# 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?

• 255/248 is not the same moneyness for 1 hour versus 1 day or 1 year. That is why the results seem counterintuitive. – Frido Rolloos Apr 2 '20 at 16:46
• @ilovevolatility That makes sense now. So the expected call prices are accurate, but just seem counterintuitive? Could you use that method to determine the price of an option intraday? Example: I expect underlying X to move Y dollars in Z time, plug in Z time for the time to maturity, and return a theoretical call price? If not, what would be the best way to do that? – user13138159 Apr 2 '20 at 16:49

## 1 Answer

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.

• Apologies for my lack of clarity. You're correct, my confusion is upon observing that decrease in contract value. If an underlying makes a dramatic move upward in one hour versus the same move in one day, would it not be the case that the call price is more valuable if the move is completed in one hour versus at the end of one day? Your second point potentially addresses this, although I am unsure if it is related also. – user13138159 Apr 2 '20 at 16:41
• The parameter that you changed tells you how long is left where the option is still alive, when Time to Expiration is 0 the option is worth it's payoff, 7 in the case of a call and 0 in the case of a put. With less time left you decrease the potential for large moves to happen, and so the value of the option also decreases. – Oscar Apr 2 '20 at 16:45
• Ah! This makes sense now. I totally misunderstood that the time to expiration parameter in the formula means, as you initially inferred. – user13138159 Apr 2 '20 at 16:47
• So just to confirm my understanding: this formula cannot be used to determine the intraday value of a call at a certain expiry (that is more than 1 day/1 hour away) by plugging in the expected sell to close time? Example: I expect underlying X to move Y dollars in Z time, plug in Z time for the time to maturity, and return a theoretical call price? If not, what would be the best way to do that? – user13138159 Apr 2 '20 at 16:52
• No, that isn't how the formula works. In your example Z is already determined as the time to maturity, if an option expires in one year then Z is 1. It's not something that you plug in based on your expectations. To do what you're talking about you would need the delta of the option, which tells you how much the option will change in value when the underlying changes. If delta is 0.5 it means that if the underlying changes by 1 dollar then your option changes by 0.5 dollars. You also have Theta which is the change of price of your option from the passing of time. (1) – Oscar Apr 2 '20 at 17:00