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I am trying to understand the stochastic model of a financial market in one period by [Föllmer, Schied]. They introduce call and put options for the primary assets, which are non-negative. They do not specify if the strike can be negative or if it should be positive. I think its economically not useful to have negative strike prices for options whose underyling is one of the primary assets. Since $(S -K)^+>0$ for $S\ge 0$ and $K<0$ it would be basically a sure payoff. Is this a fair assessment?

Next, they define basket options, whose underyling is the value of some portfolio. Some portfolio can have negative inital value. Thus, I believe that it could be useful to have a negative strike price. Is this true?

Wikipedia page on strike prices does not specify it either.

I can specify the model if necessary. Let me know. Thanks for reading.

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    $\begingroup$ This would imply that the underlying value can also be negative which is impossible a stock can't lose more than 100%. You can however have a strike that is below or above the current price of the underlying. if you buy a call under the strike price you are buying an option that is deep in the money. The exercise price (How much you get if you execute the option right now) will be positive. Naturally these are more expensive. You can look deeper in the concept of lognoram returns if you are interested why negative prices don't really work. Negative returns are another thing. $\endgroup$ – Jorisdrees Dec 20 '19 at 13:57
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    $\begingroup$ I guess if they were interest rate or currency options. $\endgroup$ – AlRacoon Dec 20 '19 at 14:27
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    $\begingroup$ Yes, if the underlying asset (whether composite or not) can have a value represented as negative, then it would be reasonable for an option contract to have negative strikes. This is not a situation you find in exchange-traded options, nor have I seen it in the exotic options, unless you count swaptions. The accounting and cashflow pattern can be done however all contract parties agree to do it. $\endgroup$ – Brian B Dec 20 '19 at 14:53
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    $\begingroup$ If you chose a model where the stock price is always positive, then indeed you always exercise the option. But think of caplets/floorlets which are options on interest rates. The latter may be negative and hence, you could use negative strikes. But in the stock realm, you won't find negative strikes $\endgroup$ – Kevin Dec 20 '19 at 14:54
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    $\begingroup$ @BrianB - the ICE has listed options on brent/wti spreads as well as calendar spread options on a number of different commodities. all of these can and regularly do have negative strikes. $\endgroup$ – will Dec 20 '19 at 16:51
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If the underlying asset cannot be negative, then an option on it with a negative strike would be meaningless. A call would always be in-the-money with no chance of being worthless, and a put would always be worthless.

You can, however, have options on assets that can have negative values, like interest rates (a relatively recent phenomenon), or spread options, more common in commodities like WTI-Brent.

Note that the payout formulas are the same - if you have a spread call option with a strike of -2, the payout would still be $\max{[(S-K),0]}$. If the spread was, say, -1, then your payout would be $\max[({-1} - {-2}),0] = 1$. If the spread were -3, the payout would be $\max[({-3} - {-2}),0] = 0$.

The models for pricing these options are different, however, since the standard Black-Scholes model assumes an always-positive underlying price and log-normal rates of change.

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If your underlying can not take negative values then an option with a negative strike would be a sure payoff. There is however no reason preventing the option from having a negative strike. Id be happy to buy many such options from you.

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  • $\begingroup$ You may buy them, but for strike price $K<0$ please consider that I'll have to ask you to pay the value of the underlying, a tansaction fee, plus an extra $|K|$ for it. $\endgroup$ – Friedrich Dec 21 '19 at 22:07
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    $\begingroup$ Not the right price : ) $\endgroup$ – roz Dec 22 '19 at 23:06
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In addition to some outrights having negative prices (as pointed out by CasusBelli and discussed in a JPM piece from 2011), spreads can take negative values and options on spreads trade (e.g. CSOs in CL).

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I've heard of OTC options with negative strikes in electricity markets. If you have a lot of generation from non-intermittent sources, like hydrological and nuclear, negative prices are almost inevitable during periods with little demand.

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