Interpreting a 60-Day Return Window in Volatility Estimation

Summary

The document examines a formula attributed to a paper on learning-based derivative pricing. The question is how to interpret a 60-day range of daily returns when estimating volatility for Black–Scholes: if the sample standard deviation is already daily volatility, the author expects annualization to use the square root of the number of trading days and asks why the paper instead divides by the square root of the window length.

One response suggests the paper may contain a typo or may use a different definition of its symbol: if it denotes the square root of the sum of squared returns, dividing by the square root of the observation count could produce daily volatility, whereas doing so to an already computed standard deviation would rescale it again. A second reply offers a general point about volatility scaling with the square root of time, but does not resolve the notation issue. The exchange is tentative and does not verify the source formula.

Key ideas

• A standard deviation of daily returns is already expressed at a daily horizon.
• Dividing an existing standard deviation by the square root of the sample size may apply an extra scaling step.
• The disputed formula could make sense if its symbol represents a root sum of squares rather than a standard deviation.
• The replies do not verify the paper’s notation or conclusively resolve the question.

Tags

# Volatility estimation based on a 60 days range


# Volatility estimation based on a 60 days range












In Hutchinson et al: A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Network (1994) paper (link), to estimate $\sigma$ for the Black-Scholes formula, it says (p. 881):

I'm not sure to understand. If $s$ is the standard deviation of the 60 last daily returns, it's the daily volatility based on a sample of 60 days. Why don't we multiply by $\sqrt{252}$ to have the annualized volatility ? I don't understand why he divides by $\sqrt{60}$.

## Answer by nbbo2 (score 3, accepted)

https://quant.stackexchange.com/a/57174

This is indeed very strange, and is probably a typo in the paper.

It would be correct if $s^2$ is the sum of squares of the last 60 days returns, and $s$ is the square root of that. Then the division by $\sqrt{60}$ would give the daily vol. But if $s$ is the standard deviation, as they claim, then we would be doing the division twice and that would be wrong.

So I believe $s$ is not what they claim. Any other ideas?

## Answer by D J Sims (score -3)

https://quant.stackexchange.com/a/57173

Volatility over N periods is roughly proportional to sqrt(n). Using the annual number will give a different result that will make the monthly appear lower.

Because volatility is ergodic it doenst just increase linearly over time.

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