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 the left corn ...

, volatility (usually denoted by ''σ'') is the degree of variation of a trading price series over time, usually measured by the standard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of logarithmic returns.
Historic volatility measures a time series of past market prices. Implied 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 ...

looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).
Volatility terminology

Volatility as described here refers to the actual volatility, more specifically: * actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. * actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past **near synonymous is realized volatility, thesquare root
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 realized varianceRealized variance or realised variance (RV, see American and British English spelling differences, spelling differences) is the sum of squared returns. For instance the RV can be the sum of squared daily returns for a particular month, which would yi ...

, in turn calculated using the sum of squared returns divided by the number of observations.
* actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an option
Option or Options may refer to:
Computing
*Option key, a key on Apple computer keyboards
*Option type, a polymorphic data type in programming languages
*Command-line option, an optional parameter to a command
*OPTIONS, an Hypertext Transfer Prot ...

)
Now turning to implied 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 ...

, we have:
* historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options)
* current implied volatility which refers to the implied volatility observed from current prices of the financial instrument
* future implied volatility which refers to the implied volatility observed from future prices of the financial instrument
For a financial instrument whose price follows a Gaussian
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an ...

random walk
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 ...

, or Wiener process
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). ...

, the width of the distribution increases as time increases. This is because there is an increasing probability
Probability 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 in which they are contained ...

that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.
Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used. These can capture attributes such as "fat tail
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and Heavy-tailed distributio ...

s".
Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.
Mathematical definition

For any fund that evolves randomly with time, volatility is defined as thestandard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times.
Thus, "annualized" volatility is the standard deviation of an instrument's yearly logarithmic returns.
The generalized volatility for time horizon
A time horizon, also known as a planning horizon, is a fixed point of time in the future at which point certain processes will be evaluated or assumed to end. It is necessary in an accounting
Accounting or Accountancy is the measurement, proce ...

''T'' in years is expressed as:
: $\backslash sigma\_\backslash text\; =\; \backslash sigma\_\backslash text\; \backslash sqrt.$
Therefore, if the daily logarithmic returns of a stock have a standard deviation of and the time period of returns is ''P'' in trading days, the annualized volatility is
: $\backslash sigma\_\backslash text\; =\; \backslash sigma\_\backslash text\; \backslash sqrt.$
A common assumption is that ''P'' = 252 trading days in any given year. Then, if = 0.01, the annualized volatility is
: $\backslash sigma\_\backslash text\; =\; 0.01\; \backslash sqrt\; =\; 0.1587.$
The monthly volatility (i.e., ''T'' = 1/12 of a year or ''P'' = 252/12 = 21 trading days) would be
: $\backslash sigma\_\backslash text\; =\; 0.1587\; \backslash sqrt\; =\; 0.0458.$
: $\backslash sigma\_\backslash text\; =\; 0.01\; \backslash sqrt\; =\; 0.0458.$
The formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk
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 ...

, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent ''α'' to extrapolate natural processes:
: $\backslash sigma\_T\; =\; T^\; \backslash sigma.\backslash ,$
If ''α'' = 2 the Wiener process
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). ...

scaling relation is obtained, but some people believe ''α'' < 2 for financial activities such as stocks, indexes and so on. This was discovered by , who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with ''α'' = 1.7. (See New Scientist, 19 April 1997.)
Volatility origin

Much research has been devoted to modeling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place. Roll (1984) shows that volatility is affected bymarket microstructure
Market microstructure is a branch of 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 ...

. Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of adverse selection
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant ...

, they adjust their trading ranges, which in turn increases the band of price oscillation.
In September 2019, JPMorgan Chase
JPMorgan Chase & Co. is an American multinational
Multinational may refer to:
* Multinational corporation, a corporate organization operating in multiple countries
* Multinational force, a military body from multiple countries
* Multinational ...

determined the effect of US President
The president of the United States (POTUS) is the head of state
A head of state (or chief of state) is the public persona
A persona (plural personae or personas), depending on the context, can refer to either the public image of ...

Donald Trump
Donald John Trump (born June 14, 1946) is an American politician
A politician is a person active in party politics
A political party is an organization that coordinates candidate
A candidate, or nominee, is the prospective reci ...

's tweets, and called it the Volfefe index
The Volfefe Index was a stock market index of Volatility (finance), volatility in market sentiment for US Treasury bonds caused by Twitter#Tweets, tweets by former President of the USA, President Donald Trump.
''Bloomberg News'' observed Volfefe w ...

combining volatility and the covfefe
Covfefe ( ) is a misspelling that Donald Trump used in a Viral phenomenon, viral Twitter, tweet when he was President of the United States, U.S. President. It instantly became an Internet meme.
Six minutes after midnight (Eastern Time Zone, ED ...

meme
A meme ( ) is an idea, behavior, or style that spreads by means of imitation from person to person within a culture and often carries symbolic meaning representing a particular phenomenon or theme. A meme acts as a unit for carrying cultural ...

.
Volatility for investors

Investors care about volatility for at least eight reasons: # The wider the swings in an investment's price, the harder emotionally it is to not worry; # Price volatility of a trading instrument can define position sizing in a portfolio; # When certain cash flows from selling a security are needed at a specific future date, higher volatility means a greater chance of a shortfall; # Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values; # Higher volatility of return when retired gives withdrawals a larger permanent impact on the portfolio's value; # Price volatility presents opportunities to buy assets cheaply and sell when overpriced; # Portfolio volatility has a negative impact on thecompound annual growth rate
Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical st ...

(CAGR) of that portfolio
# Volatility affects pricing of options, being a parameter of 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 ...

.
In today's markets, it is also possible to trade volatility directly, through the use of derivative securities such as options and variance swapA variance swap is an over-the-counter (finance), over-the-counter financial derivative that allows one to Speculation, speculate on or hedge (finance), hedge risks associated with the magnitude of movement, i.e. volatility (finance), volatility, of ...

s. See Volatility arbitrage.
Volatility versus direction

Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculatingstandard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

(or variance
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 ...

), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time.
For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. This would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a normal distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

; in reality stocks are found to be leptokurtotic.
Volatility over time

Although the Black-Scholes equation assumes predictable constant volatility, this is not observed in real markets, and amongst the models areEmanuel Derman
Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''.
He is a co-author of Black–Derman ...

and Iraj Kani
Iraj ( fa, ایرج - ʾīraj; Middle Persian, Pahlavi: ērič; from Avestan language, Avestan: 𐬀𐬌𐬭𐬌𐬌𐬀 airiia, literally "Aryan") is seventh Shah of the Pishdadian dynasty of ''Shahnameh''. Based on Persian mythology, Iranian myth ...

's and Bruno DupireBruno Dupire is a researcher and lecturer in quantitative finance. He is currently Head of Quantitative Research at Bloomberg LP. He is best known for his contributions to local volatility modeling and Functional Ito Calculus. He is also an Instructo ...

's local volatilityA local volatility model, 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 ...

, Poisson process
In probability
Probability 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 in which t ...

where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of stochastic volatility
In statistics, stochastic volatility models are those in which the variance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic ...

.
It is common knowledge that types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all. In foreign exchange market
The foreign exchange market (Forex, FX, or currency market) is a global decentralized
Decentralization or decentralisation is the process by which the activities of an organization, particularly those regarding planning and decision maki ...

, price changes are seasonally with periods of one day and one week.
Periods when prices fall quickly (a crash
Crash or CRASH may refer to:
Common meanings
* Collision, an impact between two or more objects
* Crash (computing), a condition where a program ceases to respond
* Cardiac arrest, a medical condition in which the heart stops beating
* Couch su ...

) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible bubble
Bubble or Bubbles may refer to:
Physical bubbles
* Bubble (physics), a globule of one substance in another, usually gas in a liquid
** Soap bubble, commonly referred to as a "bubble"
People
* Bubbles, a contestant on ''Real Chance of Love ( ...

) may often be followed by prices going up even more, or going down by an unusual amount.
Most typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual. This is termed autoregressive conditional heteroskedasticity
In econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A ...

. Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.
Not only the volatility depends on the period when it is measured but also on the selected time resolution. The effect is observed due to the fact that the information flow between short-term and long-term traders is asymmetric. As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.
The risk parity weighted volatility of the three assets Gold, Treasury bonds and Nasdaq acting as proxy for the Marketportfolio seems to have a low point at 4% after turning upwards for the 8th time since 1974 at this reading in the summer of 2014.
Alternative measures of volatility

Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns. Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.Implied volatility parametrisation

There exist several known parametrisations of the implied volatility surface, Schonbucher, SVI and .http://www.readcube.com/articles/10.1002/wilm.10201?locale=enCrude volatility estimation

Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down. To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of ''n'' independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation ''σ'', the expected value of the magnitude of the observations is √(2/)''σ'' = 0.798''σ''. The net effect is that this crude approach underestimates the true volatility by about 20%.Estimate of compound annual growth rate (CAGR)

Consider theTaylor series
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 an ...

:
:$\backslash log(1+y)\; =\; y\; -\; \backslash tfracy^2\; +\; \backslash tfracy^3\; -\; \backslash tfracy^4\; +\; \backslash cdots$
Taking only the first two terms one has:
:$\backslash mathrm\; \backslash approx\; \backslash mathrm\; -\; \backslash tfrac\backslash sigma^2$
Volatility thus mathematically represents a drag on the CAGR (formalized as the "volatility tax
The volatility tax is a mathematical finance term, formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses (or Volatility (finance), volatility) on Compound interest, compound returns.\mathrm \approx \mathrm - \tfrack\sigma^2
for a rough estimate, where ''k'' is an empirical factor (typically five to ten).

Criticisms of volatility forecasting models

Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility especially out-of-sample, where different data are used to estimate the models and to test them. Other works have agreed, but claim critics failed to correctly implement the more complicated models. Some practitioners andportfolio managers
Portfolio may refer to:
Objects
* Portfolio (briefcase), a type of briefcase
Collections
* Portfolio (finance), a collection of assets held by an institution or a private individual
* Artist's portfolio, a sample of an artist's work or a ca ...

seem to completely ignore or dismiss volatility forecasting models. For example, Nassim Taleb
Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness
In co ...

famously titled one of his ''Journal of Portfolio Management
''The Journal of Portfolio Management'' (also known as JPM) is a quarterly academic journal covering asset allocation, performance measurement, market trends, risk management, and portfolio optimization. The journal was established in 1974 by Peter ...

'' papers "We Don't Quite Know What We are Talking About When We Talk About Volatility". In a similar note, Emanuel Derman
Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''.
He is a co-author of Black–Derman ...

expressed his disillusion with the enormous supply of empirical models unsupported by theory.Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press. He argues that, while "theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity", we should remember that "models are metaphors – analogies that describe one thing relative to another".
See also

*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 ...

* Dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variation ...

* Financial economics
Financial economics is the branch of economics
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among i ...

* IVX
IVX is a volatility index providing an intraday, VIX-like measure for any of US securities and exchange traded instruments. IVX is the abbreviation of Implied Volatility Index and is a popular measure of the implied volatilityIn financial math ...

* Jules RegnaultJules Augustin Frédéric Regnault (; 1 February 1834, Béthencourt – 9 December 1894, Paris
Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estima ...

* Risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty
Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to ...

* VIX
VIX is the ticker symbol
A ticker symbol or stock symbol is an abbreviation
An abbreviation (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was ...

*
* Volatility tax
The volatility tax is a mathematical finance term, formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses (or Volatility (finance), volatility) on Compound interest, compound returns.Graphical Comparison of Implied and Historical Volatility

video

* ttp://staff.science.uva.nl/~marvisse/volatility.html A short introduction to alternative mathematical concepts of volatility

Volatility estimation from predicted return density

Example based on Google daily return distribution using standard density function

Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility

Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005).

video

* ttp://staff.science.uva.nl/~marvisse/volatility.html A short introduction to alternative mathematical concepts of volatility

Volatility estimation from predicted return density

Example based on Google daily return distribution using standard density function

Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility

Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005).

Further reading

# {{Use dmy dates, date=August 2014 Mathematical finance Technical analysis