# Ways of treating time in the BS formula

The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions:

• What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, d1 would involve division by zero.
• Given that $t$ is the number of days in the year, what value should we divide it by? 252 or 365? It is that important? It it worth using quantlib to take account of various calendars?
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Two quick points:

1. Recall that the derivation involves continuous time and $(t, t+\Delta t)$ arguments---so the granularity is (at the margin) infinite. And hence time zero does not really get reached until we actually are at expiry.

2. Generally speaking want the number of business days, not calendar days, and holidays do matter. So one generally uses the 'number of trading days til expiry / 252'.

Here is a quick example moving maturity from 1 year to 0.9975 years

R> EuropeanOption("call", 100, 100, 0.01, 0.03, 1, 0.4)
Concise summary of valuation for EuropeanOption
value    delta    gamma     vega    theta      rho   divRho
16.5382   0.5927   0.0096  38.2821  -8.3458  42.7367 -59.2749
R> EuropeanOption("call", 100, 100, 0.01, 0.03, 0.9975, 0.4)
Concise summary of valuation for EuropeanOption
value    delta    gamma     vega    theta      rho   divRho
16.5150   0.5926   0.0096  38.2332  -8.3578  42.6295 -59.0986
R>

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For a scholar example, one shall use 250 or 252 days.

As you noticed Dmitri, the Quantlib library includes a series of calendars.

These classes are based on various well-known financial markets conventions, such as Actual365 and ISDA.

You might be interested in the following webpage: http://quantlib.org/marketconventions.shtml

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Regarding conventions

One thing to keep in mind in all questions about "what's right and what's not?" is that conventions don't always matter as much as one would think.

When a trader marks his vols by looking up option prices on the market, he is going to mark them using the pricing model which his quants implemented. So whether he uses one convention or the other, he's still going to get the same price for these options and most likely similar prices for other options.

Now, to answer your question, when it comes to time conventions, neither calendar nor business days are really correct. If you use calendar days then your options will have 3 days time decay on Monday which you can't do anything against. And yet in some cases one can think that the market moves more between Fri close and Mon close than other days so biz days would not be appropriate either.

What some desks, especially in equities implement is an elastic time. That means they weight time depending on much they believe the market will move relative to other days. So for instance any day with foreseeable volatility due to some number announcements (results or nonfarm payroll) will have a heavier weight. That means that on these days your options will have more decay, which is fair given that the market is expected to move more as well. However these are very subtle considerations for large volume market-making desks.

Regarding expiry date

It is customary to model the option as equal to the intrinsic value on the day of expiry. Risk managing expiring options is a known headache, and again this is usually done by market-making specialists. Most options traders will avoid having options expiring in their books which are too close to the strike, and if they have them they will have to do a lot of adhoc delta hedging.

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If you assume a daily time step, then the day of expiry,the value of the option (a call in the following example) is known, by definition: $$C_T = (S_T - K)^+$$

About the number of days you want to use, keep in mind that you use the underlying BS assumptions that the underlying asset's prices are log-normally distributed, that is, that its return are normally distributed. This is highly unlikely when you consider business days values, and I believe it would be even more unlikely taking into account days where prices do not change (holidays).

So, I suggest you to use the 252 days convention. Note that if you choose not to take into account bank holidays, you should not include them in the data sample from which you estimate your daily volatility $\sigma$.

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