For path dependent and forward starting options it is important to assess Vega, the sensitivity of the option’s value to changes in volatility, and in particular to assess these sensitivities for forward buckets. A first step in this process is to determine how forward volatilities for these forward buckets are calculated from spot volatilities implied in current market option prices.
The procedure for determining these forward implied volatilities is similar to the procedure used for determining forward rates from spot prices.
Given the spot rates for a zero coupon security maturing at time 1 and a zero coupon security maturing at time 2, (where time 1 < time 2), it is possible through bootstrapping, to calculate the forward rate for the period between time 1 and time 2.
Similarly given the spot implied volatilities for the period t0 to t1 (?t0, t1) and t0 to t2 (?t0, t2) respectively, it is possible to infer the expected volatility between t1 and t2 (?t1, t2). This volatility is the forward implied volatility (also known as the forward-forward volatility) for the period [t1, t2].
In ‘Dynamic Hedging’ Nicholas Nassim Taleb presents the formula for computing the annualized forward implied volatility for the period between [tn-?, tn], , as follows:
This formula accounts for unequal non overlapping time steps in line with how spot implied volatilities and options prices are quoted in the market.
For example let us consider the following annualized spot implied volatilities for at-the-money call options of strike 272 on Barclay’s stock (BARC) for 31-Jan-2014 obtained from ivolatility.com.
Where t0 is time 0.
Using the data provided first calculate the annualized variance for the periods (t0,tn), i.e. .
Figure 1 Calculating annualized variance
Then calculate the annualized forward variance for the period [tn-?, tn], . For the first period 0-30, the forward variance will be the same as the spot variance. For later periods, 30-60, 60-90, 90-120, …, 360-720 & 720-1080, the forward variance will be calculated using Taleb’s formula as follows:
Figure 2 Implied forward variance formula
Figure 3 Excel implementation of implied forward volatility
The square root of this forward variance will be the annualized forward implied volatility for the period [tn-?, tn]:
Figure 4 Implied forward volatility final steps
For the entire data set the forward implied volatilities are as follows:
The graphical plot of the spot and forward implied volatilities is given below:
Forward implied volatility between two points is the ‘local volatility’ between (S, t) and (S, t+?t). The generalization of this formula gives Dupire-Derman-Kani’s local volatility which is a function of time to expiry and option moneyness.
Comparing local, implied and forward volatilities
Since we have spent a fair bit of time with NVDA options in earlier posts while charting the local volatility surface for NVDA, it would be instructive to see how the forward implied volatility for NVDA option plots when put through Taleb’s implied forward volatility formula.
The implied forward volatilities will change for every series of option expiries for a given strike price.
It is also useful to compare all three volatilities (implied, local and forward) on the same grid for the same series of options.
We would have expected to see convergence between local volatilities and implied forward volatilities and the plot above shows some tracking between these two volatility numbers. The raw implied volatility data is relatively flat when compared to the other two volatility estimates.