Finding a Put Spread Arbitrage from Strike Convexity
Summary
The document analyzes whether two European puts on the same non-dividend-paying stock imply an arbitrage: the lower-strike put costs eight dollars and the higher-strike put costs nine dollars, with strikes of eighty and ninety. The accepted explanation constructs a zero-cost position by buying eight higher-strike puts and shorting nine lower-strike puts. At expiry, the payoff is zero above the higher strike, positive between the strikes, and nonnegative below the lower strike, yielding an arbitrage under the stated prices.
The discussion also corrects a confusion about using convexity to compare option prices at scaled strikes. The convexity inequality has a direction that depends on whether the scaling weight lies between zero and one; applying the same direction to a factor greater than one produces a false contradiction. A payoff diagram makes the spread’s logic intuitive, while the convexity argument provides a broader pricing constraint. The example does not discuss transaction costs, margin, or market frictions that could affect whether the theoretical arbitrage is practically exploitable.
Key ideas
- The example prices violate a no-arbitrage relationship between puts at different strikes.
- Buying eight higher-strike puts and shorting nine lower-strike puts creates a zero-cost portfolio at the stated prices.
- The portfolio payoff is nonnegative at expiry and positive in some stock-price regions.
- Convexity inequalities must be applied with the correct direction for the scaling weight.
- Transaction costs and trading constraints may affect practical arbitrage execution.
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# How to Take Advantage of Arbitrage Opportunity of Two Options # How to Take Advantage of Arbitrage Opportunity of Two Options I got the following interview question and corresponding solution, but I have a different understand that might be wrong, so I really appreciate your advice on it: A European put option on a non-dividend paying stock with strike price 80 dollars is currently priced at 8 dollars and a put option on the same stock with strike price 90 dollars is priced at 9 dollars. Is there an arbitrage opportunity existing in these two options? Solution: since the price of a put option as a function of the strike price is a convex function, and since a put option with strike 0 is worthless, we always have $P(0)+aP(K) = aP(K)>P(aK)$. So we have: $(8/9)*P(90) = (8/9)*9 = 8>P(80)$ Since the put option with strike price 80 dollars is currently priced at 8, it is overpriced and we should short it. The overall arbitrage portfolio is to short 9 units of put with $K=80$ and long 8 units of put with $K=90$. At time 0, the initial cash flow is zero. At maturity date, we have three possible scenarios: $S_T>=90$,payoff=0 (no put is exercised) $90>S_T>=80$, payoff = $8*(90-S_T)>0$ (puts with K=90 are exercised) $S_T<80$, payoff = $8*(90-S_T)-9*(80-S_T)>0$ (all puts are exercised) The final payoff $>=0$ with positive probability. So it is clearly an arbitrage opportunity. But can I understand the question as follows? I think the put option with strike 80 is underpriced (instead of overpriced), why? because: by using $aP(K)>P(aK)$ mentioned above, we have:$(9/8)*p(80)>=p[(9/8)*80]=p(90)=8$ So we have $P(80)>=8$, so it is under priced. I'm wondering if I'm wrong? ## Answer by Magic is in the chain (score 4, accepted) https://quant.stackexchange.com/a/50311 I think it is far easier to understand by just drawing the payoffs. You have two put options: - A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and - a put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts payoff exceeds the 80 strike payoff by 10) when both are in the money. Additionally the 90-strike put option pays something in the region between 80 and 90 and the 80-strike put pays nothing in this region. Now two ways to proceed to create arbitrage: 1) zero cost, positive payoff, 2) negative cost, non-negative payoff. let's go with the first: Buying 8 options of 90-strike will cost 8 times 9=72, selling 9 options of 80-strike will generate the same amount (9 times 8=72). So the cost of the strategy is zero. The payoff diagram is as follows: PS: And for the convexity logic, if you plot the given option prices as a function of strike, you get a straight line. Convexity would imply the price of 80-strike should be slightly lower? ## Answer by pi_6Squre (score 0) https://quant.stackexchange.com/a/81761 I think the reasoning with the inequality $$P(aK) > aP(K)$$ only works for a in the range $[0,1]$, as the convexity inequality determines. Actually for $a > 1$ the inequality sign should be reversed. So it should not create contradictions.
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