- Implied volatility
financial mathematics, the implied volatility of an option contract is the volatility implied by the market priceof 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.
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:
where is the theoretical value of an option, and is a pricing model that depends on plus other inputs.
The function is monotonically increasing in , 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 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 volatility implied by the market price , or the implied volatility.
call optioncontract, , 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 18.7%, or:
To verify, we apply the implied volatility back into the pricing model, 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, , does not have a closed-form solution for its inverse, . Instead, a
root findingtechnique is used to solve the equation:
While there are many techniques for finding roots, two of the most commonly used are
Newton's methodand 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. , 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-Scholesmodel, 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 neutralportfolio, 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.
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 volatilityand volatility smilefor more information.
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.
* [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
Wikimedia Foundation. 2010.
См. также в других словарях:
implied volatility — Volatility of a financial instrument that is imputed by subtracting all of the other factors thought to contribute to the price of an option. The amount remaining after those subtractions is attributed to volatility. Implied volatility is not the … Financial and business terms
Implied Volatility - IV — The estimated volatility of a security s price. In general, implied volatility increases when the market is bearish and decreases when the market is bullish. This is due to the common belief that bearish markets are more risky than bullish… … Investment dictionary
Implied volatility — The expected volatility in a stock s return derived from its option price, maturity date, exercise price, and riskless rate of return, using an option pricing model such as Black/Scholes. The New York Times Financial Glossary * * * The… … Financial and business terms
IMPLIED VOLATILITY — Прогноз рынка будущих непостоянных уровней … Малая энциклопедия трейдера: глоссарий к книге
Implied Volatility — Not yet defined … International financial encyclopaedia
Volatility arbitrage — (or vol arb) is a type of statistical arbitrage that is implemented by trading a delta neutral portfolio of an option and its underlier. The objective is to take advantage of differences between the implied volatility of the option, and a… … Wikipedia
Volatility (finance) — Volatility most frequently refers to the standard deviation of the continuously compounded returns of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility… … Wikipedia
Volatility smile — In finance, the volatility smile is a long observed pattern in which at the money options tend to have lower implied volatilities than in or out of the money options. The pattern displays different characteristics for different markets and… … Wikipedia
volatility — A measurement of the change in price over a given period. It is often expressed as a percentage and computed as the annualized standard deviation of the percentage change in daily price. Chicago Board of Trade glossary The rate of change in a… … Financial and business terms
Volatility Arbitrage — Trading strategies that attempt to exploit differences between the forecasted future volatility of an asset and the implied volatility of options based on that asset. Because options pricing is determined by the volatility of the underlying asset … Investment dictionary