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When time to maturity tends to 0, like on expiry day, denominator $\sqrt t$ in becomes 0 and the first term in the formula becomes large enough to make theta of the contract more than its premium. How should this condition be dealt?

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It is true that it is common that BS theta exceeds the actual market value of an option if the time to expiry is short.

  • Therefore most systems compute theta via finite difference (FD) as a true 1 day bump and reprice theta (shifting the evaluation date one day forward and repricing).
  • An additional benefit is that holdidays and weekends can easily be included in the computation (Friday will be a 3 day theta, provided Monday is a working day).

I am using Julia to demonstrate this in an answer to a similar question found here.

A more detailed example demonstrating that Bloomberg's OVML uses this logic and how it compares to quantlib can be found in this answer.

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