Historical volatility estimates using few samples have a downward bias, since the average squared value of N standard normal deviates is 1, but the average root-mean-squared value is less than 1 and increasing in N, by Jensen's inequality. From the properties of the chi-squared distribution, the expected RMS is
sqrt(2/N) * gamma((N+1)/2) / gamma(N/2)
RMS(N) for a few N are
1 0.7979
5 0.9515
10 0.9754
20 0.9876
30 0.9917
40 0.9938
50 0.9950
To compare a 10-day historical vol to a longer-term vol you should first scale it by 1/0.9754. The effect is stronger for heavy-tailed distributions such as Student's t.
Compiling xrms.cpp and running gives
# n_trials = 1000000
N exact_RMS_normal sim_RMS_normal sim_RMS_t5_std
1 0.797885 0.799085 0.735631
2 0.886227 0.887082 0.830001
3 0.921318 0.922329 0.871714
4 0.939986 0.940664 0.895914
5 0.951533 0.952212 0.911351
6 0.959369 0.960037 0.922501
7 0.965030 0.965687 0.931166
8 0.969311 0.969964 0.937820
9 0.972659 0.973280 0.943205
10 0.975350 0.975881 0.947469
11 0.977559 0.978010 0.951371
12 0.979406 0.979732 0.954538
13 0.980971 0.981311 0.957337
14 0.982316 0.982611 0.959785
15 0.983484 0.983736 0.962003
16 0.984506 0.984795 0.964017
17 0.985410 0.985675 0.965802
18 0.986214 0.986490 0.967370
19 0.986934 0.987226 0.968731
20 0.987583 0.987926 0.970041
21 0.988170 0.988559 0.971263
22 0.988705 0.989088 0.972375
23 0.989193 0.989508 0.973338
24 0.989640 0.989923 0.974309
25 0.990052 0.990319 0.975172
26 0.990433 0.990672 0.975933
27 0.990786 0.991022 0.976670
28 0.991113 0.991318 0.977322
29 0.991418 0.991651 0.977991
30 0.991703 0.991948 0.978641
31 0.991969 0.992239 0.979206
32 0.992219 0.992476 0.979795
33 0.992454 0.992702 0.980269
34 0.992675 0.992929 0.980786
35 0.992884 0.993141 0.981279
36 0.993080 0.993338 0.981722
37 0.993267 0.993513 0.982095
38 0.993443 0.993657 0.982473
39 0.993611 0.993825 0.982809
40 0.993770 0.993959 0.983206
41 0.993922 0.994086 0.983553
42 0.994066 0.994224 0.983863
43 0.994203 0.994373 0.984168
44 0.994335 0.994488 0.984460
45 0.994460 0.994628 0.984747
46 0.994580 0.994754 0.985031
47 0.994695 0.994846 0.985302
48 0.994806 0.994969 0.985576
49 0.994911 0.995097 0.985805
50 0.995013 0.995195 0.986109