Detecting Call-Spread Arbitrage from Strike Convexity
Summary
The document explains an arbitrage test for call options sharing an expiry. Call prices should be convex as a function of strike, so the price at a midpoint strike cannot exceed the average of the prices at two equally spaced outer strikes. The example violates this bound: the middle-strike call is priced above the convexity limit implied by the lower- and higher-strike calls.
The proposed trade sells two middle-strike calls and buys one call at each outer strike. The initial premium received exceeds the cost of the purchased options. At expiry, the combined payoff is non-negative across the underlying-price regions: it rises and falls between the outer strikes and is zero outside them. This creates an upfront gain without a negative terminal payoff under the stated prices and payoff assumptions. The document does not address transaction costs, bid–ask spreads, financing, or whether the quoted prices are executable, and it does not establish that this is the only possible arbitrage portfolio.
Key ideas
- Call option prices must be convex across strikes for options with the same expiry.
- A midpoint call price above the average of equally spaced outer-strike call prices violates convexity.
- Selling two middle-strike calls and buying one call at each outer strike exploits the stated violation.
- The resulting expiry payoff is non-negative for every underlying price in the example.
- Transaction costs, quote executability, and other market frictions are outside the analysis.
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# Arbitrage opportunity interview question
# Arbitrage opportunity interview question
I have seen this interview question mentioned in a couple of places:
There are three call options on the market, with the same expiry and with strikes 10, 20, and 30. Suppose the call option with strike 10 costs \$12, the call option with strike 20 costs \$7, and the call option with strike 30 costs \$1. Is there an arbitrage opportunity?
The answer apparently is to buy two of the call options priced at \$20, and sell one of each of the call options priced at \$10 and \$30.
How does one arrive at this answer and is it unique?
Thanks.
## Answer by Gordon (score 31, accepted)
https://quant.stackexchange.com/a/22329
A similar question for put option has been discussed in this question: Finding Arbitrage in two Puts. Basically, the call option payoff is a convex function of the strike. Then the call option price is also a convex function of the strike. Specifically, let $C(K)$ denote the call option price with strike $K$. Then for $ 0 < K_1 < K_2$, \begin{align*} C\left(\frac{K_1 + K_2}{2}\right) \le \frac{1}{2}\big(C(K_1) + C(K_2) \big). \end{align*}
For the example, let $K_1 = 10$ and $K_2 = 30$. Then \begin{align*} C(20) &= C\left(\frac{K_1 + K_2}{2}\right)\\ &\le \frac{1}{2}\big(C(K_1) + C(K_2) \big)\\ &= \frac{1}{2} (12 + 1) = 6.5. \end{align*} However, $C(20) = 7$, which contradicts to the above. Therefore, there is an arbitrage opportunity.
For an arbitrage strategy, we should short (i.e., sell) the option that is over-priced, and long (i.e., buy) the option that is under-priced. Specifically, we short two options with strike 20, and long one option with strike 10 and long another option with strike 30. At the start, we have the profit \begin{align*} 2 \times 7 - 12 - 1 = 1 $. \end{align*} At the option maturity, the payoff to us is \begin{align*} (S_T-10)^+ + (S_T-30)^+ - 2 (S_T-20)^+ = \begin{cases} 0, & \mbox{if } S_T \leq 10,\\ S_T-10, & \mbox{if } 10 \le S_T \le 20, \\ 30-S_T, & \mbox{if } 20\le S_T \le 30,\\ 0 , & \mbox{if } S_T \ge 30, \end{cases} \end{align*} which is always non-negative. Then, we have a guaranteed profit at the start and potential further profit at the option maturity, while without any liabilities.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.