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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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  • $\begingroup$ What kind of asset has prices that vary between -1 and +2. Is it electricity? Is it actually the difference between the price of two assets, both of which are always positive. (As the price of an asset should always be). In any case the Black Scholes framework is probably not applicable in this situation, you will need a different model. $\endgroup$ Commented Sep 18, 2024 at 11:01
  • $\begingroup$ Hi mate...It’s quite similar, the spread between natural gas prices of two countries… if country 1 is more expensive than country 2, the spread is negative; if country 1 is cheaper, the spread is positive. I want to study the volatility of this spread. $\endgroup$ Commented Sep 18, 2024 at 11:06
  • $\begingroup$ Sounds like your spread is probably an already (relatively) stationary series, in which case you're fine just taking a simple difference between observations to calculate your volatility. You may find the Bachelier pricing model more useful than BSM, this assumes normally distributed prices rather than returns, and was used for a little bit during/after WTI went negative in 2020. $\endgroup$ Commented Sep 18, 2024 at 11:11
  • $\begingroup$ So, using the model you suggested, I could use only the difference between consecutive prices in my data series. Regarding the question of converting it to an annual format or aligning it with the expiration date mentioned in my original post? $\endgroup$ Commented Sep 18, 2024 at 11:24
  • $\begingroup$ You cannot have negative prices in Black Scholes. quant.stackexchange.com/a/74179/54838 shows this in the context of interest rate options. You should not use historical vol for options on either case, see quant.stackexchange.com/a/76367/54838. $\endgroup$ Commented Sep 18, 2024 at 18:08

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