Time Units and Expiration in the Black–Scholes Formula
Summary
The question asks whether Black–Scholes can use an expiration measured in hours after converting that duration to years, and why an option calculator returns a different value when the input changes from a day to an hour. The answer interprets the change as a shorter time to expiration and points to the role of time in option value, rather than treating the shorter interval as a larger opportunity for a dramatic move. It also suggests checking that the calculator input represents time remaining to expiration.
A side comment raises a separate modeling issue: the normality assumptions used in Black–Scholes may be less defensible over very short horizons than arguments based on daily returns suggest. The answer does not establish that the model is accurate at hourly frequencies, and the reported calculator values cannot be fully assessed without the other inputs and their units. The central practical lesson is to convert time consistently with the volatility convention and interpret the input as remaining maturity.
Key ideas
- Black–Scholes time to expiration must use units consistent with the volatility input.
- Changing a day to an hour changes the remaining maturity and can change option value.
- Check that a calculator field represents time remaining until expiration.
- The answer raises, but does not resolve, whether model assumptions hold at very short horizons.
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Full text
# Does the Black-Scholes formula work when unit of time is in hours? # Does the Black-Scholes formula work when unit of time is in hours? In the Black-Scholes formula, the unit of time is usually in years from what I understand. An online calculator I found allows the users to input the time in days and years. Would the formula still be accurate if I were to plug in say 1 hour for the time variable, by first converting it to years, given that 1 hour is roughly 0.000114155 years? I would assume yes, but the reason I am asking is because of this case scenario that I found with the above calculator. Given the following parameters, the calculator returns a call value of $7.81. However, when I change the time units to years, and plugin the above converted value of one hour to years, the call value drops to $7.00. Why is this the case? Shouldn't a dramatic move increase the price for a call option if it happens in a shorter period of time versus if it takes longer to happen? ## Answer by Oscar (score 1, accepted) https://quant.stackexchange.com/a/53004 I'm not sure what you're asking here quite, it seems to me that you are inputting a shorter time to maturity (from one day to one hour) and noticing a decrease in the contract value. Theta, the derivative of the option price with regards to time, is negative for for all options so this will always be the case no matter the time scale. Are you sure you have understood correctly what the Time to Expiration parameter in the formula means? As an interesting side note I have seen the argument made that daily returns are normal by the CLT because they are made up of sum of many small price intraday changes that are I.I.D and come from some (any) distribution. So you could reason that because of this Black-Schooles framework would not hold when considering shorter time periods than this. This has however nothing to do with what you're confused about, I believe, even though that is how I originally interpreted your title.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.