Matching Time and Volatility Conventions in Black-Scholes

Summary

The document addresses whether Black-Scholes time to expiration should be measured in trading days or calendar days when volatility is estimated using trading-day data, and asks how the risk-free rate should be treated. One response advises keeping the time convention consistent with the convention used to calculate volatility. This highlights that the annualization basis for volatility and the time input in the pricing model must align.

A second response notes that option market makers and data vendors commonly use calendar time. It illustrates the practical issue with weekend exposure: an option position can experience time decay while markets are closed, alongside uncertainty about the volatility change when trading resumes. The discussion does not provide a complete conversion procedure or specify how to convert the interest-rate input. It presents conventions and intuition rather than a quantitative comparison, and emphasizes that outcomes are considered on average.

Key ideas

• The volatility annualization convention should be consistent with the time-to-expiration convention in Black-Scholes.
• Calendar time is described as common among option market makers and data vendors.
• Weekend time decay and uncertainty about volatility at reopening illustrate calendar-time exposure.
• The discussion does not give a full parameter conversion method or resolve the risk-free-rate convention.

Tags

# Trading days or calendar days for Black-Scholes parameters?


# Trading days or calendar days for Black-Scholes parameters?












Black-Scholes requires volatility estimated in trading days. How does this affect other parameters? Specifically, should the time-to-expiration also be in trading days? And how does this affect the risk-free interest rate?

## Answer by Lucas Morin (score 5, accepted)

https://quant.stackexchange.com/a/7823

I remember this discussion here: http://www.wilmott.com/messageview.cfm?catid=3&threadid=62227

You should absolutely match your convention for time to expiration to the convention you used for calculating volatility. There seems to be other ways to proceed, as modifying the volatility to match your convention but I don't really see the point in using them.

## Answer by Vince (score 2)

https://quant.stackexchange.com/a/7888

the convention for most market makers of options is to use calender time. it is also the convention in data one gets from options data vendors. another way to see this is that market makers will fade their bids near the close on a 'normal' friday for fear of holding inventory ahead of the weekend, since it is a crap shoot whether the pick-up in vols on Monday morning will offset the time decay from a ho-hum weekend. all this holds on average.

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.