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Choosing Price Distributions That Allow Negative Values

Article Quant Q&A · Author: Question Anxiety
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Summary

The document raises a modeling question: how to represent prices that may become negative when the usual geometric Brownian motion framework keeps prices positive. It considers whether a different distribution, such as a beta distribution, could be used, and asks how to begin constructing a stochastic differential equation for that purpose. The question also notes that taking logarithms and applying Itô’s lemma is tied to the positive-price formulation and does not directly accommodate negative prices.

No derivation, proposed process, or empirical comparison is supplied. The material is therefore useful as a framing of a modeling issue rather than as a worked method. It leaves open how the distribution’s support should be chosen, whether the process should have bounded or unbounded negative values, and how the resulting model would be calibrated or checked against data. Readers should not treat it as evidence that a beta distribution is suitable for any particular market or price series.

Key ideas

  • Geometric Brownian motion maintains positive prices, which can be unsuitable when negative prices are possible.
  • A logarithmic transformation does not directly solve the problem of modeling negative price values.
  • The document asks how to construct an alternative stochastic process, including whether a beta distribution could help.
  • It provides no proposed dynamics, calibration procedure, or evidence comparing candidate distributions.

Tags

StatisticsFutures
Full text
# Alternatives to Lognormality for negative Prices


# Alternatives to Lognormality for negative Prices












If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a Geometric Brownian Motion. The classic way would be to apply the log transformation followed by Itos Lemma, but since this would not allow for prices to be negative how would I need to start?

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