I was reading the example problem they gave, and I will try to summarize it here:
Stock Price today is $30
Probability of an up move to $40 is 55%
Probability of a down move to $22.25 is 45%
There is a call option for $30 in 1 year selling for (mispriced) $6.50. Book asks how we can arbitrage this.
The book says:
Short 100 call options @ $6.50 per call
They calculate the “delta” (for hedging) as (10-0)/(40-22.50) = .5714 shares per option, so purchase 57.14 shares
“A portfolio that is long 57.14 shares of stock at $30 per share and short 100 calls at $6.50 per call has a net cost of:”
Net portfolio cost = (57.14 * $30) - (100 x $6.50) = $1,064
The book then states: “The values of this portfolio at maturity if the stock moves up to $40 or down to $22.50 are:”
portfolio value in up-move = (57.4 x $40 ) - (100 x $10) = $1,286
portfolio value in down-move = (57.4 x $22.50) - (100 x $0) = $1,286
In the up scenario, if you short a call option, the person on the other side has the right to buy at $30, correct? So they would exercise the option and you would have to deliver 100 shares of stock at $30. You only own 57.14 shares, where are they getting the other shares? How are we only losing $1,000 (100 x $10) to the short call options in this case?
Any help is appreciated. I might be missing something blatant.
It’s also the price of the option. If it had equalled the price we get by arbitrage pricing, we wouldn’t have shorted. The arbitrage price should be at a price lower than $6.50. Since this is overpriced we short the call option in such a way that we can get a positive payoff at the end of 1 year. As more and more traders see this oppotunity, they will ultimately bid the price down.
Also, the return we get by this combination is more than 20%. But if the option traded at a price calculated by the arbitrage model, we would have earned the Risk-free rate.
I understand that the price of this call option presents us with an arbitrage opportunity, but I don’t understand how when we are holding 57.14 shares of the underlying stock, and the person has the right to purchase 100 shares @ $30 from us (because of the option) that we don’t lose more money.
In my mind, if the stock goes to $40, our portfolio would be: 57.14* 40 = $2285.6 The person has the right to BUY 100 shares from us @ $30, so we would have to get another 42.86 shares in the open market @ 40, which costs us $1714.4
So we’ve gained $571.4 from owning 57.14 shares from $30 to $40, and $650 from selling the option. BUT, then we have to deliver an additional 42.86 shares @ 30, which will cost us $1285.8.
Stock Gain ($571.4) + Option Gain ($650) - Delivering 42.86 shares @ 30 ($1714.4) = -$493
It is assumed that options settle in cash so the underlying is never delivered. Only the amount of loss for the call writer will be deducted from his account i.e. S+ = $40 At initiation, portfolio value = $1,064
So, after an up-move our loss is 100 x $10 but an overall profit of 57.14 x $40 (2286 - 1000) = $1,286. Assuming a down move, an option holder will not exercise his/her option at contract expiration. We have already collected the premium of 100x$6.50 so now ( after 1 year ) the portfolio value is 57.14 x $22.50 = $1,286