Implied volatilities

The parameters of the BS formula are all considered to be $\mathscr{F}\left( 0\right)$-measurable by assumption. In reality though, although the current price, the strike price, the maturity and the interest rate are observed at time $t=0$, the volatility $\sigma$ of the asset price is not. In addition, one can also observe the actual market call price $C$ of this particular contract. This allows one to invert numerically the BS formula and construct a series of implied volatilities $\left\{ \hat{\sigma}\right\} _{t,K}$ across time and different strike prices. Bajeux and Rochet show that there is a one-to-one relationship between implied volatilities and option prices, therefore formalizing the use of option markets as instruments to hedge volatility risk. Models that incorporate stochastic volatilities, like Hull and White [HW] (1989), are sound candidates of utilizing this feature, giving a theoretical background of interpreting the implied volatility as the expectation of the average future volatility over the life of the option. This is explained below:

If the future volatility is stochastic but independent of the stock price and with zero price of risk, one can derive the HW option pricing formula, just a modification of the BS formula where $S=S\left( 0\right)$ and $%% \sigma =\sigma \left( 0\right)$

$$C_{HW}\left( S,K,T,r,\sigma \right) =\mathbf{E}C_{BS}\left( S,K,T,r,\gamma \right)$$   ,

where the random variable

$$\gamma ^{2}=\frac{1}{T}\int_{0}^{T}\sigma ^{2}\left( s\right) ds$$

is the average volatility over the life of the option, and the expectation is taken with respect to this random variable.

For instance if one considers an at-the-money [ATM] option, where $%% K_{ATM}=Se^{rT}$, the HW formula will give $d_{2}=-d_{1}$, and

$$C_{HW}\left( S,K_{ATM},T,r,\sigma \right) =\mathbf{E}S\left[ 2\Phi \left( \frac{\gamma }{2}\sqrt{T}\right) -1\right]$$   .

On the other hand, if $C_{ATM}$ is the observed price, the ATM implied volatility will solve

$$C_{ATM}=C_{BS}\left( S,K,T,r,\hat{\sigma}_{ATM}\right) =S\left[ 2\Phi \left( \frac{\hat{\sigma}_{ATM}}{2}\sqrt{T}\right) -1\right]$$   .

Assuming that the HW model is the correct model, $C_{HW}\left( S,K_{ATM},T,r,\sigma \right) =C_{ATM}$, we have the relationship

$$\Phi\left( \frac{\hat{\sigma}_{ATM}}{2}\sqrt{T}\right) =\mathbf{E}%% \Phi\left( \frac{\gamma }{2}\sqrt{T}\right)$$   .

Now, considering that the cumulative normal density is approximately linear around zero, and assuming short maturities [to keep the value close to zero] we end up with the approximate relationship

$$\hat{\sigma}_{ATM}\approx \mathbf{E}\sqrt{\frac{1}{T}\int_{0}^{T}\sigma ^{2}\left( s\right) ds}\text{.}$$

Thus the implied ATM volatility is approximately equal to the expected average volatility over the life of the option.