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I am reading Hull's Options book. He introduces a one-step binomial model and a no-arbitrage argument, using the example shown in the picture below: enter image description here

Consider a portfolio consisting of a long position in $\Delta$ shares of the stock and a short position in one call option (he does not explicitly give the strike, but he clearly assumes that it is \$20).

Question 1: Why do we short one call option? Why do we not long a call or short a put?

After some calculation, $\Delta = 0.25$, and if we assume risk-free rate is 4%, then the no-arbitrage price of the option is 0.545.

If the value of the option were more than 0.545, the portfolio would cost less than 4.455 to set up and would earn more than the risk-free rate. If the value of the option were less than 0.545, shorting the portfolio would provide a way of borrowing money at less than the risk-free rate.

Question 2: could you explain how we make a profit if the price of the option is less than 0.545? When it is less than 0.545 and we short the portfolio, do we gain \$20 $\Delta$ from the share? What about the short position of the call?

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    $\begingroup$ "How does shorting a portfolio make a profit". As usual, if you short a portfolio and the value of it goes down, you make a profit. That's obvious. What is peculiar and important here is that you make money both in the StockUp and the StockDown state, which means that the profit is riskless and (since there is no state where you lose money) it is an arbitrage. $\endgroup$
    – nbbo2
    Commented Mar 28 at 13:13

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Question 1: Why do we short one call option? Why do we not long a call or short a put?

You could do the other combinations, but then you would have to:

  • Short Put > Short Stock
  • Long Call > Short Stock
  • Long Put > Long Stock

To delta hedge the portfolio and think about the individual results in terms of the riskless rate like in your second quoted paragraph.

Question 2: could you explain how we make a profit if the price of the option is less than 0.545? When it is less than 0.545 and we short the portfolio, do we gain $20 Δ from the share? What about the short position of the call?

The "portfolio" you are talking about is a short single call and long 0.25 units of the stock. Therefore, if you short the portfolio, it becomes long a single call and short 0.25 units of the stock. If the option is less than 0.545 (say 0.5 instead of 0.545), this portfolio is worth:

\begin{equation} \Pi = 0.25*20 - 0.5 = 4.5 \\ Scenario\:U\:Profit = 4.5 - 0.25*(22-20) + 1 - 4.5 = 0.5 \\ Scenario\:D\:Profit = 4.5 - 0.25*(18-20) + 0 - 4.5 = 0.5 \end{equation}

Therefore, if you lend a portfolio at the riskless rate,

\begin{equation} Interest\:Earned = \Pi*\exp(r*\tau) - 4.5 = 4.5*exp(0.04*\tau) - 4.5 \end{equation}

I would bet that the $\tau$ of the option is such that $Interest\:Earned$ is less than \$0.5. From,

\begin{equation} min[Scenario\:U\:Profit, Scenario\:D\:Profit] = \\\$0.5 > Interest\:Earned \end{equation}

What is the $\tau$ of the option please? You need to specify.

Also, the strike of the option is not \$20, it seems to be \$21 as the payoff in $Scenario\:U$ is \$1 instead of \$2.

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