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Why High Annualized Volatility Requires a Lognormal Price Model

Article Quant Q&A · Author: options_student
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Summary

The question challenges the interpretation of annualized volatility as a normal range for a stock price, especially when volatility reaches or exceeds 100%. Applying a symmetric normal distribution directly to the price can imply negative outcomes, which conflict with the fact that a stock price cannot fall below zero.

The brief answer points to a likely terminology error in a referenced options text: the intended model is lognormal. A lognormal model applies the distribution to the logarithm of price, keeping modeled prices positive. The document does not develop the model, derive probability ranges, or discuss assumptions such as drift and changing volatility, so it serves mainly as a correction to the framing rather than a full explanation.

Key ideas

  • A normal distribution applied directly to stock prices can assign probability to negative prices.
  • The response identifies the lognormal distribution as the intended model for stock prices.
  • Annualized volatility alone does not specify a complete forecast of future prices.

Tags

EquitiesVolatilityStatistics
Full text
# Significance of annualized volatility over 100% on the normal distribution?


# Significance of annualized volatility over 100% on the normal distribution?












Assume stock is 50 dollars. From what I understand, an annualized vol of 20% means there is a ~68% chance the stock will be between 40 and 60 a year from now; a ~95% chance it will be between 30 and 70; and so on.

What if volatility is 100% or higher? Would that mean ~68% of the time it will be between 0 and 100 a year from now? A ~95% chance between 0 and 150? Am I interpreting this correctly? Can someone explain how this makes sense since stocks cannot go below zero?

## Answer by Steve Becker (score 2)

https://quant.stackexchange.com/a/68415

Shelly Natenberg has a typo. He means lognormal

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