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Two Theta Definitions for Options Under Changing Volatility

Article Quant Q&A · Author: Cris
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Summary

The document explains two ways to measure an option’s one-day theta when implied volatility varies across expiries. Fixed-expiry theta holds the option’s implied volatility constant during the time bump, matching the Black–Scholes convention. Fixed-tenor theta rolls the volatility surface forward, so the option’s implied volatility changes as its tenor shortens. The latter can include the effect of changes in the volatility term structure directly in theta.

The distinction clarifies why volatility changes may appear either in theta or in vega, depending on the chosen convention. Both definitions can apply beyond vanilla options. The discussion gives conceptual definitions rather than a numerical example or a market-wide rule for choosing between them. Its practical guidance is that Greeks are tools: practitioners can use standard definitions or adapt them to the product and risk question at hand.

Key ideas

  • Fixed-expiry theta measures a one-day time change while holding implied volatility constant.
  • Fixed-tenor theta rolls the volatility surface forward and allows the option’s implied volatility to change.
  • The chosen theta convention determines whether term-structure effects appear in theta or are left to vega.
  • The discussion presents both approaches as useful conventions rather than requiring one universal definition.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Definition of Theta in presence of time varying volatility


# Definition of Theta in presence of time varying volatility












I’m wondering what is the right or standard way to think/define Theta decay with time varying volatility such as the ones observed in the market.

One way could be to fix implied vol and use vol*sqrt(time to expiry) where appropriate. Therefore time decomposes from volatility. This appears the simplest however theta decay wouldn’t capture effects like volatility crushing. In this case it seems that one has to capture that risk through Vega. Perhaps that would be closest to A BS formula.

The other would be to account for the changing vol between expiries. Accounting for that in theta would be somewhat related to removing from the total variance what the market sort of implies for the immediate variance. This could incorporate volatility crushing in theta directly.

More generally it seems that defining Greeks in a real scenario becomes unclear. Do practitioners tend to use one definition of Greeks or do they resort to custom models.

## Answer by Soumirai (score 3, accepted)

https://quant.stackexchange.com/a/80394

There are indeed (at least) two ways to compute theta, which is defined as a 1 day time bump:

- Fixed expiry, where the implied vol of the option is kept constant: this is Black-Scholes theta;

- Fixed tenor, where the vol surface is rolled forward by 1 day, and so the implied vol of the option is effectively rolled down by one day: this captures your second point.

Note that these definitions work for any product, not just vanilla options.

Regarding Greeks in general, it's really up to the practitioner. There are standard definitions of Greeks, but then you shouldn't be constrained by definitions. They are tools, and practitioners should master their tools!

Shown 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.