Day-Count Conventions and Variance Interpolation in Volatility Curves

# Day weight conventions for vol curves in quantlib


# Day weight conventions for vol curves in quantlib












I'm trying to build a day weight 'aware' vol curve (simple term structure in time, but flat across a given time slice) in quantlib. I've supplied a list of dates & vols and built it with ql.blackVarianceCurve, trying different pre-defined daycount types (Actual365Fixed, Business252 etc). My aim would be initially to produce a vol curve that has constant variance per business day. So if I were to give it 1w vol = 10%, it would understand that (ignoring holidays) 1w at 10% represents term variance of 10%^2 * 7/365, and that is evenly distributed across the 5 business days. Overnight vol [from thurs to fri] = 10% * sqrt(7/5) = 11.83% Monday vol [from thurs to mon] = 10%*(7/5 * 2/4) = 8.366% etc ie the quoted vols are adjusted for business/calendar day fraction.

In theory I could write a function which does this, outputting each business day implied vol [appropriately adjusted] and then input that into the quantlib functions, but that seems like a heavy handed approach.

thanks in advance for any help.

## Answer by Luigi Ballabio (score 1)

https://quant.stackexchange.com/a/85690

By default, `BlackVarianceCurve` interpolates linearly in variance; this has the effect you want, i.e., depending on the day counter, constant variance per day (`Actual365Fixed`) or business day (`Business252`).

I'll use a simple setup based on June 18th 2026, a Thursday:

```
In [1]: import QuantLib as ql
   ...: import math

In [2]: today = ql.Date(18, ql.June, 2026)
   ...: ql.Settings.instance().evaluationDate = today

In [3]: dates = [ today + 7, today + 14 ]
   ...: vols = [ 0.10, 0.12 ]
   ...: vol = ql.BlackVarianceCurve(today, dates, vols, ql.Business252(ql.WeekendsOnly()))
```

So a 10% one-week volatility and a 12% two-weeks volatility.

If you ask for the variance one week from today (i.e., at `today+7`), you'll get what you expect: $\sigma^2 T$, with $\sigma$ = 10% and $T$ = 5/252 according to the business/252 day count convention:

```
In [4]: var1w = vol.blackVariance(today + 7, 1.0)
   ...: 
   ...: print(var1w)
   ...: print(0.10**2 * (5/252))
0.0001984126984126984
0.00019841269841269844
```

If you ask for the variance for Friday (one day from today), you'll get a fifth of that:

```
In [5]: var1d = vol.blackVariance(today + 1, 1.0)
   ...: 
   ...: print(var1d)
   ...: print(var1w / 5)
3.968253968253968e-05
3.968253968253968e-05
```

and if you ask for the variance at each of the other business days, you'll see that it increases by the same amount per day:

```
In [6]: var4d = vol.blackVariance(today + 4, 1.0) # Monday
   ...: var5d = vol.blackVariance(today + 5, 1.0) # Tuesday
   ...: var6d = vol.blackVariance(today + 6, 1.0) # Wednesday

In [7]: print(var4d - var1d)
   ...: print(var5d - var4d)
   ...: print(var6d - var5d)
   ...: print(var1w - var6d)
3.968253968253968e-05
3.968253968253968e-05
3.968253968253968e-05
3.968253968253968e-05
```

so I'd say that this fulfills your expectation that the variance is evenly distributed across the 5 business days. If you try it out, you'll get the same behavior (with a different variance-per-day) between the one-week and two-weeks nodes.

I'm not sure I understand how you're calculating the volatility you'd expect, though. For any time, given the variance $V$, you'll get the volatility from today to that time by inverting the same $V = \sigma^2 t$ formula you used before, so $\sigma = \sqrt{(V/t)}$. For instance, the one-day volatility ($t$=1/252) would be

```
In [8]: print(math.sqrt(var1d / (1/252)))
0.1
```

and you can also retrieve it directly as

```
In [9]: print(vol.blackVol(today + 1, 1.0))
0.1
```

If you want the volatility between two dates in the future (e.g., Monday and Tuesday) you can use the same $\sigma = \sqrt{(V/t)}$ formula, but with $V$ being the variance between Monday and Tuesday and $t$ being the time between them (1/252); or you can call

```
In [10]: print(vol.blackForwardVol(today + 4, today + 5, 1.0))
0.1
```

As long as you stay in the first week, though, you're interpolating linearly in time between $t$=0 (where the variance is also 0) and the first node ($t$=$T$=5/252). This means that the volatility will be constant and equal to $\sigma$ over any period in the first week (you can work it out easily with paper and pencil). Over the second week, you'll see the volatility change:

```
In [11]: print(vol.blackForwardVol(today + 11, today + 12, 1.0))
0.13711309200802083
```

If you use Actual/365, you'll see the same behavior, except that between Friday and Monday the variance will increase three times as much as between Monday and Tuesday, and you'll have to calculate times as $n$/365 where $n$ is the number of calendar days.