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In
financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. See Quantitative analyst. In general, there exist two separate branch ...
, the implied volatility (IV) 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 ...
contract is that value of the volatility of the
underlying In finance, the underlying of a derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equal to the current market price of said option. A non-option
financial instrument Financial instruments are monetary contracts A contract is a legally binding document between at least two parties that defines and governs the rights and duties of the parties to an agreement. A contract is legally enforceable because it me ...
that has embedded optionality, such as an
interest rate capAn interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment fo ...
, 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 Security is freedom from, or resilience against, potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics Physics is the natural science that studies matter, its El ...
. 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.

# Motivation

An option pricing model, such as Black–Scholes, uses a variety of inputs to derive a theoretical value for an option. Inputs to pricing models vary depending on the type of option being priced and the pricing model used. However, in general, the value of an option depends on an estimate of the future realized price volatility, σ, of the underlying. Or, mathematically: :$C = f\left(\sigma, \cdot\right) \,$ where ''C'' is the theoretical value of an option, and ''f'' is a pricing model that depends on σ, along with other inputs. The function ''f'' is monotonically increasing in σ, meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, by the
inverse function theorem 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 and th ...
, there can be at most one value for σ that, when applied as an input to $f\left(\sigma, \cdot\right) \,$, will result in a particular value for ''C''. Put in other terms, assume that there is some inverse function ''g'' = ''f''−1, such that :$\sigma_\bar = g\left(\bar, \cdot\right) \,$ where $\scriptstyle \bar \,$ is the market price for an option. The value $\sigma_\bar \,$ is the volatility implied by the market price $\scriptstyle \bar \,$, or the implied volatility. In general, it is not possible to give a closed form formula for implied volatility in terms of call price. However, in some cases (large strike, low strike, short expiry, large expiry) it is possible to give an
asymptotic expansionIn 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). It ha ...
of implied volatility in terms of call price.

## Example

A
European call optionIn finance, the style or family of an option (finance), option is the class into which the option falls, usually defined by the dates on which the option may be Exercise (options), exercised. The vast majority of options are either European or Ameri ...
, $C_$, on one share of non-dividend-paying XYZ Corp with a strike price of $50 expires in 32 days. The risk-free interest rate The risk-free interest rate is the rate of return 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 wit ... is 5%. XYZ stock is currently trading at$51.25 and the current market price of $C_$ is $2.00. Using a standard Black–Scholes pricing model, the volatility implied by the market price $C_$ is 18.7%, or: :$\sigma_\bar = g\left(\bar, \cdot\right) = 18.7\%$ To verify, we apply implied volatility to the pricing model, ''f ,'' and generate a theoretical value of$2.0004: :$C_ = f\left(\sigma_\bar, \cdot\right) = \2.0004$ which confirms our computation of the market implied volatility.

# Solving the inverse pricing model function

In general, a pricing model function, ''f'', does not have a closed-form solution for its inverse, ''g''. Instead, a
root finding In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often lie bel ...
technique is often used to solve the equation: :$f\left(\sigma_\bar, \cdot\right) - \bar = 0 \,$ While there are many techniques for finding roots, two of the most commonly used are
Newton's method In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

and
Brent's method In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable m ...
. Because options prices can move very quickly, it is often important to use the most efficient method when calculating implied volatilities. Newton's method provides rapid convergence; however, it requires the first partial derivative of the option's theoretical value with respect to volatility; i.e., $\frac \,$, which is also known as ''vega'' (see The Greeks). If the pricing model function yields a closed-form solution for ''vega'', which is the case for
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 ...
, then Newton's method can be more efficient. However, for most practical pricing models, such as a binomial model, this is not the case and ''vega'' must be derived numerically. When forced to solve for ''vega'' numerically, one can use the Christopher and Salkin method or, for more accurate calculation of out-of-the-money implied volatilities, one can use the Corrado-Miller model. Specifically in the case of the Black Scholes-Mertonmodel, Jaeckel's "Let's Be Rational" method computes the implied volatility to full attainable (standard 64 bit floating point) machine precision for all possible input values in sub-microsecond time. The algorithm comprises an initial guess based on matched asymptotic expansions, plus (always exactly) two Householder improvement steps (of convergence order 4), making this a three-step (i.e., non-iterative) procedure. A reference implementation in C++ is freely available. Besides the above mentioned
root finding In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often lie bel ...
techniques, there are also methods that approximate the multivariate
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
directly. Often they are based on
polynomials 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). I ...
or
rational functions 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). I ...
. For the Bachelier ("normal", as opposed to "lognormal") model, Jaeckel published a fully analytic and comparatively simple two-stage formula that gives full attainable (standard 64 bit floating point) machine precision for all possible input values.

# Implied volatility parametrisation

With the arrival of
Big Data Big data is a field that treats ways to analyze, systematically extract information from, or otherwise deal with data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of informati ...

and
Data Science #REDIRECT Data science#REDIRECT Data science Data science is an Interdisciplinarity, interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract knowledge and insights from structured and unstructured data ...

parametrising the implied volatility has taken central importance for the sake of coherent interpolation and extrapolation purposes. The classic models are the
SABR Sabr ( ar, صَبْرٌ, ṣabr) (literally 'endurance' or more accurately 'perseverance' and 'persistence'"Ṣabr", ''Encyclopaedia of Islam'') is one of the two parts of (the other being ') in . It teaches to remain steadfast and to keep doi ...
and model with their IVP extension.

# Implied volatility as measure of relative value

As stated by Brian Byrne, the implied volatility of an option is a more useful measure of the option's relative value than its price. The reason is that the price of an option depends most directly on the price of its underlying asset. If an option is held as part of a delta neutral portfolio (that is, a portfolio that is hedged against small moves in the underlying's price), then the next most important factor in determining the value of the option will be its implied volatility. Implied volatility is so important that options are often quoted in terms of volatility rather than price, particularly among professional traders.

## Example

A call option is trading at $1.50 with the underlying In finance, the underlying of a derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ... trading at$42.05. The implied volatility of the option is determined to be 18.0%. A short time later, the option is trading at $2.10 with the underlying at$43.34, yielding an implied volatility of 17.2%. Even though the option's price is higher at the second measurement, it is still considered cheaper based on volatility. The reason is that the underlying needed to hedge the call option can be sold for a higher price.

# As a price

Another way to look at implied volatility is to think of it as a price, not as a measure of future stock moves. In this view, it simply is a more convenient way to communicate option prices than currency. Prices are different in nature from statistical quantities: one can estimate volatility of future underlying returns using any of a large number of estimation methods; however, the number one gets is not a price. A price requires two counterparties, a buyer, and a seller. Prices are determined by supply and demand. Statistical estimates depend on the time-series and the mathematical structure of the model used. It is a mistake to confuse a price, which implies a transaction, with the result of a statistical estimation, which is merely what comes out of a calculation. Implied volatilities are prices: they have been derived from actual transactions. Seen in this light, it should not be surprising that implied volatilities might not conform to what a particular statistical model would predict. However, the above view ignores the fact that the values of implied volatilities depend on the model used to calculate them: different models applied to the same market option prices will produce different implied volatilities. Thus, if one adopts this view of implied volatility as a price, then one also has to concede that there is no unique implied-volatility-price and that a buyer and a seller in the same transaction might be trading at different "prices".

# Non-constant implied volatility

In general, options based on the same underlying but with different strike values and expiration times will yield different implied volatilities. This can be viewed as evidence that an underlying's volatility is not constant but instead depends on factors such as price level or time, or it can be viewed as evidence that the underlying's price changes do not follow the distribution that is assumed in the model under consideration (such as Black-Scholes). There exist few known parametrisation of the volatility surface (Schonbusher, SVI, and gSVI) as well as their de-arbitraging methodologies. See
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 ...

# Volatility instruments

Volatility instruments are financial instruments that track the value of implied volatility of other derivative securities. For instance, the
CBOE The Chicago Board Options Exchange (CBOE), located at 433 West Van Buren Street File:Bucuresti, Romania. Strada in Bucuresti.jpg, 250px, Street in downtown Bucharest (Romania) A street is a public thoroughfare in a built environment. It is a p ...
Volatility Index (
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 ...

) is calculated from a weighted average of implied volatilities of various options on the
S&P 500 Index The Standard and Poor's 500, or simply the S&P 500, is a stock market index In finance, a stock index, or stock market index, is an Index (economics), index that measures a stock market, or a subset of the stock market, that helps invest ...
. There are also other commonly referenced volatility indices such as the VXN index (
Nasdaq The Nasdaq Stock Market () is an American stock exchange A stock exchange, securities exchange, or bourse is an exchange Exchange may refer to: Places United States * Exchange, Indiana Exchange is an Unincorporated area, unincorpora ...
100 index futures volatility measure), the QQV (QQQ volatility measure),
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 ...
- Implied Volatility Index (an expected stock volatility over a future period for any of US securities and exchange-traded instruments), as well as options and futures derivatives based directly on these volatility indices themselves.

* Forward volatility

* * * * *