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They won't be the same. If you run a discrete simulation you will get the actual (or an instance of an actual path) price process for the future value of the stock using the real probability measure. If you do the same thing using the closed form solution, the path will look very similar but will drift downwards. Why are they different? To see it ...

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I though about this one more time: method of moments means that you do the following: calculate some statistics (i.e. the moments) on the sample express the moments of the distribution that you want to fit in terms of the parameters of this distribution solve the resulting system of equations. If you estimate $E[S^n]$ by averaging the ...

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as you post 3 questions on this topic and after reading them: this is homerwork/study material- right? So for comparing Fast Fourier, MC and Panjer there are tons of publications out there. For the formulas for the momemts of $S$ look here or google "moments in the collective risk model". You should notice that: If you know the distribution of $N$ and $X$ ...

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I use straightforward approach: Generate "returns"; Make cumulative sum of returns from Step 1; Take any Nth (N should be "big enough") point for series obtained on Step 2. That would be "closes"; Then take max and min between "closes" = highs and lows. In R: n <- 10000 # quantity of "ticks" inside 1 day m <- 200 # number of days rets <- ...

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You can do the following: For each $i$ in $1$ to number of Mont-Carlo runs $K$ simulate the number of losses $N_i$ simulate $N_i$ many loss-sizes $X_{i,1},\ldots,X_{i,N_i}$ calculate $L_i = \sum_{j=1}^{N_i} X_{i,j}$ Doing this you get a sample of losses $L_1,\ldots,L_K$ and you can do all sorts of hisograms, density fits, VaR, ES calculations on it. ...

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