What is the difference between Calendar Spread & Forward volatility strategy?
a calendar spread is an easily tradeble strategy which has a very complex exposure. A forward volatility is a hard-to-trade strategy with a very simple exposure.
Forward volatility can only be traded via forward volatility agreement - a futures like otc instrument. You bet on the implied vol between two future points in time, T and T+ t. The expiry time is T, and the payoff at expiry is the strike price (in vol points, determined when you entered in to the agreement) minus the atm implied vol for an option with expiry t, as on the expiry date (T). There is no static replication of this instrument. If you believe in local vol, you can dynamically hedge with a strangle of expiry T and a straddle at T+t. In certain markets where you have mid-curve options (interest rates options), you can dynamically replicate using vol triangulation. If you can settle for forward variance (instead of volatility) you can buy a variance swap for T+t and sell one for T. Variance swaps can be statically replicated (by buying a lot of strikes in inverse proportion of the squre of the strike for the expiry). So as you see, it is a very difficult strategy to trade. The exposure, on the other hand, is simple - the volatility between T+t vs T (see the pay-off). A pure forward vol position has no delta, no gamma and no time-decay (you may have some net theta because of vol surface carry).
A long calendar spread is done by buying a longer date option vs a shorter dated one at fixed strikes. Here you have a dynamic exposure to short gamma (becuase of the fixed strikes), long theta (if you have gamma, you will have theta), varying delta (usually delta-neutral at the onset) and usually long vega. A part of the vega (which does not result from smile/ skew) is vega exposoure due to forward vol. Usually your gamma will dominate in terms of dollar risk and will make it a poor substitute if you are interested in only forward vol exposire. So this is a very simple stragegy to trade with a very complex risk profile with all sorts of cross greeks.
Hope this makes things clear.