Pricing Options When Underlying Prices Can Become Negative

Summary

The document raises a modeling issue for option pricing when the underlying can take negative values. It observes that the standard Black–Scholes setup models prices as lognormal, which rules out negative prices, and points to negative oil prices as a motivating case. It asks whether a different distribution, such as a beta distribution, could accommodate negative outcomes and what changes would then be needed in the model.

The text is a question rather than a proposed method or answered analysis. It does not identify a suitable distribution, explain how to derive option prices under an alternative process, or provide calibration or empirical evidence. Its useful contribution is framing the modeling constraint: allowing negative prices requires reconsidering the dynamics and pricing assumptions together, including the return distribution, rather than simply swapping one terminal distribution into the Black–Scholes formula. The appropriate approach would depend on the underlying market and model requirements.

Key ideas

• A lognormal price model cannot represent negative underlying prices.
• Negative commodity prices motivate reconsidering standard option-pricing assumptions.
• Changing the price distribution also raises questions about the process dynamics and return model.
• The document proposes no alternative model or pricing method and supplies no supporting analysis.

Tags

# Different distributions for option pricing


# Different distributions for option pricing












So the classic BS assumption of lognormal prices imply that the stock price can not be negative. Now since recently also oil prices were negative I was wondering, whether it would be possible to change this distributional assumption somehow to something (f.e. Beta distribution) which would allow for these negative prices. How would I proceed if I wanted to implement this as the whole BS model would then not be valid anymore f.e. one could not use normal distribution of returns anymore etc.

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