Thread by @G3ni3sWish

Forward Volatility -🧵
As the name suggests, forward volatility is the implied vol between any 2 days in the future.If implied vols for any 2 dates in the future exists, then we can find the forward implied vol between them.Forward implied vol is shortened to forward vol or fvol.

How is forward volatility calculated?
Volatility is a tricky metric (not linearly additive over time) so we deal with its simpler and younger sibling: “Variance”. As you can expect from younger siblings:
Variance = Volatility^2 … (when both are annualised)

Variance for time T (Days To Expiry (dte)):
Var = σ^2 * T … (where σ is the annualised implied vol and T is dte in years)
So, if Variance for T dte is Var1 and for T + ΔT is Var2
Then the variance for "ΔT days after T days" is (Var2– Var1). Simple!
Var(ΔT) = Var(T+ΔT) - Var(T)

And using above formula, we calculate the annualised volatility for "ΔT days after T days" as:
σ(forward volatility) = √ ((Var2– Var1)/ ΔT)
or, expressing everything in terms of volatility,
σ(forward volatility) = √ ((σ2^2 * (T + ΔT) – σ1^2 * T)/ ΔT)

https://x.com/muskk/status/1583840403509641217?s=46&t=cHPfGzveShlvPhAaM_uzdA

So, if we have 3 expiries (E1, E2 & E3) trading,
then we'll have 3 possible fvols viz E1-E2, E1-E3, E2-E3
For 4 expiries (E1, E2, E3, E4)
we have 6 possible fvols viz E1-E2, E1-E3, E1-E4, E2-E3, E2-E4, E3-E4
And so, for n different expiries, we have n*(n-1)/2 unique fvols

Example:
In this image, we can see fvol between any 2 dates indicated by row and column (21 fvols).When dates are same, we have the implied vol for that expiry.
Way to read this:
ivol for 11 Jan is 12.836%
ivol for 25 Jan is 12.917%
fvol between 11 Jan'24 and 25 Jan'24 is 13.055%

How is forward volatility traded?
Forward vols are traded using hedged calendar spread.These are also called horizontal spreads (if same strike) or diagonal spreads (if using different strikes).They are called call calendars if entered using calls and put calendars if using puts.

Call Calendars:
Short ATM Call (Expiry E1) + Long ATM Call (E2)
If Expiry E1 is earlier than E2, then E1 call has lesser premium than E2 & the above structure becomes debit. Hence it is called a long call calendar (short front end long back end of term structure).
And vice versa

Put Calendars:
Short ATM Put (Expiry E1) + Long ATM Put (E2)
If Expiry E1 is earlier than E2, then E1 call has lesser premium than E2 & the above structure becomes debit. Hence it is called a long put calendar (short front end long back end of term structure).
And vice versa

The blue line in the payoff chart is for when the trade is entered and the brown line is the payoff chart at expiry. The blue line gradually contorts its shape to become the brown line as days pass, expiry nears and nothing else changes.

You see how the payoff is worse as we move away from the ATM in both directions? That’s due to delta PnL & it has to be hedged by using futures. A calendar thus becomes a true trade on fvol only when it is dynamically delta hedged during its lifetime (which is the earlier expiry)

Greeks of Calendar Spread:
A Long Calendar Spread is a long (forward) vega, short gamma (and hence long theta) spread. So ideally what it needs, to be green, is high implied vol and low realised vol. Short Calendar is vice versa.

Why is it traded?
Forward vols are a way in which market can express its opinion on what the fvol should be for a time-slice in the future. Market participants can disagree with this pricing by either longing or shorting the calendar and hedging deltas over its lifetime.

Why or when would market participants disagree with fvols and trade calendars?
A. New event that cause market volatility get added to a future time slice. E.g. Earnings announcement, Dividend ex/record date announcement, macro data release announcement, M&A announcements etc

B. Existing event causing market volatility move out of this future timeslice (advanced or delayed). E.g Earnings call got delayed, vol causing events getting advanced/delayed/cancelled.

C. No change in set of events but vols of earlier expiry got repriced, thus changing the fvol and market has to reprice later expiry to bring fvol back to earlier level. This can be caused by vol shocks due to price dislocations/liquidity holes, earlier events causing shocks etc.

Even if the nature of the events in the above options changes, fvols will re-adjust. Then of course, we have vol decay caused by veta but that's not specifically applicable here
If there are any other stimuli that can affect forward vols that I have missed, feel free to comment👇

Finally, this thread on forward volatility is meant to be a quick primer on what fvol means and what it doesn't mean (mainly, it's not a cheap regular longvol bet aided by tailwinds of theta). I hope none of this sounds like a trade recommendation (coz it's not👀)