Estimating Near-Expiry Black–Scholes Theta with a One-Day Repricing

Summary

The document addresses a near-expiry limitation of the Black–Scholes theta formula: as time to maturity approaches zero, the formula can produce a theoretical daily decay larger than the option’s market premium. It describes a practical alternative used by many systems: estimate theta by shifting the valuation date forward by one day and repricing the option, a finite-difference calculation.

Moving the date by a calendar day also lets the calculation account for non-trading days. For example, a Friday shift may span a weekend when the following Monday is a working day. The answer points to demonstrations in Julia and comparisons involving market and library calculations, but those materials are not included here. The document offers a practical convention rather than a derivation, and does not specify model inputs or explain how to handle every calendar or expiry convention.

Key ideas

• The closed-form Black–Scholes theta can exceed an option’s market value near expiry.
• A one-day forward valuation-date shift and repricing provides a finite-difference theta estimate.
• Date-based repricing can account for weekends and holidays in the elapsed period.
• The document refers to further examples but does not include their calculations or details.

Tags

# Theta using black scholes when time to maturity approaches 0


# Theta using black scholes when time to maturity approaches 0












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?

## Answer by AKdemy (score 2, accepted)

https://quant.stackexchange.com/a/77075

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.

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.