Covered Calls and the Stock Price Upper Bound for Call Options

# Upper Bound on European/American Call Option (Hull)


# Upper Bound on European/American Call Option (Hull)












I recently began reading Hull's derivatives textbook, and found a line that he didn't expand on much. Let $c$ be the price of a European call, $C$ be the price of an American call, and $S_0$ be the current price of the stock underlying the call. Hull then says:

> "No matter what happens, the option can never be worth more than the stock. Hence, the stock price is an upper bound to the option price: $c \leq S_0$ and $C \leq S_0$. If these relationships were not true, an arbitrageur could easily make a riskless profit by buying the stock and selling the call option."

I had no clue how this would end up in an arbitrage, so I tried prove it myself for the American call case. But when I did, I got something different from what Hull claimed, so I must have made a mistake somewhere:

Suppose the call has strike $K$. Initially, we buy the stock for $S_0$ and write the call for $c$, so that we currently have $C-S_0$. At maturity, the stock's price has changed to $S_T$, so that so far, we have $C + (S_T - S_0)$.

Now, we either have $K < S_T$ or $K \geq S_T$. Consider the case where $K \geq S_T$. In this case, the owner of the call will not exercise, so our final profit is $C + (S_T - S_0)$.

Isn't the final results conditional on $S_T$? How does this make the profit riskless?

## Answer by KaiSqDist (score 0, accepted)

https://quant.stackexchange.com/a/79927

I don't agree that the final results are conditional on $S_T$. The whole point of a covered call (short call long underlying), is to hold the asset in the event the option counterparty exercises the call i.e. $S_T>K$ as you have mentioned. You are not expected to sell the underlying asset when it is worth $S_T$. Therefore, the expression $C+(S_T-S_0)$ is erroneous.

The riskless profit will therefore always be given by:

$$C-S_0$$

## Answer by Herberto Silva (score 0)

https://quant.stackexchange.com/a/81716

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