Upper Bounds on Put Prices When the Underlying Can Turn Negative
Summary
The document explains why a put option on an underlying that cannot fall below zero cannot be worth more than its strike. At the lowest possible underlying value, the put payoff reaches the strike; because the option’s value is based on possible discounted payoffs, it cannot exceed that maximum payoff under the stated assumption. This provides an intuitive payoff bound for an in-the-money put.
The bound changes when the underlying can take negative values, as seen in episodes involving negative interest rates and oil futures. The discussion notes that options near zero strikes traded above their strikes when the related futures prices became negative, and that exchanges listed negative-strike options. It also points out a separate qualification: with negative interest rates, delayed cash can be worth less than cash today, so an option value may exceed its strike even without a negative underlying, given a sufficiently long expiry. The explanation is qualitative and does not derive discounted option-pricing bounds.
Key ideas
- For a nonnegative underlying, the maximum put payoff is capped by the strike.
- A payoff bound constrains option value when based on the maximum possible payoff.
- If the underlying can become negative, a put payoff can exceed its strike.
- Negative futures prices have led to options near zero strikes trading above those strikes.
- Negative interest rates can also affect strike-based price bounds through discounting.
Tags
Full text
# Can an In-the-Money Put Option's price $>$ its Strike Price? # Can an In-the-Money Put Option's price $>$ its Strike Price? The screenshot below suggests thatan ITM put option's price can't overstep its strike price? Why or why not? ## Answer by will (score 7, accepted) https://quant.stackexchange.com/a/58369 Under the assumption that the underlying cannot have a negative value, then the value of a put option cannot be greater than the strike. The reason behind this doesn't require maths, it's fairly simple: the lowest possible value of the underlying is zero. At that price, you make the maximum possible profit, of K. The value of the option is the probability weighted average of the payoffs - we have just explained that the maximum payoff is K, so there is no possible probability distribution that can average more than that. Therefore the maximum possible value is the maximum payoff, which is K. If we remove that assumption that the underlying can trade negative, then the above reasoning goes away. This happened when the possibility of rates going negative first became apparent, and suddenly zero strike swaptions became a real thing. It happened again earlier this year when oil futures traded down to negative \$40/bbl. For a couple of months after that several options with strikes near zero (ie the 50c, \$1, \$1.50) traded at prices above their strikes. The exchanges (nymex and ice) actually listed negative strike options, though the volume that traded on them was extremely small, and market makers didn't go anywhere near them. EDIT: and to complete the answer, i guess i should include @Quantuple's comment above - if you have negative interest rates, such that \$1 today is worth less than \$1 in the future, then an option of any strike can ahve a value higher than the strike, if your time to expiry is large enough.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.