<aside> ❓ First, a quick note before we start doing arithmetic

<aside> 📏 Why do we talk about “variance time” instead of “volatility time”?

• Volatility is the standard deviation of returns.
• Standard deviations are computed by taking the square root of the average of squared returns.
• The average of squared returns is known as variance.
• To compute such an average we divide by the number of returns in the sample.
• Therefore, variance (not volatility) is proportional to the number of returns or time.
• We must perform all operations (ie addition, multiplication, etc) in variance terms then we can simply take the square root to get back to volatility terms. </aside>

</aside>

We established that:

• A calendar-day model assigns each day of the year 1/365 of the variance.
• A business-day model assigns each business day of the year 1/251 of the variance.

Visually, this is how variance is distributed according to each model:

• The total variance is the same for each model but:
  • a business day model clusters the variance on 251 days, with each business day receiving 1/251 or .4% of the annual variance
  • a calendar-day model uniformly assigns .27% of the variance to each calendar day

We can zoom in on just January 2023 for a clearer picture:

Just as accrual accounting’s smoothing effect gives us a more accurate picture of a business’s performance (versus cash accounting’s lumpiness), we would be better served to specify our own calendar instead of defaulting to the overly basic assumptions of 365-day calendar and 251-day business models.

• The calendar day model smooths variance too much by assigning the same amount of variance to a business day vs a weekend/holiday.
• The business day model is too lumpy — it buckets a full year’s variance into just 251 days and pretends that zero variance time passes over a weekend.

In sum,

• the calendar day model has time passing “too fast” over a weekend. The straddle doesn’t decay as much as the 365-day model predicts but it does decay, so the market is assigning a non-zero amount of variance time to weekends but not a full 1/365th.
• the business day model has time pass “too slow” over a weekend. It acts as if zero variance time has passed but, again, the straddle does actually erode. If the weekend actually contained zero time then the straddle on Friday evening would be worth the same as the straddle on Monday morning.

Variance Schedules

We will construct a more realistic variance schedule. To do this let’s look at the naive calendar and business day models in a calendar format.

Recall the tenors of each model:

• Calendar day model: 365 days
• Business Day model: 251 days (365 less 104 weekend days less 10 holidays)

This is a sample of the calendars represented by each of the 2 models for January 2023:

The key columns to note are the time remaining columns. They normalize the number of days remaining under each counting convention to the number of days in the year.

The time remaining for the calendar day model declines linearly. For the business day model, it declines evenly on business days and not at all on weekends or holidays. We established all of this above this is just spelling it out in a schedule.