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I am studying Monte Carlo method by reading An Introduction to Financial Option Valuation and my questions come from the last paragraph of the section 15.2 on page 144.

The author, Desmond J.Higham, gave an computational example on the simulation of E(exp(z)). On the analysis of the results,he indicated the ratio of those errors. But I don't understand what information can we get from this ratio.

Here is the example:

We simulate $\mathrm{E}(\mathrm{exp}(z))$ by Monte Carlo method,where $z \sim N(0, 1)$. For different sample size, say, $2^{16}$ and $2^{17}$, the variance error are $0.00531$ and $0.00364$, respectively. The ratio of these two errors is approximate 1.5.

So my questions are:

  1. what information can we learn from this ratio?
  2. why should the ratio approaches square root of 2 as the sample size approach infinite?
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It might be simpler if you thought about it in terms of sample mean of z. The standard error of the sample mean is given by $2^{-0.5 \times 16}$ and $2^{-0.5 \times 17}$, respectively. So the ratio between is thus the square root of 2. –  John Jul 31 '13 at 17:50
    
Please avoid crossposting. Especially since this is not a strictly QF question. The basics of MC shall be well known here. –  Quartz Aug 1 '13 at 8:28
    
I asked the same question at stats.stackexchange.com/questions/66214/…, and got the answer there. –  Hebe Aug 1 '13 at 13:28
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