Separating Black–Scholes Theta from Volatility Carry
Summary
The document explains a way to distinguish ordinary time decay from changes in option value caused by the volatility surface aging. For a European vanilla option, it treats Black–Scholes theta as the one-day price change when implied volatility is held fixed. It defines total one-day theta by comparing today’s option value with tomorrow’s value using the tomorrow-dated volatility surface, while carry theta compares tomorrow’s value under the aged surface with tomorrow’s value using today’s volatility inputs.
Under these definitions, total theta is the sum of the constant-volatility component and the residual attributed to volatility carry. The distinction can help analyze a calendar spread, where a trader wants to separate time decay from changes associated with the volatility term structure. The decomposition depends on the pricing convention and the assumed volatility surface; the answer explicitly limits its explanation to European vanilla options and highlights that expiry date should remain fixed while the valuation date advances.
Key ideas
- Black–Scholes theta measures a time step’s price effect while implied volatility is held constant.
- Total one-day theta can be measured by repricing with the next day’s volatility surface.
- Volatility carry is the difference between using the aged surface and keeping today’s volatility inputs.
- The decomposition depends on pricing conventions and is presented for European vanilla options.
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Full text
# Splitting theta from vol carry
# Splitting theta from vol carry
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.
## Answer by Ezy (score 3)
https://quant.stackexchange.com/a/43261
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 seeShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.