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

**Finance** is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

In statistics, the **standard deviation** is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

- Volatility terminology
- Mathematical definition
- Volatility origin
- Volatility for investors
- Volatility versus direction
- Volatility over time
- Alternative measures of volatility
- Implied volatility parametrisation
- Crude volatility estimation
- Estimate of compound annual growth rate (CAGR)
- Criticisms of volatility forecasting models
- Volatility hedge funds
- See also
- References
- External links
- Further reading

Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).

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.

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**, the square root of the realized variance, in turn calculated using the sum of squared returns divided by the number of observations.

- near synonymous is
**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)

In mathematics, a **square root** of a number *a* is a number *y* such that *y*^{2} = *a*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *a*. For example, 4 and −4 are square roots of 16 because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *a* has a unique nonnegative square root, called the *principal square root*, which is denoted by √*a*, where √ is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 3^{2} = 3 · 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this example 9.

**Realized variance** or **realised variance** is the sum of squared returns. For instance the RV can be the sum of squared daily returns for a particular month, which would yield a measure of price variation over this month. More commonly, the realized variance is computed as the sum of squared intraday returns for a particular day.

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.

Now turning to implied volatility, 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 random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability 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.

A **random walk** is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers. An elementary example of a random walk is the random walk on the integer number line, , which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality. As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology as well as economics. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of π can be approximated by the use of random walk in an agent-based modeling environment. The term *random walk* was first introduced by Karl Pearson in 1905.

In mathematics, the **Wiener process** is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard **Brownian motion process** or **Brownian motion** due to its historical connection with the physical process known as Brownian movement or Brownian motion originally observed by Robert Brown. It is one of the best known Lévy processes and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

**Probability** is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used.^{ [1] } These can capture attributes such as "fat tails". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.

In probability theory and statistics, the **Lévy distribution**, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a **van der Waals profile**. It is a special case of the inverse-gamma distribution. It is a stable distribution.

For any fund that evolves randomly with time, the square of volatility is the variance of the sum of infinitely many instantaneous rates of return, each taken over the nonoverlapping, infinitesimal periods that make up a single unit of time.

Thus, "annualized" volatility *σ*_{annually} is the standard deviation of an instrument's yearly logarithmic returns.^{ [2] }

The generalized volatility *σ*_{T} for time horizon *T* in years is expressed as:

Therefore, if the daily logarithmic returns of a stock have a standard deviation of *σ*_{daily} and the time period of returns is *P* in trading days, the annualized volatility is

A common assumption is that *P* = 252 trading days in any given year. Then, if *σ*_{daily} = 0.01, the annualized volatility is

The monthly volatility (i.e., *T* = 1/12 of a year or *P* = 252/12 = 21 trading days) would be

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, 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:

If *α* = 2 you get the Wiener process scaling relation, but some people believe *α* < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, 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.)

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 by market microstructure.^{ [3] } 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, they adjust their trading ranges, which in turn increases the band of price oscillation.^{ [4] }

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 the compound annual growth rate (CAGR) of that portfolio
- Volatility affects pricing of options, being a parameter of the Black–Scholes model.

In today's markets, it is also possible to trade volatility directly, through the use of derivative securities such as options and variance swaps. See Volatility arbitrage.

Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating standard deviation (or variance), 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 reality stocks are found to be leptokurtotic.

Although the Black Scholes equation assumes predictable constant volatility, this is not observed in real markets, and amongst the models are Emanuel Derman and Iraj Kani's^{ [5] } and Bruno Dupire's local volatility, Poisson process where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of stochastic volatility.^{ [6] }

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.^{ [7] }

Periods when prices fall quickly (a crash) 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) 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. 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.

The risk parity weighted volatility of the three assets Gold, Treasury bonds and Nasdaq (Worldvolatility.com) 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.worldvolatility.com

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 measure of volatility was suggested.^{ [8] } This measure is defined as the standard deviation of ensemble returns instead of time series of returns.

There exist several known parametrisation of the implied volatility surface, Schonbucher, SVI and gSVI.^{ [9] }

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/π)*σ* = 0.798*σ*. The net effect is that this crude approach underestimates the true volatility by about 20%.

Consider the Taylor series:

Taking only the first two terms one has:

Volatility thus mathematically represents a drag on the CAGR (formalized as the "volatility tax"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:

for a rough estimate, where *k* is an empirical factor (typically five to ten).

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 ^{ [10] }^{ [11] } especially out-of-sample, where different data are used to estimate the models and to test them.^{ [12] } Other works have agreed, but claim critics failed to correctly implement the more complicated models.^{ [13] } Some practitioners and portfolio managers seem to completely ignore or dismiss volatility forecasting models. For example, Nassim Taleb famously titled one of his * Journal of Portfolio Management * papers "We Don't Quite Know What We are Talking About When We Talk About Volatility".^{ [14] } In a similar note, Emanuel Derman expressed his disillusion with the enormous supply of empirical models unsupported by theory.^{ [15] } 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".

Well known hedge fund managers with expertise in trading volatility include Mark Spitznagel and Nassim Nicholas Taleb of Universa Investments, Paul Britton of Capstone Holdings Group,^{ [16] } Andrew Feldstein of Blue Mountain Capital Management,^{ [17] } and Nelson Saiers from Saiers Capital.^{ [18] }

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.

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. Collectively these have also been called the **risk sensitivities**, **risk measures** or **hedge parameters**.

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.

**Modern portfolio theory** (**MPT**), or **mean-variance analysis**, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk.

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

In finance, the **Sharpe ratio** is a way to examine the performance of an investment by adjusting for its risk. The ratio measures the excess return per unit of deviation in an investment asset or a trading strategy, typically referred to as risk, named after William F. Sharpe.

A **short-rate model**, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the **short rate**, usually written .

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 mathematical finance, the **Black–Derman–Toy model** (**BDT**) is a popular short rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) #Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the lognormal distribution, and is still widely used.

The **CBOE Volatility Index**, known by its ticker symbol **VIX**, is a popular measure of the stock market's expectation of volatility implied by S&P 500 index options. It is calculated and disseminated on a real-time basis by the Chicago Board Options Exchange (CBOE), and is commonly referred to as the *fear index* or the *fear gauge.*

In statistics, **stochastic volatility** models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.

In finance, the **Chen model** is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" as it describes interest rate movements as driven by three sources of market risk. It was the first stochastic mean and stochastic volatility model and it was published in 1994 by Lin Chen, economist, theoretical physicist and former lecturer/professor at Beijing Institute of Technology, Yonsei University of Korea, and Nanyang Tech University of Singapore.

In financial econometrics, the **Markov-switching multifractal (MSM)** is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations. MSM captures the outliers, log-memory-like volatility persistence and power variation of financial returns. In currency and equity series, MSM compares favorably with standard volatility models such as GARCH(1,1) and FIGARCH both in- and out-of-sample. MSM is used by practitioners in the financial industry to forecast volatility, compute value-at-risk, and price derivatives.

**Forward volatility** is a measure of the implied volatility of a financial instrument over a period in the future, extracted from the term structure of volatility.

**Modigliani risk-adjusted performance** (also known as **M ^{2}**,

**Financial correlations** measure the relationship between the changes of two or more financial variables over time. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.

The **volatility tax** is a mathematical finance term, formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses on compound returns. It has also been called “volatility drag”.

- ↑ "Levy distribution".
*wilmottwiki.com*. - ↑ "Calculating Historical Volatility: Step-by-Step Example" (PDF). 14 July 2011. Retrieved 15 July 2011.
- ↑ Roll, R. (1984): "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market",
*Journal of Finance***39**(4), 1127–1139 - ↑ Glosten, L. R. and P. R. Milgrom (1985): "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders",
*Journal of Financial Economics***14**(1), 71–100 - ↑ Derman, E., Iraj Kani (1994). ""Riding on a Smile." RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39" (PDF). Risk. Retrieved 2007-06-01.CS1 maint: Multiple names: authors list (link)
- ↑ "Volatility".
*wilmottwiki.com*. - ↑ "Taking Advantage Of Volatility Spikes With Credit Spreads".
- ↑ Sarkissian, Jack (2016). "Express Measurement of Market Volatility Using Ergodicity Concept". SSRN 2812353 .
- ↑ Babak Mahdavi Damghani & Andrew Kos (2013). "De-arbitraging with a weak smile". Wilmott. http://www.readcube.com/articles/10.1002/wilm.10201?locale=en
- ↑ Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). "Forecasting Volatility and Correlations with EGARCH models".
*Journal of Derivatives*.**1**(2): 51–63. doi:10.3905/jod.1993.407877. - ↑ Jorion, P. (1995). "Predicting Volatility in Foreign Exchange Market".
*Journal of Finance*.**50**(2): 507–528. doi:10.1111/j.1540-6261.1995.tb04793.x. JSTOR 2329417. - ↑ Brooks, Chris; Persand, Gita (2003). "Volatility forecasting for risk management".
*Journal of Forecasting*.**22**(1): 1–22. CiteSeerX 10.1.1.595.9113 . doi:10.1002/for.841. ISSN 1099-131X. - ↑ Andersen, Torben G.; Bollerslev, Tim (1998). "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts".
*International Economic Review*.**39**(4): 885–905. CiteSeerX 10.1.1.28.454 . doi:10.2307/2527343. JSTOR 2527343. - ↑ Goldstein, Daniel and Taleb, Nassim, (28 March 2007) "We Don't Quite Know What We are Talking About When We Talk About Volatility".
*Journal of Portfolio Management***33**(4), 2007. - ↑ Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press.
- ↑ Devasabai, Kris (1 March 2010). "Interview with Paul Britton Founder CEO of Capstone".
*Hedge Funds Review*. Retrieved 26 April 2013. - ↑ Schaefer, Steve (14 February 2013). "Blue Mountain's Andrew Feldstein: Three Ways to Play a More Volatile Steel Industry".
*Forbes*. Retrieved 26 April 2013. - ↑ Creswell, Julie and Louise Story (17 March 2011). "Funds Find Opportunities in Volatility".
*New York Times*. Retrieved 26 April 2013.

- Graphical Comparison of Implied and Historical Volatility, video
- Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) "Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think"
- 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).

- Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). "Why Are U.S. Stocks More Volatile?".
*Journal of Finance*.**67**(4): 1329–1370. doi:10.1111/j.1540-6261.2012.01749.x. SSRN 2257549 .

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