Derivatives Foundation puzzles, solved step by step

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  1. 010In some stock, the 99-strike call trades at 5.60 and the 101-strike call at 4.70, same expiry. Estimate the price of a digital option that pays 1 if the stock finishes above 100 at that expiry.Option payoffs and no-arbitrageCoreExotics tradingStructured products

    Try it first

    Before any arithmetic: which combination of the two calls has a payoff that looks most like a step at 100?

    Show the worked solution

    About 0.45. Buying the 99 call and selling the 101 call pays 0 below 99, 2 above 101 and a straight ramp between. Divide that by the width of 2 and the payoff is 0 below 99, 1 above 101 and a ramp through 100: a digital with its edge smoothed over two points. Its cost is (5.60 minus 4.70) over 2, which is 0.45. The narrower the spread, the closer the ramp sits to the step, and the price converges to the digital.

    Why does a call spread stand in for a digital?

    A light switch is a step: off or on. A dimmer that goes from fully off to fully on over a tiny turn of the knob is, for every practical purpose, the same switch. A call spread over its width is a dimmer: it ramps from 0 to 1 across the two strikes, and as the strikes close in on 100 the ramp becomes the step. So the digital is the limit of a scaled call spread, and a traded call spread gives you a price for it without any model.

    A call spread over its width is a ramp; the digital is a step; shrink the width and they meetCall spread 99 / 101, scaled by 1/2959910010110510(call 99 - call 101) / 2rampstock price at expiryDigital struck at 100959910010110510pays 1 if S > 100stepstock price at expiryprice of the ramp = (5.60 - 4.70) / 2 = 0.45, so the digital is worth about 0.45
    The 99 to 101 call spread divided by its width of 2 pays 0 below 99, ramps to 1 at 101 and crosses the digital's step exactly at 100, so the two payoffs differ only inside the narrow band between the strikes and the spread's price, (5.60 minus 4.70) over 2, gives a digital value of 0.45.
    The relationship
    D(100)≈C(99)−C(101)101−99=5.60−4.702=0.45D(K)=−∂C∂KD(100) \approx \frac{C(99) - C(101)}{101 - 99} = \frac{5.60 - 4.70}{2} = 0.45 \qquad D(K) = -\frac{\partial C}{\partial K}
    C(K)the price of a call struck at K
    D(K)the price of a digital paying 1 above K
    (C(99) - C(101)) / 2the slope of the call price in strike, estimated across 100
    What it says in wordsThe digital is minus the slope of the call price with respect to strike, and a centred call spread measures that slope.

    Is 0.45 the digital's price or an approximation, and which way is it off?

    It is an approximation to the slope at 100 taken from two points either side. Because the spread is centred on 100, the first-order error cancels and what remains is small, of the order of the curvature of the call price between 99 and 101. If the digital were struck at 99 instead, the same spread would overstate it, because the call price is convex in strike and the ramp sits above the step on that side. On a desk you would quote the digital from the tightest spread the market will show you, and hedge it with that spread, so the approximation is also the hedge.

    What does 0.45 say about the market, and what is the limitation?

    A digital paying 1 above 100 at 0.45, with rates near zero, means the pricing probability of finishing above 100 is about 45%, slightly below one half. That is a risk-neutral probability, not a forecast, and it is pulled down by the skew: with a steeper put skew, out-of-the-money calls are cheaper in volatility terms and the slope in strike is steeper, which moves the digital. Say that a flat-volatility formula would miss this, and that the call spread picks the skew up automatically because it uses the two traded prices.

    Where candidates lose it

    The common error is to take the difference of the two call prices, 0.90, and present it as the digital. That is the price of a spread that pays 2 above 101, not 1. Divide by the width.

    The second loss is reaching for a lognormal formula with a guessed volatility. The question gives you two traded prices precisely so you can price the digital without a model; use them.

    What the interviewer asks next

    • The 99.5 and 100.5 calls are 5.37 and 4.93. What does that pair say about the digital, and why might it differ from 0.45?
    • How would you hedge a short digital you sold at 0.45, and what goes wrong near expiry?
    • Price a digital that pays 1 if the stock finishes below 100.
  2. 025A stock trades at 500. The one-year 500-strike call is priced at 40 and the 500-strike put at 45, and interest rates are zero. What dividend is the options market pricing in, and what would you check before trading on it?Option payoffs and no-arbitrageCoreEquity derivativesMarket making

    Try it first

    Before using any formula: the put costs more than the call at the same at-the-money strike. With zero rates, that tells you

    Show the worked solution

    A dividend of 5 per share. With zero rates, put-call parity says call minus put equals spot minus the dividend minus the strike. Here 40 minus 45 = 500 minus D minus 500, so D = 5, and the implied forward is 495. Before trading against it, check that the options are European or that early exercise is worth nothing, what it costs to borrow the stock, whether the ex-date falls before expiry, and whether all four prices are live and tradable.

    Why does a put costing more than a call point to a dividend?

    Suppose you agree today to buy a friend's scooter in a year for its fair price, but the friend will keep using it to deliver parcels and pocket the fees until then. You would pay less than today's price, by the fees they collect. Holding a call and selling a put at the same strike is an agreement to buy the stock at the strike, so call minus put must equal the forward price minus the strike, and the forward is spot minus whatever the holder collects before expiry. With zero rates there is no interest to account for, so a put dearer than the call by 5 says the forward is 5 below spot.

    The relationship
    C−P=S−D−K  ⇒  40−45=500−D−500  ⇒  D=5C - P = S - D - K \;\Rightarrow\; 40 - 45 = 500 - D - 500 \;\Rightarrow\; D = 5
    C, Pthe call and put prices at the same strike and expiry
    Sthe spot price, 500
    Kthe strike, 500
    Dthe cash the stock pays out before expiry, here the implied dividend
    What it says in wordsWith zero rates, the call minus the put equals the spot less the dividend less the strike, so the dividend is what balances the equation.
    Put-call parity: the options imply a forward of 495, so 5 comes off the stockWhat the options saycall - put = 40 - 45 = -5forward = strike + call - put= 500 + (-5)implied forward 495What the stock saysforward = spot - what the holderreceives before expiry= 500 - Dforward 500 - D=500 - D = 495, so D = 5494495496497498499500501dividend 32 unexplainedIf the companypays only 3:spot500forward 495
    Call minus put is minus 5, so the options imply a forward of 495, and since the stock's forward is 500 minus the dividend, the market is pricing a dividend of 5; if the company pays only 3, the remaining 2 has to be explained by something else, such as the cost of borrowing the stock.

    What would you check before trading on it?

    Say your own estimate of the dividend is 3. Then a reversal looks attractive: short the stock, buy the call, sell the put, collecting 500 + 45 minus 40 = 505; at expiry the options deliver the stock back at 500 and you owe the 3 dividend, 503 in all, a locked-in 2 per share. That 2 is only yours if you can borrow the stock for less than 0.4% of its price over the year; a hard-to-borrow stock shows up in parity exactly as an extra dividend. Then check the rest: American calls can be exercised just before a dividend, the ex-date must fall before expiry, and four bid-offer spreads can eat a gap of 2 on their own.

    What is the general lesson the interviewer wants?

    That parity is an accounting identity between three things you can trade, and any gap between what it implies and what you believe is a claim about something you have not yet priced. The options market does not quote a dividend; it quotes a forward, and the dividend, the borrow cost and the interest rate are the pieces you split it into. A good answer gives the 5, names the implied forward of 495, and then lists what could make the 5 something other than a dividend. The limitation to say plainly: with non-zero rates, the strike is discounted and the arithmetic shifts, so state the zero-rate assumption before quoting the number.

    Where candidates lose it

    The common answer is that the put is dearer because the market expects the stock to fall. At one strike and one expiry, direction cannot make the put dearer than parity allows; anyone could sell the put, buy the call and short the stock against it. The gap is a forward, not a view.

    The second loss is stopping at 5. The question asks what you would check, and the borrow cost is the one interviewers wait for: an implied dividend above the announced one is often a stock that is expensive to short.

    What the interviewer asks next

    • Interest rates are now 6% a year. Redo the implied dividend, and say which way it moves.
    • The company announces a dividend of 8. Which trade would you put on with these four prices, and what is the risk?
    • Why might an American call on this stock be worth more than its European twin, and how would that distort the implied dividend?
  3. 044Using only calls, build a payoff that is zero below 90, rises one for one to 10 at 100, falls back to zero at 110, and stays at zero above that.Option payoffs and no-arbitrageCoreEquity derivativesStructured products

    Try it first

    Which call portfolio gives the tent?

    Show the worked solution

    Long one 90 call, short two 100 calls, long one 110 call: a call butterfly. Read the slope of the target from left to right: 0, +1, -1, 0. A long call adds +1 to the slope at its strike, so the changes of +1 at 90, -2 at 100 and +1 at 110 give the weights. Check the corners: 0 at 90, 10 at 100, 0 at 110 and above. At 20% volatility and three months it costs about 3.69.

    How do you read a payoff picture as a list of calls?

    A road that is flat, then climbs, then drops, then is flat again can be described by where the gradient changes and by how much. A payoff made of straight pieces is the same: each call adds one unit of slope from its strike onwards, so the number of calls at a strike is simply the change of slope at that strike. The tent has slope 0 below 90, +1 from 90 to 100, -1 from 100 to 110, and 0 above 110. The changes are +1 at 90, -2 at 100 and +1 at 110. So you buy one 90 call, sell two 100 calls and buy one 110 call. Above 110 the three legs pay (S - 90) - 2(S - 100) + (S - 110) = 0, which confirms the payoff returns to zero and stays there.

    Every kink is a change of slope, and every change of slope is a number of calls-40-20010203090100110120stock at expiry, from 80 to 120payofflong 90 callshort two 100 callslong 110 callpeak 10 at 100+1 at 90-2 at 100+1 at 110At 20% volatility and 3 months: 10.71 - 2 x 3.99 + 0.95 = 3.69, and never below zero
    The long 90 call, the two short 100 calls and the long 110 call add up to a tent that is zero below 90, peaks at 10 at 100 and returns to zero from 110 onwards, because the slope changes by +1, -2 and +1 at the three strikes.
    The relationship
    Payoff(S)=(S−90)+−2(S−100)++(S−110)+,weight at K=slope just above K−slope just below K\text{Payoff}(S) = (S - 90)^+ - 2(S - 100)^+ + (S - 110)^+, \qquad \text{weight at } K = \text{slope just above } K - \text{slope just below } K
    (S - K)+the payoff of a call struck at K, the larger of S - K and zero
    weight at Kthe number of calls to hold at that strike; negative means sell
    Sthe stock price at expiry
    What it says in wordsThe weight on each strike is the jump in slope there, which is how any straight-line payoff is built from calls.

    What does the butterfly cost, and what does its price tell you?

    Take an illustrative stock at 100 with 20% volatility and three months to expiry. The calls cost 10.71, 3.99 and 0.95, so the butterfly costs 10.71 - 2 x 3.99 + 0.95 = 3.69. The payoff is a tent of height 10 and base 20, and its price is close to the chance of finishing near 100 times the peak. Divide the cost by the peak payoff of 10 and you get 0.369, close to the 0.383 risk-neutral chance that the stock ends between 95 and 105, which is why a butterfly is the market's way of pricing the probability of a narrow range. Shrink the strike gap towards zero and the scaled butterfly becomes the risk-neutral density itself, a result traders use to read the distribution off a strip of call prices.

    Where does the no-arbitrage check come in?

    The tent never pays less than zero, so it can never cost less than zero. That means C(90) - 2C(100) + C(110) must be at least 0: call prices must be convex in strike. Suppose a screen shows the 90 call at 11.00, the 100 call at 7.00 and the 110 call at 2.50. The butterfly costs 11.00 - 14.00 + 2.50 = -0.50: you are paid 0.50 to hold a payoff that is never below zero. A negative butterfly price is a free lunch, and spotting it on a quote sheet is the reason the question is asked. The limitation in practice is that each leg has a bid and an offer, so the check must use the prices you can actually trade at, buying at offers and selling at bids, and small apparent violations usually vanish once the spread is paid.

    Where candidates lose it

    The common loss is short one 100 call instead of two. One short call only cancels the slope of the 90 call, which flattens the payoff at 10 forever; it takes two to turn the slope down to -1 and bring it back to zero.

    The second is building the tent and stopping. The interviewer usually follows with the price: a butterfly must cost more than zero because it never pays less than zero, and a candidate who connects that to convexity in strike has answered the real question.

    What the interviewer asks next

    • Build the same tent with puts only. Is the cost the same?
    • Build a payoff that is 0 below 90, rises to 10 at 100 and stays at 10 above.
    • The 90, 100 and 110 calls trade at 11.00, 7.00 and 2.50. What do you do?
    • As the gap between the strikes shrinks, what does the scaled butterfly price approach?
  4. 059On a stock that pays no dividend, the three-month 100-strike call trades at 6.50 and the six-month 100-strike call trades at 6.10. Is there an arbitrage, and how would you capture it?Option payoffs and no-arbitrageCoreMarket makingVolatility trading

    Try it first

    Two calls, same strike, the longer one cheaper. What do you do?

    Show the worked solution

    Yes. Sell the three-month call at 6.50, buy the six-month call at 6.10, and lock in 0.40 today with no risk. On a non-dividend stock, a call with more time to expiry is worth at least as much as the same strike with less, because at three months the six-month call is still worth at least its intrinsic value, which is exactly what the expiring call pays out. The position never owes money, so the 0.40 is a pure profit.

    Why must the longer call be worth at least the shorter one?

    A ticket that lets you buy a train seat any time in the next six months is worth at least as much as one that expires in three; you can always do with the long ticket exactly what you would have done with the short one, and then keep it. At the three-month expiry the short call pays max(S - 100, 0), and the six-month call at that moment is worth at least max(S - 100, 0) too, because a call on a non-dividend stock is never worth less than its intrinsic value and is usually worth more. So holding the long and being short the short can never leave you with a negative balance at three months, and the 0.40 you took in today is yours.

    A longer-dated call cannot be worth less: the six-month quote sits below the floor0m3m6m9m12m0.004.008.0012.00months to expiry100-strike call value, stock at 100value at one volatilityfloor for any longer expiry: 6.503m call: 6.506m should be near 9.186m quoted: 6.10The trade, todaySell the 3m 100 call+ 6.50Buy the 6m 100 call- 6.10Cash locked in+ 0.40and the spread can only pay moreAt three monthsstock above 100: deliver,the 6m call is worth morestock at or below 100: theshort expires, the long lives
    At-the-money call value rises with time to expiry, so the six-month call must be worth at least the 6.50 the three-month call trades at; the quoted 6.10 sits below that floor, and selling the three-month while buying the six-month locks in 0.40 with a spread that can only pay more.

    What happens at three months in each case?

    Walk through both branches. If the stock is above 100, the short call is exercised against you, you deliver stock for 100, and you still hold a six-month call worth more than S - 100, so you exercise or sell it and finish ahead; if the stock is at or below 100, the short call expires worthless and you are left holding a live six-month call for free. Either way you keep the 0.40 from today and own something worth zero or more. The limitation is the non-dividend assumption: with a large dividend before six months, early exercise of an American three-month call could matter, and the comparison needs the dividend's present value subtracted.

    The relationship
    C(K,T2)≥C(K,T1)for T2>T1 on a non-dividend stock6.10<6.50⇒sell T1, buy T2, keep 0.40C(K, T_2) \ge C(K, T_1) \quad \text{for } T_2 > T_1 \text{ on a non-dividend stock} \qquad 6.10 < 6.50 \Rightarrow \text{sell } T_1,\ \text{buy } T_2,\ \text{keep } 0.40
    C(K, T)the price of a call with strike K and time to expiry T
    T_1, T_2three and six months
    0.40the cash locked in today, the smallest profit the trade can make
    What it says in wordsA longer-dated call at the same strike can never be worth less than a shorter one, so a longer call quoted cheaper is sold against the shorter one for a riskless profit.

    Why would a market maker see this quote and what does it say about the vol surface?

    The three-month price of 6.50 corresponds to an implied volatility of about 33% with zero rates, and at that volatility a six-month call would be worth around 9.18. A quote of 6.10 for the six-month implies a term structure of volatility so inverted that no volatility at all could justify it, which is why this is an arbitrage and not merely a view on calendar spreads. Real quotes rarely break the bound outright; what you see instead is a long-dated call a few ticks above the floor on an illiquid name, and the question is then whether the bid-ask spread swallows the edge before you can trade both legs.

    Where candidates lose it

    Candidates treat it as a volatility question and start talking about the term structure. Implied volatility cannot push a longer call below a shorter one at the same strike, so the answer is a bound, not a view.

    The second loss is getting the direction wrong under pressure. Sell what is dear, the three-month at 6.50; buy what is cheap, the six-month at 6.10. Then walk the two branches at the first expiry out loud.

    What the interviewer asks next

    • Does the same bound hold for puts on a non-dividend stock?
    • The stock pays a large dividend in month four. Can the three-month call now be worth more than the six-month?
    • What if the two calls had different strikes, say 100 and 105?
    • Both quotes are mid prices and the bid-ask is 0.30 on each. Is the arbitrage still there?