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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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    $\begingroup$ Have you considered shifted lognormal models, e.g. quant.stackexchange.com/q/43785/848 ? $\endgroup$ – Bob Jansen Aug 27 '20 at 15:01
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    $\begingroup$ What are you trying to model? An example that allows for negative prices is the Ornstein-Uhlenbeck mean-reverting process, which is extensively used for commodity pricing. $\endgroup$ – gte Aug 27 '20 at 15:32
  • $\begingroup$ More info needed. Are you looking for a closed-form solution, or just want to simulate? Practitioners would use shifted lognormal or some kind of normal model (e.g. Vasicek). The good thing about the shifted lognormal is that you actually control how much negative the prices can go. This can be quite useful because many financial processes aren't expected to go too much negative (e.g. interest rates). $\endgroup$ – user2743931 Aug 27 '20 at 19:46
  • $\begingroup$ @gte I am trying to model electricity prices. But how would the OU allow for negative commodity prices. Especially when afterwards trying to price futures with the power prices as the underlying. $\endgroup$ – Question Anxiety Aug 30 '20 at 18:01
  • $\begingroup$ In this case, you are probably looking for spikes, hence a mean-reverting jump-diffusion model might be more accurate (for example, Geman and Roncoroni, 2006). Also, I bumped into this (sciencedirect.com/science/article/pii/S0140988311001721) comparison of different models that consider negative prices that might be helpful. $\endgroup$ – gte Aug 31 '20 at 10:38

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