# 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.

• I dont see a clear definition of $K_1,K_2$, if it is $K_1\leq K_2$, then $C(K_1)\geq C(K_2)$ is true. – emcor Nov 20 '14 at 10:26
• I'm sorry. $K_1$ is defined to be strictly less than $K_2$ – user1937237 Nov 20 '14 at 16:48