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In financial mathematics, the implied volatility of an option contract is the volatility implied by the market price of the option based on an option pricing model. In other words, it is the volatility that, given a particular pricing model, yields a theoretical value for the option equal to the current market price. Non-option financial instruments that have embedded optionality, such as an interest rate cap, can also have an implied volatility. Mathematical finance is the branch of applied mathematics concerned with the financial markets. ...
An option contract is an agreement in which the buyer (holder) has the right (but not the obligation) to exercise by buying or selling an asset at a set price (strike price) on (European style option) or before (American style option) a future date (the exercise date or expiration); and...
Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. ...
Market price is an economic concept with commonplace familiarity; it is the price that a good or service is offered at, or will fetch, in the marketplace; it is of interest mainly in the study of microeconomics. ...
Option contracts are complex to value. ...
Interest rate cap An interest rate cap is a series of European call options or caplets on a specified interest rate, usually the LIBOR interest rate. ...
Motivation
An ordinary option pricing model, such as Black-Scholes, uses a variety of inputs to derive a theoretical value for an option. These inputs may 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 volatility,  , of the underlying. Or, mathematically: The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...
In finance, an underlying is an investment from which a derivative security is derived. ...
 where  is the theoretical value of an option, and  is a pricing model that depends on plus other inputs.
The function is a monotonically increasing function for , meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, this also means that there can be at most one value for , that, when applied as an input to  , will result in a particular value for .
Put in other terms, assume that there is some inverse function  , such that
 where  is the market price for an option. The value  is the volatitily implied by the market price , or the implied volatility.
Example A standard call option contract,  , on 100 shares of non-dividend-paying XYZ Corp. stock is struck at $50 and expires in 32 days. The risk-free interest rate 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 20.6%, or: A call option is a financial contract between two parties, the buyer and the seller of this type of option. ...
 To verify, we apply the implied volatility back into the pricing model, and we generate a theoretical value of $2.002:  which confirms our computation of the market implied volatility.
Solving the inverse pricing model function In general, a pricing model function, , does not have a closed-form solution for its inverse,  . Instead, a root finding technique is used to solve the equation: A root-finding algorithm is a numerical method or algorithm for finding a value x such that f(x) = 0, for a given function f. ...
 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. In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...
In numerical analysis, Brents method is a complicated but popular root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. ...
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, 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 vega numerically, it usually turns out that Brent's method is more efficient as a root-finding technique. In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives. ...
The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...
In finance, the binomial options pricing model provides a generalisable numerical method for the valuation of options. ...
Implied volatility as measure of relative value Often, the implied volatility of an option is a more useful measure of the option's relative value than its price. This is because the price of an option depends most directly on the price of its underlying security. If an option is held as part of a delta neutral portfolio, that is, a portfolio that is hedged against small moves in the underlier's price, then the next most important factor in determining the value of the option will be its implied volatility. Delta neutral refers to a portfolio containing options, that has been hedged so that its overall value will not change for small changes in the underliers price. ...
Implied volatility is so important that options are often quoted in terms of volatility rather than price, particularly between professional traders.
Example A call option is trading at $1.50 with the underlier 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 underlier 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 on a volatility basis. This is because the underlier needed to hedge the call option can be sold for a higher price.
Non-constant implied volatility In general, options based on the same underlier but with different strike value and expiration times will yield different implied volatilities. This is generally viewed as evidence that an underlier's volatility is not constant, but, instead depends on factors such as the price level of the underlier, the underlier's recent variance, and the passage of time. See stochastic volatility and volatility smile for more information. Stochastic volatility models are used in the field of quantitative finance to evaluate derivative securities, such as options. ...
Volatility Smile refers to the long-observed pattern in which at-the-money options tend to have lower implied volatilities than other options. ...
Chicago Board Options Exchange Volatility Index The Chicago Board Options Exchange Volatility Index (VIX) measures how much premium investors are willing to pay for options as insurance to hedge their equity positions. The VIX is calculated using a weighted average of implied volatility of At-The-Money and Near-The-Money striked in options on the S&P 500 Index futures. The Chicago Board Options Exchange (CBOE), located at 400 South LaSalle Street in Chicago, is one of the worlds largest options exchanges with an annual trade of over 15 billion shares of stock options in more than 1200 companies, 50 stock indexes, and 50 exchange-traded funds (ETFs) [citation...
VIX Index from inception to Jan. ...
There also exists the VXN index (Nasdaq 100 index futures volatility measure) and QQV (QQQQ volatility measure).
Computer implementations - Real-time calculator of implied volatilities when the underlying follows a Mean-Reverting Geometric Brownian Motion, by Razvan Pascalau, Univ. of Alabama
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