Sampling Effects on Realized Volatility Estimates for GBM Paths

Summary

The document asks how realized volatility behaves when estimated from daily price increments generated by a geometric Brownian motion (GBM). It focuses on the distribution of the estimate across simulated paths and asks whether the estimate is unbiased for the model’s volatility parameter, as well as what can be said about higher moments.

The response interprets the question as asking about the variance of pathwise volatility estimates across many paths, each built from a finite number of time steps. It suggests that coarser sampling, with fewer steps, should increase the estimate’s dispersion and notes a possible connection to discrete-observation adjustments for variance swaps. However, the reply does not prove unbiasedness, derive a distribution or variance, or provide references beyond suggesting further research. It is a useful pointer to the sampling issue, but not a complete quantitative treatment.

Key ideas

• Realized volatility from GBM increments can be studied as an estimator across simulated paths.
• The response expects estimate variance to increase when each path uses fewer time steps.
• The document raises unbiasedness and higher moments but does not resolve either question.
• Discrete observation adjustments for variance swaps may offer a related line of inquiry.

Tags

# Distribution of realized volatility for stock prices from a GBM


# Distribution of realized volatility for stock prices from a GBM












If you generate random stock price paths according to a GBM with daily increments, what will be the distribution of the realized volatility? Assume that the realized volatility is measured over daily increments for the whole year.

I assume that the realized volatility is an unbiased estimator of the parameter sigma in the GBM, but I've never seen it proved. As for the higher moments, what can be said ?

## Answer by James Spencer-Lavan (score -2)

https://quant.stackexchange.com/a/34017

I assume the OP means what is the variance, across N paths of M steps each, of the realised volatility of the GBM.

You would imagine that it is related to the dt-step size implied by M. i.e. for lower M, the variance of the pathwise-estimated volatility increases.

Feels like is related to strike adjustments for variance swaps with discrete observations, not sure if there is literature out for that but worth googling

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