Implied volatility
Encyclopedia
In financial mathematics, the implied volatility of an option
contract is the volatility
of the price of the underlying security
that is implied by the market price
of the option based on an option pricing
model. In other words, it is the volatility that, when used in a particular pricing model, yields a theoretical value for the option equal to the current market price of that option. 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 returns of a security.
. 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 σ, 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 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. 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.
, on 100 shares of non-dividend-paying XYZ Corp. The option 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, f and we generate a theoretical value of $2.0004:
which confirms our computation of the market implied volatility.
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.,
, 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 for vega numerically, it usually turns out that Brent's method is more efficient as a root-finding technique.
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 between professional traders.
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.
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.
and volatility smile
for more information.
Volatility Index (VIX
) is calculated from a weighted average of implied volatilities of various options on the S&P 500 Index. There are also other commonly referenced volatility indices such as the VXN index (Nasdaq 100 index futures volatility measure), the QQV (QQQ volatility measure), IVX
- 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.
Option (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
contract is the volatility
Volatility (finance)
In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
of the price of the underlying security
Security (finance)
A security is generally a fungible, negotiable financial instrument representing financial value. Securities are broadly categorized into:* debt securities ,* equity securities, e.g., common stocks; and,...
that is implied by the market price
Market price
In economics, market price is the economic price for which a good or service is offered in the marketplace. It is of interest mainly in the study of microeconomics...
of the option based on an option pricing
Valuation of options
In finance, a price is paid or received for purchasing or selling options. This price can be split into two components.These are:* Intrinsic Value* Time Value-Intrinsic Value:...
model. In other words, it is the volatility that, when used in a particular pricing model, yields a theoretical value for the option equal to the current market price of that option. 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 returns of a security.
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 underlyingUnderlying
In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this 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 σ, meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, by the inverse function theorem
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
, 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. 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.
Example
A European call option,, on 100 shares of non-dividend-paying XYZ Corp. The option is struck at $50 and expires in 32 days. The risk-free interest rateRisk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....
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, f and we 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 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
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
and Brent's method
Brent's method
In numerical analysis, Brent's method is a complicated but popular 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 methods...
. 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, then Newton's method can be more efficient. However, for most practical pricing models, such as a binomial modelBinomial options pricing model
In finance, the binomial options pricing model provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979. Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying...
, this is not the case and vega must be derived numerically. When forced to solve for 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. 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 neutralDelta neutral
In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged due to small changes in the value of the underlying security...
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 between professional traders.
Example
A call option is trading at $1.50 with the underlyingUnderlying
In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...
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.
Implied volatility 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.
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 is generally viewed as evidence that an underlying's volatility is not constant, but, instead depends on factors such as the price level of the underlying, the underlying's recent price variance, and the passage of time. See stochastic volatilityStochastic volatility
Stochastic volatility models are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the...
and volatility smile
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 results from the probability of extreme moves...
for more information.
Volatility instruments
Volatility instruments are financial instruments that track the value of implied volatility of other derivative securities. For instance, the CBOEChicago Board Options Exchange
The Chicago Board Options Exchange , located at 400 South LaSalle Street in Chicago, is the largest U.S. options exchange with annual trading volume that hovered around one billion contracts at the end of 2007...
Volatility Index (VIX
VIX
VIX is the ticker symbol for the Chicago Board Options Exchange Market Volatility Index, a popular measure of the implied volatility of S&P 500 index options. Often referred to as the fear index or the fear gauge, it represents one measure of the market's expectation of stock market volatility over...
) is calculated from a weighted average of implied volatilities of various options on the S&P 500 Index. There are also other commonly referenced volatility indices such as the VXN index (Nasdaq 100 index futures volatility measure), the QQV (QQQ volatility measure), IVX
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 individual stock...
- 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.
Computer implementations
- Real-time calculator of implied volatilities when the underlying follows a Mean-Reverting Geometric Brownian Motion, by Razvan Pascalau, Univ. of Alabama
- Test online implied volatility calculation by Christophe Rougeaux, ESILV
External links
- Interactive Java Applet "Implied Volatility vs. Historic Volatility"
- Implied Volatility Explanation, January 13, 2007
- 2 Real Trade Examples