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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:

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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.

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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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  • $\begingroup$ 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. $\endgroup$
    – emcor
    Nov 20, 2014 at 10:26
  • $\begingroup$ I'm sorry. $K_1$ is defined to be strictly less than $K_2$ $\endgroup$ Nov 20, 2014 at 16:48

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