In

/ref> (sometimes rhova) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate. The word 'Vera' was coined by R. Naryshkin in early 2012 when this sensitivity needed to be used in practice to assess the impact of volatility changes on rho-hedging, but no name yet existed in the available literature. 'Vera' was picked to sound similar to a combination of Vega and Rho, its respective first-order Greeks. This name is now in a wider use, including, for example, the

/ref> which uses quoted call option prices to estimate the Risk-neutral measure, risk-neutral probabilities implied by such prices. For call options, it can be approximated using infinitesimal portfolios of

Vanilla Options - Espen Haug

* Volga, Vanna, Speed, Charm, Color

Vanilla Options - Uwe Wystup

Vanilla Options - Uwe Wystup

Step-by-step mathematical derivations of option Greeks

Derivation of European Vanilla Call Price

Derivation of European Vanilla Call Delta

Derivation of European Vanilla Call Gamma

Derivation of European Vanilla Call Speed

Derivation of European Vanilla Call Vega

Derivation of European Vanilla Call Volga

Derivation of European Vanilla Call Vanna as Derivative of Vega with respect to underlying

Derivation of European Vanilla Call Vanna as Derivative of Delta with respect to volatility

Derivation of European Vanilla Call Theta

Derivation of European Vanilla Call Rho

Derivation of European Vanilla Put Price

Derivation of European Vanilla Put Delta

Derivation of European Vanilla Put Gamma

Derivation of European Vanilla Put Speed

Derivation of European Vanilla Put Vega

Derivation of European Vanilla Put Volga

Derivation of European Vanilla Put Vanna as Derivative of Vega with respect to underlying

Derivation of European Vanilla Put Vanna as Derivative of Delta with respect to volatility

Derivation of European Vanilla Put Theta

Derivation of European Vanilla Put Rho

Online tools

Surface Plots of Black-Scholes Greeks

Chris Murray

Online real-time option prices and Greeks calculator when the underlying is normally distributed

Razvan Pascalau, Univ. of Alabama

R package to compute Greeks for European-, American- and Asian Options {{DEFAULTSORT:Greeks (Finance) Mathematical finance Financial ratios Options (finance)

mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics
Physics is the natu ...

, the Greeks are the quantities representing the sensitivity of the price of derivatives
Derivative may refer to:
In mathematics and economics
*Brzozowski derivative in the theory of formal languages
*Derivative in calculus, a quantity indicating how a function changes when the values of its inputs change.
*Formal derivative, an opera ...

such as options to a change in underlying parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s on which the value of an instrument or portfolio
Portfolio may refer to:
Objects
* Portfolio (briefcase), a type of briefcase
Collections
* Portfolio (finance)
In finance, a portfolio is a collection of investments
To invest is to allocate money
Image:National-Debt-Gillray.jpeg, ...

of financial instrument
Financial instruments are monetary contracts
A contract is a legally binding document between at least two parties that defines and governs the rights and duties of the parties to an agreement. A contract is legally enforceable because it me ...

s is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...

(as are some other finance measures). Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.
Use of the Greeks

The Greeks are vital tools inrisk management
Risk management is the identification, evaluation, and prioritization of risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty
Uncertainty refers to Epistemology, epistemic situations involving ...

. Each Greek measures the 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 hedgingIn finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a Portfolio (finance), portfolio typically contains ...

.
The Greeks in the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

are relatively easy to calculate, a desirable property of financial
Finance is a term for the management, creation, and study of money and investments. Pamela Drake and Frank Fabozzi (2009)What Is Finance?/ref>
Specifically, it deals with the questions of how an individual, company or government acquires money ...

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 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
The risk-free interest rate is the rate of return
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and inv ...

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
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, a landform at the mouth of a river
* D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet
* Delta Air Lines, an Ame ...

, vega
Vega is the brightest star
A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun ...

, theta
Theta (, ; uppercase Θ or ϴ, lowercase θ or ϑ; grc, ''thē̂ta'' ; Modern
Modern may refer to:
History
*Modern history
Human history, also known as world history, is the description of humanity's past. It is informed by archaeo ...

and rho
Rho (uppercase Ρ, lowercase ρ or ; el, ῥῶ) is the 17th letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

as well as gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter rep ...

, 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.
Names

The use of Greek letter names is presumably by extension from the common finance termsalpha
Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', modern pronunciation ''álfa'') is the first letter
Letter, letters, or literature may refer to:
Characters typeface
* Letter (alphabet)
A letter is a segmental symbol
A s ...

and beta
Beta (, ; uppercase , lowercase , or cursive
Cursive (also known as script, among other names) is any style of penmanship
Penmanship is the technique of writing
Writing is a medium of human communication that involves the represen ...

, and the use of sigma
Sigma (uppercase
Letter case is the distinction between the letters
Letter, letters, or literature may refer to:
Characters typeface
* Letter (alphabet)
A letter is a segmental symbol
A symbol is a mark, sign, or word that ind ...

(the standard deviation of logarithmic returns) and tau
Tau (uppercase Τ, lowercase τ; el, ταυ ) is the 19th letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

(time to expiry) 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
A quark () is a type of elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundam ...

in particle physics
Particle physics (also known as high energy physics) is a branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which rel ...

.
First-order Greeks

Delta

Delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, a landform at the mouth of a river
* D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet
* Delta Air Lines, an Ame ...

, $\backslash Delta$, 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 longcall
Call or Calls may refer to:
Arts, entertainment, and media Games
* Call, a type of betting in poker
* Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage
Music and dance
* Call (band), from Lahore, Pakis ...

(or a short put) and 0.0 and −1.0 for a long (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 parityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

, 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
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money avail ...

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 percentmoneyness
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

of an option, i.e., the ''implied'' probability that the option will expire in-the-money
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative (finance), derivative, most commonly a call option or a put option. Money ...

(if the market moves under Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physica ...

in the risk-neutral measure
In mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive an ...

). 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
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availabl ...

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
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls. The actual 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 – more precisely, the delta of the call (positive) minus the delta of the put (negative) equals 1. This is due toput–call parityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

: 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.
$\backslash Delta(call)\; -\; \backslash Delta(put)\; =\; 1$, therefore: $\backslash Delta(call)\; =\; \backslash Delta(put)\; +\; 1$ and $\backslash Delta(put)\; =\; \backslash Delta(call)\; -\; 1$.
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, $\backslash nu$, written as $\backslash mathcal$. 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 symbolkappa
Kappa (uppercase Κ, lowercase κ or cursive
Cursive (also known as script, among other names) is any style of penmanship
Penmanship is the technique of writing
Writing is a medium of human communication that involves the repre ...

, $\backslash kappa$, is sometimes used (by academics) instead of vega (as is tau ($\backslash tau$)
or capital lambda
Lambda (; uppercase , lowercase ; el, λάμ(β)δα, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the sound Dental, alveolar and postalveolar lateral approximants, /l/. In the system of Greek numerals, lambda has a ...

($\backslash Lambda$),
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
A percentage point or percent point is the unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit ...

. 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 , for example, is extremely dependent on changes to volatility.
Theta

Theta
Theta (, ; uppercase Θ or ϴ, lowercase θ or ϑ; grc, ''thē̂ta'' ; Modern
Modern may refer to:
History
*Modern history
Human history, also known as world history, is the description of humanity's past. It is informed by archaeo ...

, $\backslash Theta$, measures the sensitivity of the value of the derivative to the passage of time (see Option time valueIn finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available w ...

): the "time decay."
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 an option's price will drop, in relation to the underlying stock's price. 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
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative (finance), derivative, most commonly a call option or a put option. Moneyn ...

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

Rho
Rho (uppercase Ρ, lowercase ρ or ; el, ῥῶ) is the 17th letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

, $\backslash rho$, 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.
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).
Lambda

Lambda
Lambda (; uppercase , lowercase ; el, λάμ(β)δα, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the sound Dental, alveolar and postalveolar lateral approximants, /l/. In the system of Greek numerals, lambda has a ...

, $\backslash lambda$, omega
Omega (; capital
Capital most commonly refers to:
* Capital letter
Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (o ...

, $\backslash Omega$, or elasticity is the percentage
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

change in option value per percentage change in the underlying price, a measure of leverage
Leverage or leveraged may refer to:
*Leverage (mechanics), mechanical advantage achieved by using a lever
*Leverage (album), ''Leverage'' (album), a 2012 album by Lyriel
*Leverage (dance), a type of dance connection
*Leverage (finance), using giv ...

, sometimes called gearing.
It holds that $\backslash lambda\; =\; \backslash Omega\; =\; \backslash Delta\backslash times\backslash frac$.
Epsilon

Epsilon
Epsilon (, ; ; uppercase
Letter case is the distinction between the letters
Letter, letters, or literature may refer to:
Characters typeface
* Letter (alphabet)
A letter is a segmental symbol
A symbol is a mark, sign, or word th ...

, $\backslash epsilon$ (also known as psi, $\backslash psi$), is the percentage change in option value per percentage
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

change in the underlying dividend
A dividend is a distribution of profit
Profit may refer to:
Business and law
* Profit (accounting), the difference between the purchase price and the costs of bringing to market
* Profit (economics), normal profit and economic profit
* Profit ...

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
Equity may refer to:
Finance, accounting and ownership
*Equity (finance), ownership of assets that have liabilities attached to them
** Stock, equity based on original contributions of cash or other value to a business
** Home equity, the differe ...

products.
Second-order Greeks

Gamma

Gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

, $\backslash Gamma$, measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the value function with respect to the underlying price.
Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 (long call option) and 0 from −1 (long put option). The inverse is true for short options. 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, also referred to as DvegaDspot and DdeltaDvol, is a second order derivative of the option value, once to the underlyingspot price
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, ...

and once to volatility. It is mathematically equivalent to DdeltaDvol, 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.
If the underlying value has continuous second partial derivatives, then $\backslash text\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac$,
Charm

Charm or delta decay measures the instantaneous rate of change of delta over the passage of time. Charm has also been called DdeltaDtime. 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 oftheta
Theta (, ; uppercase Θ or ϴ, lowercase θ or ϑ; grc, ''thē̂ta'' ; Modern
Modern may refer to:
History
*Modern history
Human history, also known as world history, is the description of humanity's past. It is informed by archaeo ...

with respect to the underlying's price.
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.
Vomma

Vomma, volga, vega convexity, or DvegaDvol 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 asimplied volatilityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

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 long 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 d1 < 0 or d2 > 0.)
Veta

Veta or DvegaDtime measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time. It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day.Vera

VeraDerivatives – Second-Order Greeks – The Financial Encyclopedia/ref> (sometimes rhova) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate. The word 'Vera' was coined by R. Naryshkin in early 2012 when this sensitivity needed to be used in practice to assess the impact of volatility changes on rho-hedging, but no name yet existed in the available literature. 'Vera' was picked to sound similar to a combination of Vega and Rho, its respective first-order Greeks. This name is now in a wider use, including, for example, the

Maple
''Acer'' is a genus
Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank
In biological classification
In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscr ...

computer algebra software (which has 'BlackScholesVera' function in its Finance package).
Second order partial derivative with respect to $K$

This partial derivative has a fundamental role in the Breeden-Litzenberger formula,Breeden, Litzenberger, Prices of State-Contingent Claims Implicit in Option Price/ref> which uses quoted call option prices to estimate the Risk-neutral measure, risk-neutral probabilities implied by such prices. For call options, it can be approximated using infinitesimal portfolios of

butterfly
Butterflies are insect
Insects (from Latin ') are pancrustacean Hexapoda, hexapod invertebrates of the class (biology), class Insecta. They are the largest group within the arthropod phylum. Insects have a chitinous exoskeleton, a three ...

strategies.
Third-order Greeks

Speed

Speed 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 or DgammaDspot. 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.Zomma

Zomma measures the rate of change of gamma with respect to changes in volatility. Zomma has also been referred to as DgammaDvol. 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.Color

Color, gamma decay or DgammaDtime 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. The mathematical result of the formula for color (see below) is expressed in gamma per 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.Ultima

Ultima measures the sensitivity of the option vomma with respect to change in volatility. Ultima has also been referred to as DvommaDvol. Ultima is a third-order derivative of the option value to volatility.Greeks for multi-asset options

If the value of a derivative is dependent on two or moreunderlying
In finance, the underlying of a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...

s, its Greeks are extended to include the cross-effects between the underlyings.
Correlation delta measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. 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.
Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying.
Formulas for European option Greeks

The Greeks ofEuropean optionsIn finance, the style or family of an option (finance), option is the class into which the option falls, usually defined by the dates on which the option may be Exercise (options), exercised. The vast majority of options are either European or Ameri ...

( and ) under the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

are calculated as follows, where $\backslash phi$ (phi) is the standard normal
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

probability density function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

and $\backslash Phi$ is the standard normal
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

cumulative distribution function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...

. Note that the gamma and vega formulas are the same for calls and puts.
For a given:
* Stock price $S\; \backslash ,$,
* Strike price $K\; \backslash ,$,
* Risk-free rate $r\; \backslash ,$,
* Annual dividend yield $q\; \backslash ,$,
* Time to maturity $\backslash tau\; =\; T\; -\; t\; \backslash ,$ (represented as a unit-less fraction of one year), and
* Volatility $\backslash sigma\; \backslash ,$.
where
:$\backslash begin\; d\_1\; \&=\; \backslash frac\; \backslash \backslash \; d\_2\; \&=\; \backslash frac\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\; \backslash \backslash \; \backslash phi(x)\; \&=\; \backslash frac\; e^\; \backslash \backslash \; \backslash Phi(x)\; \&=\; \backslash frac\; \backslash int\_^x\; e^\; \backslash ,dy\; =\; 1\; -\; \backslash frac\; \backslash int\_^\backslash infty\; e^\; \backslash ,dy\; \backslash end$
Under the Black model
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond option
In finance
Finance is the study of financial i ...

(commonly used for commodities and options on futures) the Greeks can be calculated as follows:
where
:$\backslash begin\; d\_1\; \&=\; \backslash frac\; \backslash \backslash \; d\_2\; \&=\; \backslash frac\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\; \backslash end$
(*) It can be shown that $F\backslash phi(d\_1)\; =\; K\backslash phi(d\_2)$
Micro proof:
let $x=$ $d\_1\; =\; \backslash frac$ $d\_1*x\; =\; \backslash ln(F/K)\; +\; \backslash fracx^2$ $\backslash ln(F/K)\; =\; d\_1*x\; -\; \backslash fracx^2$ $\backslash frac\; =\; e^$ Then we have: $\backslash frac\; *\; \backslash frac\; =\; \backslash frac\; *\; e^$ $=e^\; *\; e^\; =\; e^\; =\; e^\; =\; 1$ So $F\backslash phi(d\_1)\; =\; K\backslash phi(d\_2)$

Related measures

Some related risk measures of financial derivatives are listed below.Bond duration and convexity

In trading of fixed income securities (bonds), various measures ofbond duration
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availabl ...

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
A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad
A basis point (o ...

(i.e. 0.01% per annum) in the yield (the yield is the underlying variable).
Analogous to the lambda is the modified duration
Modified may refer to:
*Modified (album), ''Modified'' (album), the second full-length album by Save Ferris
*Modified racing, or "Modifieds", an American automobile racing genre
See also
* Modification (disambiguation)
* Modifier (disambiguatio ...

, 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
Elasticity often refers to:
*Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress
Elasticity may also refer to:
Information technology
* Elasticity (data store), the flexibility of the data model and the clu ...

(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.
Bond convexity
In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (Bond duration, duration is the first derivative). In ...

is a measure of the sensitivity of the duration to changes in interest rate
An interest rate is the amount of interest
In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...

s, the second derivative
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study o ...

of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.
For a bond with an embedded option
An embedded option is a component of a financial bond or other security, and usually provides the bondholder or the issuer the right to take some action against the other party. There are several types of options that can be embedded into a bond. S ...

, the standard yield to maturity
The yield to maturity (YTM), book yield or redemption yield of a bond
Bond or bonds may refer to:
Common meanings
* Bond (finance)
In finance
Finance is the study of financial institutions, financial markets and how they operate within ...

based calculations here do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and effective convexity are introduced. These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see .
Beta

The beta (β) of astock
In finance, stock (also capital stock) consists of all of the shares
In financial markets
A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities i ...

or portfolio
Portfolio may refer to:
Objects
* Portfolio (briefcase), a type of briefcase
Collections
* Portfolio (finance)
In finance, a portfolio is a collection of investments
To invest is to allocate money
Image:National-Debt-Gillray.jpeg, ...

is a number describing the volatility of an asset in relation to the volatility of the benchmark that said asset is being compared to. This benchmark is generally the overall financial market and is often estimated via the use of representative indices, such as the S&P 500
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the performance of 500 large companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices. As of Decembe ...

.
An asset has a Beta of zero if its returns change independently of changes in the market's returns. A positive beta means that the asset's returns generally follow the market's returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their respective averages together. A negative beta means that the asset's returns generally move opposite the market's returns: one will tend to be above its average when the other is below its average.
Fugit

The fugit is the expected time to exercise an American or Bermudan option. It is useful to compute it for hedging purposes—for example, one can represent flows of an Americanswaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swap
In finance
...

like the flows of a swap starting at the fugit multiplied by delta, then use these to compute sensitivities.
See also

*Alpha (finance)
Alpha is a measure of the active return on an investment
Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort.
I ...

* Beta (finance)
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

* Delta neutral
* Greek letters used in mathematics, science, and engineering
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for Mathematical constant, constants, special functions, and also conventionally for Variable (mathematics), variables repr ...

*
* Vanna–Volga pricing
References

External links

Theory * Delta, Gamma, GammaP, Gamma symmetry, Vanna, Speed, Charm, Saddle GammaVanilla Options - Espen Haug

* Volga, Vanna, Speed, Charm, Color

Vanilla Options - Uwe Wystup

Vanilla Options - Uwe Wystup

Step-by-step mathematical derivations of option Greeks

Derivation of European Vanilla Call Price

Derivation of European Vanilla Call Delta

Derivation of European Vanilla Call Gamma

Derivation of European Vanilla Call Speed

Derivation of European Vanilla Call Vega

Derivation of European Vanilla Call Volga

Derivation of European Vanilla Call Vanna as Derivative of Vega with respect to underlying

Derivation of European Vanilla Call Vanna as Derivative of Delta with respect to volatility

Derivation of European Vanilla Call Theta

Derivation of European Vanilla Call Rho

Derivation of European Vanilla Put Price

Derivation of European Vanilla Put Delta

Derivation of European Vanilla Put Gamma

Derivation of European Vanilla Put Speed

Derivation of European Vanilla Put Vega

Derivation of European Vanilla Put Volga

Derivation of European Vanilla Put Vanna as Derivative of Vega with respect to underlying

Derivation of European Vanilla Put Vanna as Derivative of Delta with respect to volatility

Derivation of European Vanilla Put Theta

Derivation of European Vanilla Put Rho

Online tools

Surface Plots of Black-Scholes Greeks

Chris Murray

Online real-time option prices and Greeks calculator when the underlying is normally distributed

Razvan Pascalau, Univ. of Alabama

R package to compute Greeks for European-, American- and Asian Options {{DEFAULTSORT:Greeks (Finance) Mathematical finance Financial ratios Options (finance)