Why Call Option Prices Must Decrease with Strike
Summary
The document explains why otherwise identical call options should not become more expensive as their strike prices rise. If a lower-strike call costs no more than a higher-strike call, an investor can buy the lower-strike option and sell the higher-strike option, receiving cash up front. The options’ expiration payoffs then leave a nonnegative payoff in every spot-price range, in addition to the invested proceeds.
This payoff comparison establishes a strike-ordering condition and shows how a violation can create an arbitrage. The argument assumes the options are comparable, the trades can be executed at the observed prices, and the proceeds can be invested as described. The answer notes that sufficiently negative interest over a long horizon can undermine the simple cash-investment argument; a zero-cost ratio of option positions can address that special case. The document does not discuss transaction costs, funding constraints, or market frictions that could affect whether a theoretical violation is exploitable.
Key ideas
- A call with a lower strike should be worth at least as much as an otherwise identical call with a higher strike.
- Buying the lower-strike call and selling the higher-strike call produces a payoff that is nonnegative across expiration prices.
- A violation of this price ordering can create an arbitrage under the stated trading and investment assumptions.
- Very negative interest rates can require adjusting the simple trade structure.
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Full text
# Breaking the monotonity of option pricing?
# Breaking the monotonity of option pricing?
Suppose there are two almost identical option, their only difference is their strike price. Otherwise, all other properties -expiry, underlying, etc.- are identical. The two options are traded on a market, with observerd prices P1 and P2, P2 < P1. Can you imagine a situation, when such a price ordering would reverse, P2 > P1? (Assume you actually observe both prices on the marketplace.)
I could not really come up with meaningful scenarios, other than a sudden dump of Option 2 or a sudden buy surge of Option 1 could temporarily break the ordering, which would be quickly arbed away.
The above is an irrational explanation, I would be curious for any rational and other irrational explanations.
This question was asked from me. It seems like a trick question and somehow the monotonity of option pricing can be broken, so here it is, be creative! :)
## Answer by Chris Taylor (score 2)
https://quant.stackexchange.com/a/82525
If you have two call options with strike $K_1 < K_2$ then you expect the first one to be worth more. If the first option is not worth more, i.e. they have prices $P_1 \leq P_2$, then you can construct an arbitrage by buying the first option and selling the second, which generates an up-front cash payment of $P_2 - P_1 > 0$.
Now wait a period of time $T$ for the options to expire (during which you can invest your cash at rate $r$). Your final profit, as a function of the final spot price $S$, is
$$ P\&L = \begin{cases} (P_2-P_1)(1 + rT) & {\rm if\;} S \leq K_1\\ (P_2-P_1)(1+rT) + S - K_1 & {\rm if\;} K_1 < S \leq K_2\\ (P_2-P_1)(1+rT) + K_2-K_1 & {\rm if\;} S > K_2 \\ \end{cases} $$
This is positive in all cases, so there is an arbitrage and there cannot be any "rational" reason for this situation to exist (since arbitrages do not exist in a "rational" market).
Edit: If $1 + rT<0$ this isn't strictly an arbitrage, since you could lose money in the first case. Albeit this would be an extremely unusual situation (requiring a long time to expiry and a very negative interest rate) and it can be fixed up by buying and selling the two options in a ratio which results in zero up-front cost, and still generates zero or positive P&L at expiration.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.