Why Simulated Simple Returns Can Be Skewed Despite Lognormal Prices

Summary

The document asks whether one-year returns from simulated SPX paths should be normally distributed when the simulated terminal price is lognormally distributed. It describes generating 100,000 paths with Euler discretization, then comparing the distribution of terminal prices with the distribution of returns measured from the initial price to the one-year endpoint. The reported terminal prices appear lognormal, while the returns appear significantly left-skewed.

The central learning point is the distinction between a price level and a return definition: a lognormal price does not imply that the simple return, calculated as the price ratio minus one, is normally distributed. Under the standard geometric Brownian motion model, log returns are normally distributed, while simple returns are shifted lognormal. The post contains no model parameters, plotted evidence, or answer explaining the observed skew, so it does not establish whether the result reflects that distinction or an implementation issue.

Key ideas

• A lognormal terminal price does not imply normally distributed simple returns.
• Under geometric Brownian motion, log returns are normal, while simple returns are shifted lognormal.
• The post reports left skew in simulated one-year returns but gives no parameters or diagnostic evidence.
• Checking the exact return definition and simulation setup is necessary to interpret the reported distribution.

Tags

# Monte Carlo simulation returns not normal distributed


# Monte Carlo simulation returns not normal distributed












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.