Overview

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 (usually Black–Scholes), will return a theoretical value equal to the 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.

Implied volatility is also used as a predictor in models that forecast future realized volatility. 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

where C is the theoretical value of an option, and f is a pricing model that depends on σ, along with other inputs.

Put in other terms, assume that there is some inverse function g = f^(−1), such that

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 of implied volatility in terms of call price. A different approach based on closed form approximations has been also investigated.

Solving the inverse pricing model function

While there are many techniques for finding roots, two of the most commonly used are Newton's method and Brent's method. Because options prices can move very quickly, it is often important to use the most efficient method when calculating implied volatilities.

Specifically in the case of the Black[-Scholes-Merton] model, 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 techniques, there are also methods that approximate the multivariate inverse function directly. Often they are based on polynomials or rational functions.

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 and data science, parametrising the implied volatility has taken central importance for the sake of coherent interpolation and extrapolation purposes. The classic models are the SABR 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 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.