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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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  • $\begingroup$ Maybe start with Ornstein Uhlenbeck process or Normal model, and then take it from there. Normal model is well developed so you can find analytical formulae for different metrics that would be of interest. Re-OU process, oil prices might not look like OU process over shorter period, but they do look similar, if not exactly like, OU process over a longer period. You can also try shifted version of log normal. $\endgroup$ Jun 20, 2020 at 17:28
  • $\begingroup$ The domain of the beta distribution is $[0,1]$ and thus positive. The negative price of WTI comes from two processes, the oil price and the storage price, which you can model. You can always have a mixture, or a convex combination of two distributions, with one of the distribution defined for negative prices. $\endgroup$
    – Hans
    Jun 20, 2020 at 17:38
  • $\begingroup$ A few days before the oil futures price went negative, the exchange already sent out a memo saying they would switch from BS to the Bachelier (i.e. normal instead of lognormal prices) model for the calculation of implied volatilities for oil options on futures. cmegroup.com/content/dam/cmegroup/notices/clearing/2020/04/… $\endgroup$
    – nbbo2
    Jun 20, 2020 at 20:21
  • $\begingroup$ BTW, you mention "distribution of returns". If prices can go non-positive ($\le0$) the notion of returns becomes meaningless and cannot be used any more. You should work with distributions of prices (or of price changes) instead. $\endgroup$
    – nbbo2
    Jun 21, 2020 at 16:28

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