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# Implied volatility

In financial mathematics, the implied volatility of some derivative security is the value of the volatility parameter that must be put into Black formula to give the same price as the quote price of the instrument. Here the right Black formula to use is dependent on the exact security in question. For instruments where the underlying spot price is being modelled as log-normal[?] the Black-Scholes formula is used. For instruments where the forward price is assumed to be log-normally distributed, the Black-76 model is used.

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 2y into 1year 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:

$691.60 = 10,000*0.9(0.045*N(d_1) - 0.04*N(d_2))$
where
$d_1 = \frac{log(\frac{0.045}{0.04}) + \sigma^2}{\sigma\sqrt 2}$
$d_2 = d_1 - \sigma\sqrt 2$
$\sigma$ is the implied volatilty of the forward rate
and $N(.)$ 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 $\sigma$ 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%.

Interestingly the volatility implied by options in the market rarely corresponds to the historical volatility (i.e. the actual volatility of a forward rate experienced over past times). This shows that Black model is not a perfect model of reality.

By computing the volatility for various different strikes on a particular underlying we obtain the volatility smile[?]. The implied volatility is typically significantly higher for in-the-money options, i.e. options with strikes lower than the current forward rate, than at-the-money options. Out-the-money options also typically have a somewhat higher implied volatility than at the money. The reasons for this artefact of the market is still the subject of active academic research.

As the knowing the price of an option is, once the precise Black-type formula to use has been specified, equivalent to knowing the implied volatility practioners prefer to give quotes in terms of implied volatility as this value is constant with respect to the notional of the option (the price is proportional to the notional). This practice is common place in major options markets.

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