Implied volatility
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.
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(d1) - 0.04 * N(d2)),
where
,
,
where σ 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 σ 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 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.
By computing the volatility for all strikes on a particular underlying we obtain the volatility smile. The implied volatility is typically significantly higher for in-the-money stock and interest-rate call options, i.e. call options with strikes lower than the current forward rate, than at-the-money options. Out-of-the-money call options on stocks and interest rates typically have a higher implied volatility than at-the-money.
For stock options, the increased volatility for in-the-money call options is due to financial distress factors; there is a greater likelyhood of bankruptcy when the stock price gets lower. The increased volatility for out-of-the-money calls not well understood.
Categories: Financial mathematics