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I am generating 100,000 paths of SPX out to 1 year using Euler discretization. I look at how S is distributed for 100,000 paths at the 1 year point and I find it is lognormally distributed. I look at the distribution of the return of S for 100,000 paths between the two points S(0) and S(t), where t is 1 year, I find that the return is not normally distributed. The return is significantly left skewed. Shouldn't the return be normally distributed?

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  • $\begingroup$ Geometric Brownian motion is lognormally distributed in levels. en.wikipedia.org/wiki/Geometric_Brownian_motion $\endgroup$ – John Apr 17 '15 at 21:24
  • $\begingroup$ That would explain the lognormal distribution of the index level. Wouldn't the index return be normally distributed? $\endgroup$ – Karthik Balasubramaniam Apr 17 '15 at 21:51
  • $\begingroup$ If you divide a lognormally distributed variable by a constant, then it will still be lognormally distributed. $\endgroup$ – John Apr 18 '15 at 0:37
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    $\begingroup$ $S_t$ is lognormal iff $\log S_t$ is normal so you probably made a small interpretation mistake at some point. Either $S_t$ is not really lognormally distributed; did you test for it? Or you are not computing the log returns but $\frac{S_t-S_0}{S_0}$? Or you are looking at daily log-returns when usng a discretization of $S$ instead of $\log S$ etc... All of these information should probably have included in your question if you want a clear answer. $\endgroup$ – AFK Apr 18 '15 at 8:11
  • $\begingroup$ Thanks AFK. I do see that St is lognormally distributed. I was expecting the log returns log(St/S0) to be normally distributed, considering considering that is the underlying assumption in Euler discretization using GBM. I am following the paper frouah.com/finance%20notes/…, equation 7 to discretize my paths. $\endgroup$ – Karthik Balasubramaniam Apr 20 '15 at 18:24

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