# Implied volatility

(Redirected from Volatility smile)

In financial mathematics, the implied volatility of a financial instrument is the volatility implied by the market price of a derivative security based on a theoretical pricing model. For instruments with log-normal prices, the Black-Scholes formula or Black-76 model is used.

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## Examples

Example: suppose the price of a £10,000 notional interest rate cap struck at 4% on the 1 year GBP LIBOR rate maturity 2 years from now is £691.60. (The 1 year LIBOR rate is an annualised interest rate for borrowing money for three months). Suppose also that the forward rate for 2 year into 1 year LIBOR is 4.5% and the current discount factor for value of money in two years' time is 0.9, then the Black formula shows that:

[itex] 691.60 = 10,000*0.9(0.045*N(d_1) - 0.04*N(d_2)) [itex],

where

[itex]d_1 = \frac{log(\frac{0.045}{0.04}) + \sigma^2}{\sigma\sqrt 2}[itex],
[itex]d_2 = d_1 - \sigma\sqrt 2[itex],

where

[itex]\sigma[itex] (pronounced "sigma") is the implied volatility of the forward rate and [itex]N(.)[itex] is the standard cumulative Normal distribution function.

Note that the only unknown is the volatility. We can not invert the function analytically (see inverse function) however we can find the unique value of [itex]\sigma[itex] that makes the equation above hold by using a root finding algorithm such as Newton's method. In this example the implied volatility is 0.2 or 20%.

## Observations

Interestingly the implied volatility of options rarely corresponds to the historical volatility (i.e. the volatility of a historical time series). This is because implied volatility includes future expectations of price movement, which are not reflected in historical volatility.

## Expanded

By computing the volatility for all strikes on a particular underlying we obtain the Volatility Smile.

## 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.

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