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Option Pricing Models for Assets with Negative Prices

Article Quant Q&A · Author: Flux
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Summary

The document considers how to price options when the underlying can fall below zero, a situation that conflicts with the standard lognormal price assumption in Black–Scholes–Merton. It presents two alternatives: the Bachelier model, which uses a normal price distribution and can accommodate negative values, and a shifted lognormal model, which retains a Black–Scholes style framework after adjusting the price level.

The answer notes that a normal approximation is more reasonable when price fluctuations are modest and the strike is not far from at the money. It also stresses using normal volatility with Bachelier pricing rather than lognormal volatility. The shifted lognormal approach has been used for interest rate options when rates are negative, but its shift must be chosen carefully: an unsuitable shift can distort the model or make it unusable. The document offers model choices and cautions, but no empirical comparison or calibration procedure.

Key ideas

  • Lognormal price models cannot directly represent negative underlying prices.
  • The Bachelier model uses a normal distribution and supports negative prices.
  • A normal approximation is most suitable for moderate fluctuations and strikes near the money.
  • A shifted lognormal model can preserve a Black–Scholes style approach, but results depend on the shift.
  • Bachelier pricing requires normal volatility rather than lognormal volatility.

Tags

OptionsDerivatives pricingVolatilityFixed income
Full text
# Should a normal distribution be used for valuing options on assets that can potentially have negative prices?


# Should a normal distribution be used for valuing options on assets that can potentially have negative prices?












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?

## Answer by Andreas (score 2)

https://quant.stackexchange.com/a/53552

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.

## Answer by Jan Stuller (score 1)

https://quant.stackexchange.com/a/53551

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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