Derivatives Foundation puzzles, solved step by step

It pays exactly 20 at any stock price, so 19.20 today implies a one-year rate of 20/19.20 - 1, about 4.17%. The 90 call and short 90 put make a long forward at 90; the short 110 call and long 110 put make a short forward at 110. The stock cancels and 110 - 90 = 20 is left. Buying the box is lending at 4.17% and selling it is borrowing at that rate, so it suits a lender whose other return is lower, or a borrower whose funding costs more.

Why does the payoff not depend on the stock?

Agree to buy a scooter from one friend for 90 and to sell it to another for 110, both next year, and you will make 20 whatever scooters cost by then. The box is the same pair of agreements: long the 90 call and short the 90 put is a promise to buy at 90, and short the 110 call and long the 110 put is a promise to sell at 110, so the stock comes in and goes out and only the gap between the strikes, 20, is left. Seen as spreads, the call spread pays from 0 below 90 up to 20 above 110, and the put spread pays the mirror image, 20 below 90 down to 0 above 110. Wherever the stock ends, the two add to 20.

How do you turn the price into a rate?

You pay 19.20 today and receive 20.00 in a year with no market risk, so it is a deposit. The simple rate is 20 / 19.20 - 1 = 4.17%, and the continuously compounded rate is ln(20 / 19.20) = 4.08%. A box is a zero-coupon bond built from options, and its price is the strike gap times the discount factor, so any box that trades away from the market's interest rate is mispriced. Prices consistent with this rate, for an illustrative stock at 100 with 20% volatility, are a 90 call at 16.11, a 110 call at 5.69, a 110 put at 11.29 and a 90 put at 2.51: 16.11 - 5.69 + 11.29 - 2.51 = 19.20. Volatility does not enter the box price at all, because every volatility effect in the calls is cancelled by the puts.

When is it attractive, and what can go wrong?

Compare 4.17% with your own rates. If cash would otherwise earn 3.5%, buying the box lends at a better rate: 19.20 at 3.5% grows to only 19.87, against 20 from the box. If your funding costs 5%, selling the box borrows more cheaply: you receive 19.20 today and owe 20, while 20 owed at 5% would have raised only 19.05. Whether a box is cheap or dear is never a property of the box alone; it depends on the rate you can otherwise lend or borrow at, which is why boxes are a funding trade. The risks to name are practical. American options can be exercised early, which breaks the box, so use European index options. Four bid-offer spreads and fees can eat the 0.80 of interest. And the counterparty, or the clearing house, must still be there in a year.

Yes. Buy the 90 call, sell two 100 calls, buy the 110 call, and you are paid 1 to own a payoff that is never below zero. The butterfly costs 14 - 16 + 1 = -1, so you receive 1 today. At expiry it pays nothing below 90, rises to 10 at 100 and falls to nothing above 110, never negative. Call prices must be convex in strike: the 100 call cannot exceed (14 + 1)/2 = 7.5, and it trades at 8.

Where does the 7.5 come from?

Picture three houses on one street at 90, 100 and 110 square metres, priced at 14, 8 and 1. The middle house should be worth no more than the average of its neighbours if each extra metre is worth less than the last; if it is priced above the average, you sell it and buy the two neighbours. A call's price falls as the strike rises, and it falls at a decreasing rate, so the price at the middle strike must sit on or below the straight line between its neighbours. The line from 14 at 90 to 1 at 110 passes through 7.5 at 100. The market says 8. That half point is the mispricing.

How do you prove the payoff is never negative?

Walk the stock price up. Below 90 nothing is in the money, payoff zero. Between 90 and 100 only the 90 call pays, so the payoff is the stock price minus 90, rising to 10. Between 100 and 110 the two short 100 calls start paying out, and the slope turns to 1 - 2 = -1, so the payoff falls from 10 back to zero at 110. Above 110 the long 110 call kicks in and the slope is 1 - 2 + 1 = 0: flat at zero. The payoff is zero or positive everywhere, you were paid 1 to hold it, so you have a riskless profit of at least 1 and up to 11.

Check the other bounds too, out loud, so the interviewer sees you are not pattern matching. A call spread can never be worth more than the gap between its strikes: the 90/100 spread costs 6 and the 100/110 spread costs 7, both under 10, so those pass. Prices fall as strike rises: 14, 8, 1, pass. Only convexity fails. The limitation: in a real screen you trade at bids and offers, not mids, and a half point of theoretical edge can vanish inside the spread. The structure is an arbitrage at these prices; whether you can execute it is a separate question.