Grant,
The forward implied vol is analogous to the forward rate calculated off a yield curve. It tells you the market's current price of implied vol for expiry time 2 (t2) as traded at t1 calculated at t0.
For example. If, today, we calculate the forward implied vol for a 3 month option in 1 month this tells us what, right now, the market expects to be the implied vol for a 3 month option in 1 months time. If you lock in the forward vol you are getting the certainty that you will not pay more for the vol later.
The forward vol curve is obviously a derivative of the spot vol curve so you can see that, in the case of index options for example, you will usually be paying more for the forward vol than the spot for the same tenor. The closer to t0 the forward vol you are calculating is, the greater the convergence with spot vol.
Your second question:
A volatility surface is just an interpolation of the points between the available implied vols for the strikes and expiries which trade.
For example, say we know the implied vol for a 100 strike option and for a 90 strike option with expiry in 1 month and a 100 strike option and a 90 strike option with expiry in 2 months.
For whatever reason we decide that we wish to know the implied vol for a 6 week 95 strike option. We can take the 4 points and interpolate between them to obtain an approximation for what the vol 'should' be. This is necessary if there is no market in the options at that strike.
For a quick result you could just do a linear interpolation between the points but if we extend the idea out to a whole surface (of many strikes and several expiries) this is not very neat. There are lots of interpolation methods that people use but a cubic spleen is also easily implemented and will give much smoother results.
Usually a vol surface is used for pricing exotics. I reference my surfaces when trying to get a quick 'fair value' for binary bets which are not at a liquid strike in the equivalent options market.
A well documented problem is that of extending the surface closer to t0 when the nearest liquid expiry is > a day away. Often we find in equity and index options that it is many days away and we are left 'guessing' the near term implied vol as a function of numerous difficult assumptions and inputs. The historic spreads between the implied vol of options expiring at t0 and the next expiry month can be some guide but these are of course volatile themselves.
The concept and the maths are very simple - the assumptions that one is forced to make are more troublesome.
Please let me know if this is insufficiently articulate so that I can clarify anything.
All the best,
NQR
