How Historical and Implied Volatility Relate in Option Analysis

Summary

The document distinguishes historical volatility, calculated from past stock returns, from Black–Scholes implied volatility, inferred from traded option prices. Historical volatility summarizes observed movements, while implied volatility reflects the option market’s price for exposure to future uncertainty. The two measures can complement one another, but the text cautions against assuming a fixed relationship between them.

It notes that historical volatility has often been lower than implied volatility in broad terms, while emphasizing that comparisons depend on definitions and measurement choices. Implied volatility is model-dependent: Black–Scholes derives it under geometric Brownian motion assumptions, and other pricing models can produce different implied volatility values from the same option price. The document also connects implied volatility with expectations of future realized volatility, citing volatility and variance swap trading as context. It gives no dataset or quantitative evidence for a general historical-versus-implied spread, so the directional observation should not be treated as a universal rule or guaranteed forecast.

Key ideas

• Historical volatility is estimated from realized past price changes, while implied volatility is backed out from option prices.
• The measures describe different information and can be used together.
• Their comparison depends on definitions and measurement choices.
• Black–Scholes implied volatility depends on the model assumptions used to interpret option prices.
• Implied volatility can be viewed as related to expected future realized volatility, but the document provides no universal relationship.

Tags

# what's the relationship between forecasted stock volatility and implied volatility?(option)


# what's the relationship between forecasted stock volatility and implied volatility?(option)












what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied volatility and underlying asset volatility?(forecasted or historical)

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/10147

One is computed from historical stock time-series, that is, from observed past, the other is computed from traded option prices (prices of bets made on the stock with payoff at a future time), that is, from a view on the future paid for with cash.

They are both equally important and useful (if one has enough data to compute them). Loosely speaking, historical one tends to be lower than the implied one, but the comparison needs to be done carefully, given their specific definition elements. A priori there is no expectation for an actual "relationship" between the two. They should rather be viewed as complementing each other.

BS formula is the way to relate traded option prices and implied volatility. This is done assuming a specific geometric Brownian motion dynamics for stock, and many other things. That's why one refers to implied volatility as BS-implied volatility not to be confused with possibly other model implied volatility (like say, Bachelier model, whose dynamics are assumed arithmetic Brownian motion). Implied volatilities can be viewed as expectations of future realized volatilities (see also trading of variance/volatility swaps).

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