Estimating Instantaneous Volatility from Daily Prices

Summary

The document asks how to estimate the time-varying instantaneous volatility in a stock price model from historical daily observations. It distinguishes the constant-volatility assumption from models that allow volatility to change over time. Under constant volatility, daily log returns provide a simple estimate: their standard deviation is used as the volatility estimate.

If empirical evidence indicates volatility is not constant, the answer points to the GARCH model family as an alternative for estimating changing volatility. It does not specify a particular GARCH model, fitting procedure, sampling adjustment, or empirical example. Daily observations also cannot directly reveal an instantaneous quantity without modeling assumptions, so the suggested approaches should be understood as model-based estimates rather than direct measurements of each day's true volatility.

Key ideas

• Under constant volatility, the standard deviation of daily log returns can estimate volatility.
• A time-varying volatility assumption calls for a model that represents changes over time.
• GARCH-family models are suggested when the data provide evidence against constant volatility.
• Daily price data support model-based volatility estimates rather than direct observation of instantaneous volatility.

Tags

# Instantaneous Volatility Estimator


# Instantaneous Volatility Estimator












Suppose a Stock follows an Itô process with instantaneous volatility $\sigma(S(t),t)$. Precisely

$$dS(t)=\mu S(t)dt+\sigma(S(t),t)S(t)dW(t)$$

I have a historical data for the values of $S(t)$.How can I estimate the instantaneous volatilities $\sigma(S(t),t)$ that took place on each day in this historical daily data series?

## Answer by Neeraj (score 0)

https://quant.stackexchange.com/a/35353

This is upto you how you defined $\sigma(S(t), t)$. As @james42 has pointed out, if volatility is constant, ie $\sigma(S(t), t) = \sigma $, then you can compute $\sigma$ by taking standard deviation of daily log return $r_t$ defined $r_t = ln(S_t / S_{t-1})$, where $ln$ is natural logarithm. If there are empirical evidences to suggest that volatility is not constant then you can use plethora of GARCH family models available in the literature.

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