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The Black-Scholes-Merton model assumes that the prices of the underlying asset at maturity are log-normally distributed. I understand that this assumes that the prices can never go below zero.

However, there are cases where the underlying asset's price can be negative. For example:

  • With (hypothetical) unlimited liability companies, the stock price can go below zero.
  • With commodity futures, the futures price can go below zero.

In these cases, is a normal distribution a better assumption than a log-normal distribution?

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  • $\begingroup$ If there is a reasonable negative lower bound to the values your asset can take, then the shifted lognormal is a potential approach to consider. Largely retains the tractable nature of BSM but allows for some degree of negative values. $\endgroup$ – James Spencer-Lavan Apr 23 '20 at 9:57
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A normal distribution is reasonable as long as the price fluctuations are not too large and the strike is not very far in or out of the money.

Aside from a shifted log-normal model you can try the Bachelier Model. It does not require an arbitrary shift (which, if chosen too large skews the model and if too small breaks it) and works for negative prices of the underlying. Just make sure you use the correct parameters in your calculations, especially the "normal" instead of the "lognormal" volatility.

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You can use a shifted log-normal model and still stay within the Black-Scholes framework, to allow for negative prices. Shifted Log-normal is for example used to price options on Interest Rates under the Libor Market Model framework (rates have been negative for a while now in the Eurozone).

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