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.
In general, there exist two separate branches of finance that require ...
, the implied volatility (IV) of an
option contract is that value of the
volatility of the
underlying
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 use ...
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 between parties. They can be created, traded, modified and settled. They can be cash (currency), evidence of an ownership interest in an entity or a contractual right to receive or deliver in the form ...
that has embedded optionality, such as an
interest rate cap An 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 for ...
, 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 protection from, or resilience against, potential harm (or other unwanted coercive change) caused by others, by restraining the freedom of others to act. Beneficiaries (technically referents) of security may be of persons and social ...
. 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:
:
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 mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
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, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
, there can be at most one value for σ that, when applied as an input to
, will result in a particular value for ''C''.
Put in other terms, assume that there is some inverse function ''g'' = ''f''
−1, such that
:
where
is the market price for an option. The value
is the volatility implied by the market price
, or the implied volatility.
In general, it is not possible to give a closed form formula for implied volatility in terms of call price (for a review see ). However, in some cases (large strike, low strike, short expiry, large expiry) it is possible to give an
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
of implied volatility in terms of call price. A different approach based on closed form approximations has been also investigated.
Example
A
European call option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These optionsâ ...
,
, 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 rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations.
Since the risk-free ra ...
is 5%. XYZ stock is currently trading at $51.25 and the current market price of
is $2.00. Using a standard Black–Scholes pricing model, the volatility implied by the market price
is 18.7%, or:
:
To verify, we apply implied volatility to the pricing model, ''f ,'' and generate a theoretical value of $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 mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbe ...
technique is often used to solve the equation:
:
While there are many techniques for finding roots, two of the most commonly used are
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
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 ...
. 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.,
, 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 investment instruments. From the parabolic partial differential equation in the model, known as the Blac ...
, then Newton's method can be more efficient. However, for most practical pricing models, such as a
binomial model
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
, 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 mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbe ...
techniques, there are also methods that approximate the multivariate
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
directly. Often they are based on
polynomials
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
or
rational functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rati ...
.
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
Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe Big data is the one associated with large body of information that we could not comprehend when used only in smaller am ...
and
Data Science
Data science is an interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract or extrapolate knowledge and insights from noisy, structured and unstructured data, and apply knowledge from data across a br ...
parametrising the implied volatility has taken central importance for the sake of coherent interpolation and extrapolation purposes. The classic models are the
SABR
The Society for American Baseball Research (SABR) is a membership organization dedicated to fostering the research and dissemination of the history and record of baseball primarily through the use of statistics. Established in Cooperstown, New ...
and
SVI 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 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 c ...
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, 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 use ...
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 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 ...
and
volatility smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
for more information.
Volatility instruments
Volatility instruments are financial instruments that track the value of implied volatility of other derivative securities. For instance, the
CBOE Volatility Index (
VIX
VIX is the ticker symbol and the popular name for the Chicago Board Options Exchange's CBOE Volatility Index, a popular measure of the stock market's expectation of volatility based on S&P 500 index options. It is calculated and disseminated ...
) 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 tracking the stock performance of 500 large companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices. As of D ...
. There are also other commonly referenced volatility indices such as the VXN index (
Nasdaq
The Nasdaq Stock Market () (National Association of Securities Dealers Automated Quotations Stock Market) is an American stock exchange based in New York City. It is the most active stock trading venue in the US by volume, and ranked second ...
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 volatility of each individ ...
- 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.
See also
*
Forward volatility 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 (which refers to how implied volatility differs for related financial instruments with ...
References
Further reading
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External links
Implied volatility calculationby Serdar SEN
Calculate Beta in Excel
{{Volatility
Derivatives (finance)
Mathematical finance