Hedge Funds puzzles, solved step by step

Puzzles: 100
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009How would you price a digital option that pays Rs 100 if the index is above 11,000 at expiry, using only the prices of ordinary call options?Options and payoffsHardVolatility and relative value fundsProp and quant trading firms
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A digital paying Rs 100 above 11,000 is closest to which position?
Show the worked solution
Replicate it with a tight call spread: buy 5 calls at 10,990 and sell 5 at 11,010. The position pays 0 below 10,990 and 100 above 11,010, a steep ramp standing in for the step. So the digital costs about 5 x (C at 10,990 minus C at 11,010). With calls at 212.40 and 203.60 that is 5 x 8.80 = Rs 44. In the limit, the price is minus 100 times the slope of call prices against strike.
Why does a call spread look like a step?
A steep enough ramp can stand in for a stair. A call spread's payoff is a ramp: nothing below the lower strike, rising point for point between the strikes, flat above the upper strike. Narrow the strikes and scale up the size, and the ramp tightens into the step a digital pays. Here the strikes are 20 points apart, so each spread pays at most 20, and five spreads pay at most 100, the digital's payout.
Five 10,990 / 11,010 call spreads pay 0 below 10,990 and 100 above 11,010, overpaying the digital just below 11,000 and underpaying just above it, and with illustrative calls at 212.40 and 203.60 the position costs Rs 44.
What does the price of the spread tell you?
The spread costs the difference in call prices, so the digital costs five times that. As the strikes close in, the price becomes minus 100 times the slope of the call price against strike, and that slope is the discounted market-implied chance of finishing above the strike. With the illustrative quotes, 8.80 across 20 points is a slope of 0.44, a digital worth Rs 44 and an implied chance of about 44% before discounting.
The relationship
$D \approx 100 \times \frac{C(K-h) - C(K+h)}{2h} \;\longrightarrow\; -100\,\frac{\partial C}{\partial K}$
D the digital's price
C(K) the price of a call struck at K
h half the gap between the strikes, here 10
What it says in wordsA digital is a call spread scaled up as it narrows, so its price is the slope of call prices with strike.
Which spread does a desk that sold the digital actually buy?
A desk that has sold the digital wants a hedge that pays at least 100 wherever the digital does. The centred spread overpays just below the strike and underpays just above it, so a seller hedges with five 10,980 / 11,000 spreads, which pay the full 100 by the strike and cost a little more. That difference is what the desk charges for an index that settles right at the strike. One more point marks a strong answer: the slope of call prices includes the change in implied volatility across strikes, so with the usual equity skew, where lower strikes carry higher volatility, the digital is worth more than a flat-volatility model says.
Where candidates lose it
Candidates reach for a pricing formula straight away. The question said using only call prices, and the interviewer wants the replication argument; the formula comes after, if at all.
The second loss is the size. A call spread 20 points wide pays at most 20, so it takes five of them to pay 100; a candidate who buys one spread prices the digital at a fifth of its value.
What the interviewer asks next
How would you replicate a digital that pays 100 below 11,000?
What do the digital call and the digital put at the same strike cost together?
Why is a digital close to expiry, with the index at the strike, so hard to hedge?
034A stock trades at Rs 1,000 and its one-month options are priced at 30% implied volatility. Without a calculator, roughly what does a one-month at-the-money straddle cost, and how far must the stock move for the buyer to break even at expiry?Options and payoffsCoreVolatility and relative value fundsProp and quant trading firms
Try it first
Closest estimate for the straddle:
Show the worked solution
About Rs 69, so the stock must move about 6.9% either way. Scale the 30% annual volatility to one month by the square root of time: 30% x root(1/12) is about 8.7%. An at-the-money straddle is worth about 0.8 times that move times the price: 0.8 x 8.7% x 1,000 is about Rs 69. At expiry the buyer profits only below about Rs 931 or above Rs 1069.
Where does the 0.8 x sigma x root T rule come from?
A straddle pays the size of the move, whichever way it goes, so its value is the expected absolute move. For a normal distribution, the average absolute move is the standard deviation times root(2/pi), which is about 0.8. Over one month the standard deviation of the price is S x sigma x root T = 1,000 x 0.30 x root(1/12), about Rs 87. Times 0.8 gives about Rs 69. Black-Scholes with zero rates gives Rs 69.1, so the shortcut is good to within a rupee for short-dated at-the-money options.
A one month straddle struck at Rs 1,000 costs about Rs 69, so at expiry it loses money anywhere between about Rs 931 and Rs 1069 and profits only on a move of more than about 6.9% either way.
Why is the breakeven move bigger than the typical move traders expect?
Buying a straddle is like buying insurance against a big move in either direction: you pay the average claim up front. The breakeven at expiry is the premium, about 6.9%, while the average absolute move is the same 6.9% by construction, so a buyer at fair implied volatility breaks even only on average. The buyer makes money when realised volatility turns out higher than the 30% priced in, and the seller makes money when it turns out lower. That is why desks talk about straddles as a bet on volatility, not on direction.
The relationship
$\text{Straddle} \approx \sqrt{\tfrac{2}{\pi}}\,\sigma\sqrt{T}\,S \approx 0.8 \times 0.30 \times \sqrt{1/12} \times 1{,}000 \approx 69$
sigma implied volatility, 30% a year
T time to expiry in years, 1/12
S the stock price, Rs 1,000
What it says in wordsAn at-the-money straddle costs about 0.8 times the one standard deviation move over its life.
Say the limits. The rule assumes the option is at the money and short dated, and rates are small; for long-dated or out-of-the-money options it drifts. A single call or put is half the straddle, about 0.4 x sigma x root T x S, which is a useful second number to have ready.
Where candidates lose it
The most common loss is forgetting to scale volatility to one month, which gives an answer near Rs 240 or Rs 300. Implied volatility is quoted per year; divide by root 12 for a month.
The second loss is forgetting the 0.8 and pricing the straddle at the full one standard deviation move, Rs 87. Say where the 0.8 comes from, the average absolute value of a normal, and the answer sounds reasoned rather than memorised.
What the interviewer asks next
What does a one-week straddle cost on the same stock?
Roughly what is the one month at-the-money call worth on its own?
The stock moves 5% by expiry. Did the straddle buyer or seller win, and what does that say about realised volatility?
059A stock trades at Rs 1,000 and pays no dividend. A six-month European call struck at Rs 1,000 costs Rs 60 and the matching put costs Rs 45. Six-month simple interest is 3%. Is there an arbitrage, and how do you lock it in?Options and payoffsCoreVolatility and relative value fundsProp and quant trading firms
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What should the call minus the put be worth?
Show the worked solution
Yes: buy the call, sell the put, short the stock and lend Rs 970.87, which collects about Rs 14.13 today with nothing owed at expiry. Put-call parity says the call less the put should equal the stock less the present value of the strike, 1,000 minus 1,000/1.03, about Rs 29.13. The market prices that gap at Rs 15, so the call is cheap against the put by about Rs 14.
What does a call minus a put actually pay?
Imagine agreeing today to buy a used car in six months at a fixed price. If car prices rise you gain the difference; if they fall you lose it. A long call with a short put at the same strike does exactly that: at expiry it pays the stock price minus the strike, whichever way the stock moved. So it must cost the same as buying the stock today with money borrowed against the strike, 1,000 minus 970.87, which is Rs 29.13. That identity is put-call parityFor European options on the same stock, strike and expiry, call minus put equals the stock price minus the present value of the strike, less the value of any dividends..
The relationship
$C - P = S - \frac{K}{1 + r} = 1{,}000 - \frac{1{,}000}{1.03} \approx 29.13$
C, P the call and put prices, Rs 60 and Rs 45
S the stock price today, Rs 1,000
K the strike, Rs 1,000
r six-month simple interest, 3%
What it says in wordsA call less a put is worth the stock less the strike discounted to today.
How do you build the trade?
Buy what is cheap and sell what is dear. The option version of the forward, long call and short put, costs Rs 15; the cash version, long stock bought with borrowed money, is worth Rs 29.13, so buy the option version and sell the cash version. Buy the call for 60, sell the put for 45, short the stock for 1,000 and lend 970.87 at 3%. The cash left over today is Rs 14.13, worth about Rs 14.55 at expiry.
At expiry the long call and short put pay the stock price minus Rs 1,000 while the short stock and the maturing Rs 1,000 loan pay Rs 1,000 minus the stock price, so the package nets to zero at every price and the Rs 14.13 collected at the start is kept.
What could stop the arbitrage working?
Say what you assumed. The lock holds only if the options are European, the stock pays no dividend before expiry, and you can short the stock without a borrowing fee that eats the Rs 14. A dividend lowers the forward and so the fair gap; an American put can be exercised against you early; a hard-to-borrow stock charges a fee. A gap this size in a liquid name usually means one of those is present, so your first question back is which one.
Where candidates lose it
The common loss is comparing the call and the put directly, seeing the call is dearer and deciding it must be the one to sell. Parity says the call should be dearer, by about Rs 29; the trade depends on whether the actual gap is too wide or too narrow.
The second is forgetting to discount the strike and calling zero fair, or using 3% of the stock, Rs 30, which is close but wrong. Say the formula before the numbers, then build the four legs.
What the interviewer asks next
The stock will pay a Rs 20 dividend before expiry. Is there still an arbitrage, and which way?
The call is Rs 80 and the put Rs 45. What trade do you put on now?
Why does parity hold exactly for European options but only as a band for American ones?
084You own a stock bought at Rs 500 and sell a call struck at Rs 550 for a premium of Rs 12. At expiry, what is your maximum profit and where is your breakeven?Options and payoffsWarm upVolatility and relative value fundsProp and quant trading firms
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What is the most you can make?
Show the worked solution
Maximum profit is Rs 62 and the breakeven is Rs 488. Above the Rs 550 strike the stock is called away, so you keep the Rs 50 rise from 500 to 550 plus the Rs 12 premium. Below 550 the call expires worthless and you keep the premium, which cushions the first Rs 12 of any fall. You lose money only below 500 minus 12, which is Rs 488.
What have you actually sold?
Think of renting out a flat you own with an agreement that the tenant may buy it at a fixed price within the year. You collect rent now, but if flat prices soar, the tenant buys at the agreed price and the extra gain is theirs. A covered call swaps the upside above the strike for cash today: the premium is the rent, and the strike is the agreed sale price.
The covered call earns Rs 12 more than the plain stock at every price up to Rs 550, is capped at a profit of Rs 62 from Rs 550 upwards, breaks even at Rs 488, and falls behind the plain stock above Rs 562.
How do you find the two numbers quickly?
Take the two regions separately. At or above Rs 550 the position is worth 550 plus the 12 kept, against 500 paid, so Rs 62 whatever the stock does. Below 550 the call is worthless and the position is just the stock plus 12, so it loses money only once the stock falls more than 12 below the purchase price, at Rs 488. At a stock price of 520 you make 32: 20 on the stock and 12 of premium. At 450 you lose 38 instead of 50.
The relationship
$\text{profit} = \min(S_T, K) - S_0 + c \qquad \max = 550 - 500 + 12 = 62 \qquad \text{breakeven} = 500 - 12 = 488$
S_T the stock price at expiry
K the strike of the call sold, Rs 550
S_0 the price paid for the stock, Rs 500
c the premium received, Rs 12
What it says in wordsYou keep the stock's value up to the strike, plus the premium, minus what you paid for the stock.
What is the trade-off in plain terms?
The premium improves every outcome below Rs 562 and worsens every outcome above it. Above Rs 562 the plain stock position beats the covered call, because the gain you gave away exceeds the premium you took in. On the downside the cushion is thin: the stock can fall all the way from 500 and the premium covers only 12 of it. That is the limitation to state: a covered call is income with a cap, not protection.
Where candidates lose it
The common slip is quoting the premium, Rs 12, as the maximum profit, which forgets that you still own the stock and keep its rise up to the strike. The opposite slip is saying unlimited, which forgets the call you sold.
For the breakeven, candidates often add the premium to the purchase price and say Rs 512. The premium is money received, so it lowers the breakeven, to Rs 488.
What the interviewer asks next
At what stock price are the covered call and the plain stock worth the same at expiry?
Why might the same call fetch more than Rs 12 just before a results announcement?
Which position gives the same payoff at expiry as a covered call without owning the stock?
094You buy an at-the-money option at 20% implied volatility and delta-hedge it. Over its life the stock realises 30% volatility. Where does your P&L come from, and does the direction of the moves matter?Options and payoffsHardVolatility and relative value fundsProp and quant trading firms
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What drives the hedged position's P&L?
Show the worked solution
From gamma: the hedged option earns on the size of moves, not their direction, and pays theta for the privilege. Each day the P&L is about half gamma times the squared move, minus theta. Theta is priced for 20% volatility, so a daily move of about 1.26% breaks even. At 30%, typical moves are 1.89%, and the squared gain is 2.25 times the decay. Up or down makes no difference; the path still does.
What is left after the delta hedge?
Think of a street vendor who sells umbrellas and sunglasses from the same cart: she no longer cares whether it rains or shines, only whether the weather changes enough to bring people out. A delta hedge sells enough stock to cancel the option's first-order bet on direction, so what remains is the curvature: the option gains a little more on the way up than the hedge loses, and loses a little less on the way down than the hedge gains. That curvature is gamma, and it pays on moves either way.
The option's value curves above the straight delta hedge line on both sides of Rs 1,000, so a move either way earns the shaded gap; a day at 30% realised volatility earns about Rs 0.71 of it per option against Rs 0.32 of time decay priced at 20%.
How big is the daily P&L in numbers?
Take a stock at Rs 1,000 and a three-month at-the-money call priced at 20% volatility, with zero interest rates. Its gamma is about 0.0040 per rupee and its time decay about Rs 0.32 a day over 252 trading days. A day's move of 1.89%, typical of 30% volatility, is Rs 18.9, and half of 0.0040 times 18.9 squared is a gain of Rs 0.71, against Rs 0.32 of decay: about Rs 0.40 a day per option. The decay is exactly what a move of 1.26%, typical of 20% volatility, would pay back.
The relationship
$\text{P\&L}_{\text{day}} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 - \Theta = \tfrac{1}{2}\,\Gamma S^2\left(\sigma_r^2 - \sigma_i^2\right)\Delta t$
Gamma how fast the option's delta changes with the stock price
Delta S the day's move in the stock price
Theta the option's daily time decay
sigma_r, sigma_i realised and implied volatility, 30% and 20%
What it says in wordsEach day a hedged long option earns half its gamma times the squared move and pays its time decay; on average that is the gap between realised and implied variance.
Why does direction not matter, but the path does?
The move enters squared, so plus Rs 18.9 and minus Rs 18.9 earn the same. The P&L depends on realised volatility against implied, not on where the stock ends up. But gamma is largest near the strike and shrinks as the stock drifts away, so 30% realised in a trend that carries the stock far from the strike early earns less than 30% realised while the stock chops around the strike. Hedging frequency matters too: hedge rarely and the P&L becomes noisy, even if its average barely changes.
Where candidates lose it
The common loss is saying the option made money because the stock went up. With a delta hedge in place, direction has been sold away; a candidate who talks about direction has not understood what the hedge does.
The second loss is saying a hedged position has no P&L. It has exactly one exposure left, realised against implied volatility, and naming it is the whole answer.
What the interviewer asks next
Realised volatility comes in at 15% instead. What happens to the P&L, and why?
Why is the same realised volatility worth more while the stock stays near the strike?
You think implied volatility is too low but have no view on direction. What position expresses that?

D	the digital's price
C(K)	the price of a call struck at K
h	half the gap between the strikes, here 10

sigma	implied volatility, 30% a year
T	time to expiry in years, 1/12
S	the stock price, Rs 1,000

Gamma	how fast the option's delta changes with the stock price
Delta S	the day's move in the stock price
Theta	the option's daily time decay
sigma_r, sigma_i	realised and implied volatility, 30% and 20%

Try it first

Why does a call spread look like a step?

What does the price of the spread tell you?

Which spread does a desk that sold the digital actually buy?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the 0.8 x sigma x root T rule come from?

Why is the breakeven move bigger than the typical move traders expect?

Where candidates lose it

What the interviewer asks next

Try it first

What does a call minus a put actually pay?

How do you build the trade?

What could stop the arbitrage working?

Where candidates lose it

What the interviewer asks next

Try it first

What have you actually sold?

How do you find the two numbers quickly?

What is the trade-off in plain terms?

Where candidates lose it

What the interviewer asks next

Try it first

What is left after the delta hedge?

How big is the daily P&L in numbers?

Why does direction not matter, but the path does?

Where candidates lose it

What the interviewer asks next