# Implied volatility

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Implied volatility

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. Implied volatility, a forward-looking measure, differs from historical volatility because the latter is calculated from known past prices of a security.

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, $sigma ,$, of the underlying. Or, mathematically:

:$C = f\left(sigma, cdot\right) ,$

where $C ,$ is the theoretical value of an option, and $f\left(\right) ,$ is a pricing model that depends on $sigma ,$ plus other inputs.

The function $f$ is monotonically increasing in $sigma ,$, meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, by the inverse function theorem, there can be at most one value for $sigma ,$ 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 $g\left(\right) = f^\left\{-1\right\}\left(\right),$, such that

:

where is the market price for an option. The value is the volatility implied by the market price , or the implied volatility.

Example

A standard call option contract, $C_\left\{XYZ\right\} ,$, 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 $C_\left\{XYZ\right\} ,$ is \$2.00. Using a standard Black-Scholes pricing model, the volatility implied by the market price $C_\left\{XYZ\right\} ,$ is 18.7%, or:

:

To verify, we apply the implied volatility back into the pricing model, $f\left(\right) ,$ and we generate a theoretical value of \$2.0004:

:

which confirms our computation of the market implied volatility.

olving the inverse pricing model function

In general, a pricing model function, $f\left(\right) ,$, does not have a closed-form solution for its inverse, $g\left(\right) ,$. Instead, a root finding technique is used to solve the equation:

:

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.

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. $frac\left\{partial C\right\}\left\{partial sigma\right\} ,$, 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.

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.

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.

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) is calculated from a weighted average of implied volatilities of various options on the S&P 500 Index futures. There also exists the VXN index (Nasdaq 100 index futures volatility measure) and QQV (QQQQ volatility measure), as well as options and futures derivatives based directly on these volatility indices themselves.

Computer implementations

* [http://www.cba.ua.edu/~rpascala/impliedvol/BSOPMSForm.php 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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