What is the best way to splitting theta and vol carry on say a long calendar trade?
Basically trying to split the "good" carry component of a trade from the "bad" carry (theta) which could be earned from just being short vol instead of putting on a calendar.
Well it all depends how theta is calculated in the first place. Depending on your pricing scheme those could be very different things.
Anyways assuming that you are dealing with european vanilla then the BS theta is an instantaneous quantity that assumes that volatility does not change so you definetely don’t get any carry effect from this quantity.
Now usually people look at say 1 day theta. So calling $p(t,\sigma(t,T),T)$ the option premium as of date $t$ with expiry date $T$ and i just indicated the term structure dependence of the volatility $\sigma$ you could define total theta as
$$\theta_{total}= p(t+1,\sigma(t+1,T),T) - p(t,\sigma(t,T),T) $$
And carry theta as
$$\theta_{carry} = p(t+1,\sigma(t+1,T),T) - p(t+1,\sigma(t,T),T) $$
And the 1 day equivalent of bs thera as
$$\theta_{BS} = p(t+1,\sigma(t,T),T) - p(t,\sigma(t,T),T) $$
You see that the 2 theta components add up to total theta, that BS theta assumes the volatility is constant just like in BS case (thus the name) and so the “carry theta” is what is left over that takes into account the “vol decay” due to the volatility surface aging 1 day tomorrow.
NB: note that i purposefully defined $T$ as a date and not as a duration so that the effect of bumping the start date $t$ for pricing purpose (theta) is different from the effect of changing the expiry date $T$ of the option contract. This is useful to actually distinguish in theta calculation as you can see
