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Let $C_K(t,T)$ be the value of a European call with strike $K$ and maturity $T$ on a stock with value $S_t$ at time $t$. Then for all $t\leq T$ we have $$C_K(t,T)\leq S_t.$$

$\textbf{Proof}$: We derive an arbitrage opportunity by contradiction.

Assume $$C_K(t,T)>S_t$$ for some $t\leq T$. Consider the following:

At this time $t$: write a call option with strike $K$ and maturity $T$ on the stock and buy said stock. Since we cash the premium $C_K(t,T)$ of the call we have $C_K(t,T)-S_t>0$ left.

At time $t=T$, we do the following:

  1. If $S_T<K$, the call is worthless and nothing happens.
  2. If $S_T \geq K$, the call is made and we sell the holder the stock (which we still own).

In both cases, no money is lost and we end up with money. This is an arbitrage opportunity.

Can someone check if this proof is decent and correct?

$\textbf{Edit:}$ I did not think this was necessary, but since a comment was made I feel obliged to add the following.

This is a proof by contradiction. Since the principle of arbitrage is an axiom to financial theory, all true statements (in the theory) are implied by this principle. In other words: it is enough to show that the negation ($C_K(t,T)>S_t$) is in contradiction with the principle of arbitrage.

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  • $\begingroup$ why is $C_K(t,T)-S_t>0$ ? This implies that $C_K(t,T) > S_t$, which is exactly the opposite of the original relation you're tring to prove... $\endgroup$ – will Jan 18 at 18:45
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    $\begingroup$ @will It's a proof by contradiction. Proving $P\implies Q$ is equivalent to showing $P\land \neg Q$ is not true. $\endgroup$ – PaleBlueDot Jan 18 at 18:55
  • $\begingroup$ @will in this case $P$ is the principle of arbitrage and $Q$ is the proposition in the yellow box. $\endgroup$ – PaleBlueDot Jan 18 at 18:56
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    $\begingroup$ The proof is fine; however, it does rely on some implicit assumptions- e.g., limited liability associated with stock ownership, meaning stock price can’t go negative. $\endgroup$ – Magic is in the chain Jan 19 at 17:00
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    $\begingroup$ Ah yes, my bad, should have read more typed less! $\endgroup$ – will Jan 19 at 20:55
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What you need to note is the following: \begin{align*} S_T - \max(S_T-K, \,0) &= S_T + \min(K-S_T, \,0)\\ &=\min(K, \, S_T) >0. \end{align*}

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  • $\begingroup$ Ah, this too is a nice proof. But I'm not asking for another proof (though I do appreciate it), instead, I am asking to verify my proof. At this point I'm pretty sure my proof is correct though. $\endgroup$ – PaleBlueDot Jan 18 at 19:03
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The call price is decreasing with respect to the strike so for every strike the value of the option is inferior than the value for strike equal to zero which is the spot price.

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  • $\begingroup$ Yes, this too is a nice proof. But I was asking not for another proof (though I appreciate it, but instead was looking for critique on my proof. Though at this point I am sure the proof is correct. $\endgroup$ – PaleBlueDot Jan 18 at 19:07
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    $\begingroup$ You will not pay more money to have less gain at maturity!!! $\endgroup$ – Valometrics.com Jan 18 at 19:10
  • $\begingroup$ Yes, that can more or less be made into a formal proof. It is indeed a corollary of the law of one price that if $K_1<K_2$, then for all $t\leq T: C_{K_2}(t,T)<C_{K_1}(t,T)$. $\endgroup$ – PaleBlueDot Jan 18 at 19:14

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