HedgeFundGuy meinte am 15. Dec, 23:48:
ok...
average stock has a volatility of 30% ... how much is the bias?
Mahalanobis antwortete am 16. Dec, 01:26:
Let's say
that depends on the distribution of the returns and on the sample size...I've heard that when r ~ N(0,σ2), then E[|r|] = sqrt(2/π)σ.
But unbiasedness isn't the only issue and I leave calculating the distribution of E[|r|]/sqrt(2/π) as an exercise...
Mahalanobis antwortete am 16. Dec, 14:43:
Couldn't resist...
Ok, for r ~ N(0,1) I get the following bias for the sample standard deviation:Return Sample Size (Bias)
2 (0.20)
3 (0.11)
4 (0.08)
5 (0.06)
.
10 (0.03)
.
20 (0.013)
.
30 (0.009)
Here is the R output:
> n <- 10
> v <- n - 1
> sampvar <- rchisq(10000000,v)/v
> mean(sampvar)
[1] 0.9997442
> mean(sqrt(sampvar))
[1] 0.9725524
> approx.bias <- 1 - mean(sqrt(sampvar))
> approx.bias
[1] 0.02744761
