Monte Carlo Stock Returns with Asymmetry and Fat Tails
Summary
The answer outlines a workflow for simulating stock or index returns with volatility clustering, asymmetric risk, skew, and heavy tails. First, an autoregressive GARCH model captures serial dependence and changing volatility; a GJR-GARCH variant can represent asymmetry, while Student's t residuals allow heavier tails than a normal distribution. The standardized residuals can then be modeled with a smoothed kernel density to retain distributional features such as skew and kurtosis.
For more realistic extremes, the proposed approach replaces the central portion of that density with separate left- and right-tail models, such as generalized Pareto distributions. Monte Carlo draws from the combined residual distribution are transformed back into returns through the fitted time-series model. This is a suggested modeling framework, not a universal industry standard. Results depend on model choice, threshold selection, data quality, and estimation, and the discussion offers no comparative validation or quantified forecast performance.
Key ideas
- An autoregressive GARCH model can represent return dependence and time-varying volatility.
- GJR-GARCH can capture asymmetric volatility responses, and Student's t residuals can represent heavy tails.
- A smoothed kernel density can preserve skew and kurtosis in standardized residuals.
- Extreme Value Theory can model the left and right tails separately, for example with generalized Pareto distributions.
- Monte Carlo scenarios are generated by sampling residuals and mapping them through the fitted return model.
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# How to model stock price for a monte carlo simulation with fat tails and asymmetric risk # How to model stock price for a monte carlo simulation with fat tails and asymmetric risk I'd like to create a monte carlo simulation to determine the future price of a stock or index with a certain confidence level. I've seen examples of this described using lognormal returns but I'd like to include the fat tail risk and the asymmetry of price falls vs price rises. Is there a industry standard way to model this, or best practice that is widely used and accepted? ## Answer by Helin (score 6, accepted) https://quant.stackexchange.com/a/64286 I did a related project years ago and an example from Matlab's website, Using Extreme Value Theory and Copulas to Evaluate Market Risk, proved particularly helpful as the starting point. At a high level, you would - - Fit an Autoregressive-GARCH model to the returns. The autoregressive component accounts for autocorrelation in the time series, and the GARCH component captures heteroskedasticity. Different GARCH model can be used, but the GJR-GARCH variant can introduce asymmetry; further, and the residuals can be modeled using the Student's t-distribution, allowing for fat tails. - After the step above, you have a series of i.i.d. residuals. We can fit any density function to them; a smoothed kernel density function is a good choice, allowing you to capture skew and kurtosis profiles in a very parametric way. - The density function can be further refined using some ideas from the Extreme Value Theory. In particular, we let the middle chunk of the density function be the smoothed kernel density function from step 2, but we use a different set of density functions for the left- and right-tails. These tails can be modeled using something like a Generalized Pareto Distribution. - You're now ready to run MC simulations. You can draw normalized residuals using the density function from step 3 and convert them back into a time series of returns using the AR-GARCH model from step 1. Again, I recommend that you look at the Matlab example, which lays out everything step-by-step.
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