Question 3
MA (volatility) 2.0%
MA (variance) 0.0004
Window (days) 30
Sum of squared returns 0.01200
Dropped return 10%
Dropped return^2 0.01
Latest (new) return 2.0%
Latest (new) return ^2 0.0004
Updated sum of return^2 0.00240
Updated variance (average of return^2) 0.00008
Updated MA volatility 0.89%
This is called GHOSTING PROBLEM/FEATURE.
Question 4
Question Hull 19.3
lag variance 0.00010 0.00023
lag return 0.03279 0.01653 << for return, it would be okay to use 1/30 ins
lambda 0.94 0.94
Est Variance 0.000159 0.000228
Est Volatility 1.26% 1.51%
Question 5
Hull 19.8
lag variance 0.00010
lag return 0.02667
alpha 0.06
beta 0.92
w 0.00000200
Est Variance 0.000137
Est Volatility 1.169%
Question 6
Hull 19.8
Annual Vol 30.0%
Confidence Interval 99.0%
Trading days / year 252
Daily Volatility 1.89%
Daily Variance 0.000357
Hull's (Incorrect) answer uses normal
Normal Deviate 2.58
Lower Bound -4.868%
Upper Bound 4.868%
Sample variance uses chi-square David Harper:
Sample/d.f. 10
sample = d.f. because
Lower (.9x) 0.5%
sample mean return = 0
Upper (.0x) 99.5%
CHIINV(.9x,d.f.) 2.1559
CHIINV(.0x,d.f.) 25.1882
Lower Bound, Variance 0.000142
Upper Bound, Variance 0.001657
Lower Bound, Volatility 1.19%
Upper Bound, Volatility 4.07%
Question 7
Hull 19.8
lag variance 0.0001
lag return 0.0100
alpha (weight) 0.05
beta (weight) 0.92
gamma (weight) 0.03
LR variance 0.000133
LR volatility 1.155%
omega 0.00000400
Est Variance 0.000101
Est Volatility 1.005%
Sum of weights = 1.0 1.00
rn, it would be okay to use 1/30 instead, which is near enough
2%
0.0004
30
0.01200
10%
0.01
2%
0.0004
0.00240
0.00008 0.00240
0.89%