Volatility Models: Implied volatility (IV)
The implied volatility is the attempt to estimate FV on the basis of Black-Shcoles model. It can be made if option is a traded asset.
While the BS equation was originally intended to be used in calculat-ing derivative prices from fixed volatility estimates, it can also be
used in the opposite way to calculate volatilities from actual derivative prices in the market. If we substitute in the BS formula option
mar-ket price  and other parameters we get the nonlinear equation for volatility  :
(2.5)
Provided that the market value of the option is  there is one value for s that
makes the theoretical option value and the market price the same. The solution is simply enough to find, for example, by bisection
method. A problem is that more than a few assumptions incorporated in the Black-Scholes model are not carried out in practice. That
shows out some “holes” (using ex-pression of F. Black) in the model. For these are experimentally established fact that:
the distributions of the assets returns have nonzero third and fourth moments;
the value of IV changes with time to maturity;
the value of IV changes with strike (smile effect).
Volatility smile reflects non-lognormal distribution of the returns. Two numbers: mean and volatility parameterise lognormal distribution.
The volatility smile is a payment for this simplicity. It is neces-sary to correct all distribution for everyone strike. Naturally, the second
moment of underlying distri-bution should not change with the strike. And it is so actually in the stochastic volatility models with the
nonzero 3rd and 4th moments. In other words IV is the volatility, substituted in the wrong formula to obtain the actual option price asked
by financial markets. But IV is the decisive factor for the option trade in spite of the fact that BS model is inaccurate.
Figure 1: the performance of the DAX index (right scale), annualized 60-day historical volatility based on the DAX close daily data (left
scale) and VDAX implied volatility index (left scale) over the repre-sentative ten-years period.
|
