When Call Options with Different Strikes Have Equal Prices

Summary

The document considers the no-arbitrage relationship between prices of call options on the same stock with the same expiry but different strikes. A lower-strike call should be worth at least as much as a higher-strike call, since its payoff is never smaller. The question is why this relationship is stated as a weak inequality rather than a strict one.

The response gives a limiting case in which equal prices are possible: if both options finish out of the money and volatility is zero, both have zero value. It adds that the price inequality is strict when there is positive probability that the lower-strike option pays off. This clarifies that the weak inequality allows degenerate cases, while strictness requires a chance of a payoff difference. The discussion is brief and does not develop the arbitrage proof or examine effects such as dividends, rates, or trading frictions.

Key ideas

• For calls with the same expiry, a lower strike cannot have a lower no-arbitrage value.
• Equal prices can occur when both calls are certain to expire worthless.
• A positive probability that the lower-strike call pays off makes the price inequality strict.
• The explanation relies on standard option payoff ordering and does not analyze market frictions.

Tags

# When $C(K_2) = C(K_1)$ for call options with the same expiration date


# When $C(K_2) = C(K_1)$ for call options with the same expiration date












The exercise is to show $C(K_1) \geq C(K_2)$ where C(K) denotes the value of a call option on a stock price S with strike price K. We assume the expiry is the same for both.

I have proved this by assuming the contrary ($C(K_2) > C(K_1)$) and then shown it creates an arbitrage opportunity. My argument is similar to the following:

What i don't understand is why the original statement includes the equality. In the case where $C(K_1) = C(K_2)$ we would still have the same cashflows at the expiry, while the cashflow would be zero at time 0. In other words there is possible to make money without losing any . The only thing I can think of is that this doesn't hold if there are trading costs but in that case, depending on the trading costs it wouldn't hold for the strict inequality where the cashflows were smaller than the trading costs either.

## Answer by Mark Joshi (score 5, accepted)

https://quant.stackexchange.com/a/15544

If both options are out of the money and volatility is zero then both are worth zero.

If there is a positive probability that the lower strike option pays off then the inequality is indeed strict.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.