# Greeks (finance)

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In mathematical finance, the Greeks are the quantities representing the sensitivity 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 denoted by Greek letters (as are some other finance measures). Collectively these have also been called the risk sensitivities, [1] risk measures [2] :742 or hedge parameters. [3]

Mathematical 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 extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation or getting access to otherwise hard-to-trade assets or markets. Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the New York Stock Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges. Derivatives are one of the three main categories of financial instruments, the other two being stocks and debt. The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed a century ago, are a more recent historical example.

In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

## Use of the Greeks

Spot
price (S)
Volatility
(${\displaystyle \sigma }$)
Time to
expiry (${\displaystyle \tau }$)
Value (V) ${\displaystyle \Delta }$ Delta ${\displaystyle {\mathcal {V}}}$ Vega ${\displaystyle \Theta }$ Theta
Delta (${\displaystyle \Delta }$) ${\displaystyle \Gamma }$ Gamma Vanna Charm
Vega (${\displaystyle {\mathcal {V}}}$)  Vanna Vomma Veta
Theta (${\displaystyle \Theta }$)  Charm Veta
Gamma (${\displaystyle \Gamma }$)  Speed Zomma Color
Vomma  Ultima
 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. Note that 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.

Financial risk management is the practice of economic value in a firm by using financial instruments to manage exposure to risk: operational risk, credit risk and market risk, foreign exchange risk, shape risk, volatility risk, liquidity risk, inflation risk, business risk, legal risk, reputational risk, sector risk etc. Similar to general risk management, financial risk management requires identifying its sources, measuring it, and plans to address them.

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry.

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 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 Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by options market participants.

A financial market is a market in which people trade financial securities and derivatives such as futures and options at low transaction costs. Securities include stocks and bonds, and precious metals.

The risk-free interest rate is the rate of return of a hypothetical investment with no risk of financial loss, over a given period of time.

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.

## 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' 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.

Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the market during that same period; a negative alpha means the investment underperformed the market. Alpha, along with beta, is one of two key coefficients in the capital asset pricing model used in modern portfolio theory and is closely related to other important quantities such as standard deviation, R-squared and the Sharpe ratio.

In finance, the beta of an investment indicates whether the investment is more or less volatile than the market as a whole.

In finance, volatility is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns.

## First-order Greeks

### Delta

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

Delta , [4] ${\displaystyle \Delta }$, 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 ${\displaystyle V}$ of the option with respect to the underlying instrument's price ${\displaystyle S}$.

Delta is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter dalet 𐤃, Letters that come from delta include Latin D and Cyrillic Д.

#### 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 close to but not in general equal to one, but instead is equal to the inverse of the discount factor. 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 factor the inverse of the discount factor, so the derivative dF/dS is this factor.

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). [5] 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. 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. [6]

#### 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.

${\displaystyle \Delta (call)-\Delta (put)=1}$, therefore: ${\displaystyle \Delta (call)=\Delta (put)+1}$ and ${\displaystyle \Delta (put)=\Delta (call)-1}$.

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

 ${\displaystyle {\mathcal {V}}={\frac {\partial V}{\partial \sigma }}}$

Vega [4] 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, ${\displaystyle \nu }$, written as ${\displaystyle {\mathcal {V}}}$. 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 , ${\displaystyle \kappa }$, is sometimes used (by academics) instead of vega (as is tau (${\displaystyle \tau }$) or capital lambda (${\displaystyle \Lambda }$), [7] :315 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%. 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 option straddle, for example, is extremely dependent on changes to volatility.

### Theta

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

Theta , [4] ${\displaystyle \Theta }$, measures the sensitivity of the value of the derivative to the passage of time (see Option time value): 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 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

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

Rho , [4] ${\displaystyle \rho }$, 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

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

Lambda , [4] ${\displaystyle \lambda }$, omega , [8] ${\displaystyle \Omega }$, or elasticity [4] is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.

### Epsilon

 ${\displaystyle \epsilon =\psi ={\frac {\partial V}{\partial q}}}$

Epsilon , [9] ${\displaystyle \epsilon }$ (also known as psi, ${\displaystyle \psi }$), 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.

## Second-order Greeks

### Gamma

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

Gamma , [4] ${\displaystyle \Gamma }$, 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. 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. [10]

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, [4] also referred to as DvegaDspot [12] and DdeltaDvol, [12] 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, 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 ${\displaystyle {\text{Vanna}}={\frac {\partial \Delta }{\partial \sigma }}={\frac {\partial {\mathcal {V}}}{\partial S}}={\frac {\partial ^{2}V}{\partial S\partial \sigma }}}$,

### Charm

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

Charm [4] or delta decay [13] measures the instantaneous rate of change of delta over the passage of time. Charm has also been called DdeltaDtime. [12] 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.

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

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

Vomma, [4] volga, [14] vega convexity, [14] or DvegaDvol [14] 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 d1 < 0 or d2 > 0.)

### Veta

 ${\displaystyle {\text{Veta}}={\frac {\partial {\mathcal {V}}}{\partial \tau }}={\frac {\partial ^{2}V}{\partial \sigma \,\partial \tau }}}$

Veta [15] or DvegaDtime [14] 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

 ${\displaystyle {\text{Vera}}={\frac {\partial \rho }{\partial \sigma }}={\frac {\partial ^{2}V}{\partial \sigma \,\partial r}}}$

Vera [16] (sometimes rhova) [16] 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. Vera can be used to assess the impact of volatility change on rho-hedging.

## Third-order Greeks

### Speed

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

Speed [4] 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 [2] :799 or DgammaDspot. [12] 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

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

Zomma [4] measures the rate of change of gamma with respect to changes in volatility. Zomma has also been referred to as DgammaDvol. [12] 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

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

Color, [12] [note 1] gamma decay [17] or DgammaDtime [12] 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/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

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

Ultima [4] measures the sensitivity of the option vomma with respect to change in volatility. Ultima has also been referred to as DvommaDvol. [4] 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 more underlyings, 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. [18] It is also commonly known as cega [19] [20]

Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying. [21]

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. [18]

Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying. [21]

## Formulas for European option Greeks

The Greeks of European options (calls and puts) under the Black–Scholes model are calculated as follows, where ${\displaystyle \phi }$ (phi) is the standard normal probability density function and ${\displaystyle \Phi }$ 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 ${\displaystyle S\,}$,
• Strike price ${\displaystyle K\,}$,
• Risk-free rate ${\displaystyle r\,}$,
• Annual dividend yield ${\displaystyle q\,}$,
• Time to maturity ${\displaystyle \tau =T-t\,}$ (represented as a unit-less fraction of one year), and
• Volatility ${\displaystyle \sigma \,}$.
CallsPuts
fair value (${\displaystyle V}$)${\displaystyle Se^{-q\tau }\Phi (d_{1})-e^{-r\tau }K\Phi (d_{2})\,}$${\displaystyle e^{-r\tau }K\Phi (-d_{2})-Se^{-q\tau }\Phi (-d_{1})\,}$
delta (${\displaystyle \Delta }$)${\displaystyle e^{-q\tau }\Phi (d_{1})\,}$${\displaystyle -e^{-q\tau }\Phi (-d_{1})\,}$
vega (${\displaystyle {\mathcal {V}}}$)${\displaystyle Se^{-q\tau }\phi (d_{1}){\sqrt {\tau }}=Ke^{-r\tau }\phi (d_{2}){\sqrt {\tau }}\,}$
theta (${\displaystyle \Theta }$)${\displaystyle -e^{-q\tau }{\frac {S\phi (d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\Phi (d_{2})+qSe^{-q\tau }\Phi (d_{1})\,}$${\displaystyle -e^{-q\tau }{\frac {S\phi (d_{1})\sigma }{2{\sqrt {\tau }}}}+rKe^{-r\tau }\Phi (-d_{2})-qSe^{-q\tau }\Phi (-d_{1})\,}$
rho (${\displaystyle \rho }$)${\displaystyle K\tau e^{-r\tau }\Phi (d_{2})\,}$${\displaystyle -K\tau e^{-r\tau }\Phi (-d_{2})\,}$
lambda (${\displaystyle \lambda }$)${\displaystyle \Delta {\frac {S}{V}}\,}$
gamma (${\displaystyle \Gamma }$)${\displaystyle e^{-q\tau }{\frac {\phi (d_{1})}{S\sigma {\sqrt {\tau }}}}=Ke^{-r\tau }{\frac {\phi (d_{2})}{S^{2}\sigma {\sqrt {\tau }}}}\,}$
vanna${\displaystyle -e^{-q\tau }\phi (d_{1}){\frac {d_{2}}{\sigma }}\,={\frac {\mathcal {V}}{S}}\left[1-{\frac {d_{1}}{\sigma {\sqrt {\tau }}}}\right]\,}$
charm${\displaystyle 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 }}}}\,}$${\displaystyle -qe^{-q\tau }\Phi (-d_{1})-e^{-q\tau }\phi (d_{1}){\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{2\tau \sigma {\sqrt {\tau }}}}\,}$
vomma${\displaystyle Se^{-q\tau }\phi (d_{1}){\sqrt {\tau }}{\frac {d_{1}d_{2}}{\sigma }}={\mathcal {V}}{\frac {d_{1}d_{2}}{\sigma }}\,}$
veta${\displaystyle -Se^{-q\tau }\phi (d_{1}){\sqrt {\tau }}\left[q+{\frac {\left(r-q\right)d_{1}}{\sigma {\sqrt {\tau }}}}-{\frac {1+d_{1}d_{2}}{2\tau }}\right]\,}$
speed${\displaystyle -e^{-q\tau }{\frac {\phi (d_{1})}{S^{2}\sigma {\sqrt {\tau }}}}\left({\frac {d_{1}}{\sigma {\sqrt {\tau }}}}+1\right)=-{\frac {\Gamma }{S}}\left({\frac {d_{1}}{\sigma {\sqrt {\tau }}}}+1\right)\,}$
zomma${\displaystyle e^{-q\tau }{\frac {\phi (d_{1})\left(d_{1}d_{2}-1\right)}{S\sigma ^{2}{\sqrt {\tau }}}}=\Gamma \cdot \left({\frac {d_{1}d_{2}-1}{\sigma }}\right)\,}$
color${\displaystyle -e^{-q\tau }{\frac {\phi (d_{1})}{2S\tau \sigma {\sqrt {\tau }}}}\left[2q\tau +1+{\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{\sigma {\sqrt {\tau }}}}d_{1}\right]\,}$
ultima${\displaystyle {\frac {-{\mathcal {V}}}{\sigma ^{2}}}\left[d_{1}d_{2}(1-d_{1}d_{2})+d_{1}^{2}+d_{2}^{2}\right]}$
dual delta${\displaystyle -e^{-r\tau }\Phi (d_{2})\,}$${\displaystyle e^{-r\tau }\Phi (-d_{2})\,}$
dual gamma${\displaystyle e^{-r\tau }{\frac {\phi (d_{2})}{K\sigma {\sqrt {\tau }}}}\,}$

where

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

Some related risk measures of financial derivatives are listed below.

### Bond duration and convexity

In trading of fixed income 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.

Bond convexity is a measure of the sensitivity of the duration to changes in interest rates, the second derivative 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, the standard yield to maturity 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 Lattice model (finance)#Interest rate derivatives.

### Beta

The beta (β) of a stock or portfolio 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.

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

## Notes

1. This author has only seen this referred to in the British spelling "colour", but has written it here in the U.S. spelling to match the style of the existing article.

## Related Research Articles

In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model will return a theoretical value equal to the current market price of the option. A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an implied volatility. Implied volatility, a forward-looking and subjective measure, differs from historical volatility because the latter is calculated from known past returns of a security. To understand where Implied Volatility stands in terms of the underlying, implied volatility rank is used to understand its implied volatility from a one year high and low IV.

In finance, moneyness is the relative position of the current price of an underlying asset with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a three-fold classification: if the derivative would have positive intrinsic value if it were to expire today, it is said to be in the money; if it would be worthless if expiring at the current price it is said to be out of the money, and if the current price and strike price are equal, it is said to be at the money. There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc.

Volatility risk is the risk of a change of price of a portfolio as a result of changes in the volatility of a risk factor. It usually applies to portfolios of derivatives instruments, where the volatility of its under lyings is a major influencer of prices.

Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

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. In general, the higher the duration, 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.

In finance, a foreign exchange option is a derivative financial instrument that gives the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date. See Foreign exchange derivative.

Volatility smiles are implied volatility patterns that arise in pricing financial options. It corresponds to finding one single parameter that is needed to be modified for the Black-Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money.

A variance swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the magnitude of movement, i.e. volatility, of some underlying product, like an exchange rate, interest rate, or stock index.

In finance, volatility arbitrage is a type of statistical arbitrage that is implemented by trading a delta neutral portfolio of an option and its underlying. The objective is to take advantage of differences between the implied volatility of the option, and a forecast of future realized volatility of the option's underlying. In volatility arbitrage, volatility rather than price is used as the unit of relative measure, i.e. traders attempt to buy volatility when it is low and sell volatility when it is high.

In 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 typically contains options and their corresponding underlying securities such that positive and negative delta components offset, resulting in the portfolio's value being relatively insensitive to changes in the value of the underlying security.

In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise., though methods now exist for solving this problem.

In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

PnL Explained also called P&L Explain, P&L Attribution or Profit and Loss Explained is a type of report commonly used by traders, especially derivatives traders and produced by Product control, that attributes or explains the daily fluctuation in the value of a portfolio of trades to the root causes of the changes.

The Vanna–Volga method is a mathematical tool used in finance. It is a technique for pricing first-generation exotic options in foreign exchange market (FX) derivatives.

In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.

## References

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