Volatility Estimation for Prices That Can Be Zero or Negative

Summary

The document raises a practical problem in measuring historical volatility when an instrument’s prices fluctuate around zero and may be negative. Log returns are undefined for nonpositive prices, while simple percentage changes can become extremely large when the previous price is near zero. The writer describes adding a constant to prices before calculating log returns, but notes that the chosen constant changes the resulting volatility and asks what alternative method is appropriate.

It also asks how to match historical volatility estimates to an option’s expiration, and whether to express volatility over the option’s life or annualize it. These are useful questions about return definitions, measurement windows, and time scaling. However, the document contains no replies or worked analysis, so it does not recommend a particular volatility measure, resolve the arbitrary-offset problem, or explain how to select a lookback period. Its value is identifying the limitations of conventional return calculations for prices that can cross zero.

Key ideas

• Log returns cannot be calculated directly when prices are zero or negative.
• Percentage changes can become unstable when the starting price is close to zero.
• Adding a constant before computing log returns makes the volatility estimate depend on that arbitrary choice.
• The document asks how to align a historical volatility window with an option’s expiration.
• It raises the distinction between annualized volatility and volatility over a specific option horizon.

Tags

# Volatility in BS model, prices zero or negative


# Volatility in BS model, prices zero or negative












I would like to raise the following question:

I need to analyze the historical volatility of some prices. These prices fluctuate approximately between -1 and +2. The issue is that when calculating the logarithmic or arithmetic return, I get very high annual volatility values (e.g., 640%). If I use the relative rate of increase, due to values being close to 0, the returns also come out very large when dividing by a number close to zero.

For the calculation of the logarithmic return, I added a k (constant) to the price to eliminate the mathematical error. The truth is that this k can be arbitrary; if I set a very large k, the volatility is significantly reduced… but… what number k should be used? Do you have any other solutions?

On the other hand, I have historical data for quite a few days, but if my option expires in 1 month, wouldn’t it be logical to analyze the volatility using the historical data from 1 month ago until today?

Furthermore, if my option expires in 6 months… Do I have to annualize it? Isn’t it better to convert it to 1/2 years?

I hope you can help me; these may be basic questions, but I am just starting with options.

Thank you!

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