] Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. Zomma can be a useful sensitivity to monitor when maintaining a gamma-hedged portfolio as zomma will help the trader to anticipate changes to the effectiveness of the hedge as volatility changes.

## 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 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 $e^{-q \tau} \Phi(d_1) \,$ $-e^{-q \tau} \Phi(-d_1)\,$
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 $K \tau e^{-r \tau}\Phi(d_2)\,$ $-K \tau e^{-r \tau}\Phi(-d_2) \,$
gamma $e^{-q \tau} \frac{\phi(d_1)}{S\sigma\sqrt{\tau}} \,$
vanna $-e^{-q \tau} \phi(d_1) \frac{d_2}{\sigma} \, = \frac{\nu}{S}\left[1 - \frac{d_1}{\sigma\sqrt{\tau}} \right]\,$
charm $qe^{-q \tau} \Phi(d_1) - e^{-q \tau} \phi(d_1) \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}} \,$ $-qe^{-q \tau} \Phi(-d_1) - e^{-q \tau} \phi(d_1) \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{2\tau \sigma \sqrt{\tau}} \,$
speed $-e^{-q \tau} \frac{\phi(d_1)}{S^2 \sigma \sqrt{\tau}} \left(\frac{d_1}{\sigma \sqrt{\tau}} + 1\right) = -\frac{\Gamma}{S}\left(\frac{d_1}{\sigma\sqrt{\tau}}+1\right) \,$
zomma $e^{-q \tau} \frac{\phi(d_1)\left(d_1 d_2 - 1\right)}{S\sigma^2\sqrt{\tau}} = \Gamma\cdot\left(\frac{d_1 d_2 -1}{\sigma}\right) \,$
color $-e^{-q \tau} \frac{\phi(d_1)}{2S\tau \sigma \sqrt{\tau}} \left[2q\tau + 1 + \frac{2(r-q) \tau - d_2 \sigma \sqrt{\tau}}{\sigma \sqrt{\tau}}d_1 \right] \,$
DvegaDtime $Se^{-q \tau} \phi(d_1) \sqrt{\tau} \left[ q + \frac{ \left( r - q \right) d_1 }{ \sigma \sqrt{\tau} } - \frac{1 + d_1 d_2}{2 \tau} \right] \,$
vomma $Se^{-q \tau} \phi(d_1) \sqrt{\tau} \frac{d_1 d_2}{\sigma} = \nu \frac{d_1 d_2}{\sigma} \,$
Ultima $\frac{-\nu}{\sigma^2} \left[ d_1 d_2 (1 - d_1 d_2) + d_1^2 + d_2^2 \right]$
dual delta $-e^{-r \tau} \Phi(d_2) \,$ $e^{-r \tau} \Phi(-d_2) \,$
dual gamma $e^{-r \tau} \frac{\phi(d_2)}{K\sigma\sqrt{\tau}} \,$

where

$d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)\tau}{\sigma\sqrt{\tau}}$
$d_2 = \frac{\ln(S/K) + (r - q - \sigma^2/2)\tau}{\sigma\sqrt{\tau}} = d_1 - \sigma\sqrt{\tau}$
$\phi(x) = \frac{e^{- \frac{x^2}{2}}}{\sqrt{2 \pi}}$
$\Phi(x) = \frac{1}{\sqrt{2\pi }} \int_{-\infty}^x e^{- \frac{y^2}{2}} \,dy =1- \frac{1}{\sqrt{2\pi }} \int_{-x}^\infty e^{- \frac{y^2}{2}} \,dy$

## Other quantities

Some other basic quantities used in managing financial derivatives are:

### Bond duration

In trading of fixed interest securities (bonds), various measures of bond duration are used analogously to the delta of an option. The closest analogue to the delta is DV01, which is the reduction in price (in currency units) for an increase of one basis point (i.e. 0.01% per annum) in the yield (the yield is the underlying variable).

Analogous to the lambda is the modified duration, which is the percentage change in the market price of the bond(s) for a unit change in the yield (i.e. it is equivalent to DV01 divided by the market price). Unlike the lambda, which is an elasticity (a percentage change in output for a percentage change in input), the modified duration is instead a semi-elasticity—a percentage change in output for a unit change in input.

### Fugit

The fugit is the optimal date to exercise an American or Bermudan option. It is useful to compute it for hedging purposes—for example, one can represent flows of an American swaption like the flows of a swap starting at the fugit multiplied by delta, then use these to compute sensitivities.

• Alpha (finance)
• Beta coefficient
• Delta neutral
• Greek letters used in mathematics

## References

1. ^ Banks, Erik; Siegel, Paul (2006). The options applications handbook: hedging and speculating techniques for professional investors. McGraw-Hill Professional. p. 263. ISBN 0071453156, 9780071453158.
2. ^ Macmillan, Lawrence G. (1993). Options as a Strategic Investment (3rd ed.). New York Institute of Finance. p. 742. ISBN [[Special:BookSources/978-0-13-636002-5 0-13-099661-0|978-0-13-636002-5 0-13-099661-0]].
3. ^ Chriss, Neil (1996). Black–Scholes and beyond: option pricing models. McGraw-Hill Professional. p. 308. ISBN [[Special:BookSources/0786310251, 9780786310258 but no the roer yan pooo rit|0786310251, 9780786310258 but no the roer yan pooo rit]].
4. ^ Haug, Espen Gaardner (2007). The Complete Guide to Option Pricing Formulas. McGraw-Hill Professional. ISBN 0071389970, 9780071389976.
5. ^ Suma, John. "Options Greeks: Delta Risk and Reward". Retrieved 7 Jan 2010.
6. ^ There is a slight bias for a greater probability that a call will expire in-the-money than a put at the same strike when the underlying is also exactly at the strike. This bias is due to the much larger range of prices that the underlying could be within at expiration for calls (Strike...+inf) than puts (0...Strike). However, with large strike and underlying values, this asymmetry can be effectively eliminated. Yet the "bias" to the call remains (ATM delta > 0.50) due to the expected value of the lognormal distribution (namely, the (1/2)σ2 term). Also, in markets that exhibit contango forward prices (positive basis), the effect of interest rates on forward prices will also cause the call delta to increase.[citation needed]
7. ^ a b c d e f g h i j k Haug, Espen Gaardner (2007). The Complete Guide to Option Pricing Formulas. McGraw-Hill Professional. ISBN 0071389970, 9780071389976.
8. ^ a b c Espen Gaarder Haug, "Know Your Weapon, Part 1", Wilmott Magazine, May 2003, p. 51
9. ^ This author has only seen this referred to in the English spelling "Colour", but has written it here in the US spelling to match the style of the existing article.
10. ^ Espen Gaarder Haug, "Know Your Weapon, Part 1", Wilmott Magazine, May 2003, p. 56
11. ^ Espen Gaarder Haug, "Know Your Weapon, Part 2", Wilmott Magazine, July/August 2003, p. 53
12. ^ Macmillan, Lawrence G. (1993). Options as a Strategic Investment (3rd ed.). New York Institute of Finance. p. 799. ISBN 978-0-13-636002-5.
13. ^ a b Espen Gaarder Haug, "Know Your Weapon, Part 1", Wilmott Magazine, May 2003, p. 55
14. ^ a b c Espen Gaarder Haug, "Know Your Weapon, Part 2", Wilmott Magazine, July/August 2003, p. 52

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