Kurtosis of Simple Returns Under Geometric Brownian Motion

Summary

The document poses a question about the kurtosis of simple returns when the asset price follows geometric Brownian motion. It writes the return over an interval as the ratio of two asset prices minus one, then asks whether its kurtosis can be simplified for arbitrary interval lengths. The price process is lognormal, so the return distribution is a shifted and scaled lognormal expression rather than a normal distribution at finite horizons.

The only evidence offered is a simulation observation: for small intervals, estimated kurtosis fluctuates around three. The document does not derive an analytical formula or resolve whether this behavior holds beyond short intervals. Its note about Brownian increments points toward using their independence and distribution when simplifying the expression. The observed value should be treated as a small-horizon simulation result, not a general conclusion about GBM return kurtosis.

Key ideas

• Simple returns under GBM are derived from ratios of lognormal asset prices.
• The document asks whether return kurtosis has a closed-form simplification for arbitrary time intervals.
• Simulations are reported to fluctuate around kurtosis three at short horizons.
• The question points to Brownian increment properties but does not provide a derivation or general result.

Tags

# What is the Kurtosis of Returns in Geometric Brownian Motion?


# What is the Kurtosis of Returns in Geometric Brownian Motion?












Suppose that $dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t})$ which has solution $$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$ such that $W_t$ is a Wiener process, $\mu$ is drift, and $\sigma$ is volatility. So can \begin{split} K(\delta)&=\text{kurt}\left(\frac{S_{t+\delta}-S_t}{S_t}\right),\\ &=\text{kurt}\left(e^{\frac{\delta \sigma^{2}-2\sigma W_t}{2}+\delta\mu+\sigma W_{t+\delta}}-1\right), \end{split} be simplified for all $\delta$? In simulations, $K$ bounces around $3$ (for small $\delta$). Note that $W_{t+s}-W_s$ is independent of $s$. Any help would be much appreciated.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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