Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

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010An options book has a vega of Rs 5 lakh per volatility point. Implied volatility drops from 22% to 18% overnight. What is the P&L?Options and GreeksWarm upBank market risk
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What is the P&L on the book?
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A loss of about Rs 20 lakh. Positive vega means the book is long volatility and gains Rs 5 lakh for each point implied volatility rises. Volatility fell from 22% to 18%, four points, so the book loses four times Rs 5 lakh. That is a first-order estimate: vega itself changes as volatility and time move.
How do you turn a Greek into rupees?
Think of an electricity tariff quoted per unit. The bill is the rate times the units used, and you need to know which unit the rate refers to. VegaThe change in an option position value for a one point change in implied volatility. is a rate quoted per volatility point, so the P&L is vega times the number of points volatility moved, with the sign. Here the book has positive vega, so it is long options, and volatility moved minus 4 points: Rs 5 lakh times minus 4 is minus Rs 20 lakh.
Implied volatility falls four points from 22% to 18%, and at a vega of Rs 5 lakh per point the book loses Rs 5 lakh on each point, a loss of Rs 20 lakh in total.
The relationship
$\Delta V \approx \nu \times \Delta\sigma = 5 \text{ lakh} \times (18 - 22) = -20 \text{ lakh}$
nu vega, Rs lakh per volatility point
Delta sigma the change in implied volatility, in points
What it says in wordsMultiply the book's sensitivity per point by how many points volatility moved.
When is the Rs 20 lakh estimate wrong?
In three ways worth naming. Vega is a local slope, so a four point move is large enough for vega itself to change, and the true loss can differ from the straight-line estimate. Second, a single vega number assumes every strike and maturity moved by the same four points; in practice short-dated volatility often moves more than long-dated, and the skew can twist. Third, overnight the book also loses or gains time decay and any delta and gamma from the underlying move, so the full P&L explain has more than one line.
On a risk desk, say how you would check it. Compare the vega estimate with a full revaluation at 18%, and bucket vega by maturity so a twist in the volatility surface is visible. If the explained P&L and the actual P&L differ by much, the gap is what the risk team investigates next.
Where candidates lose it
The trap is treating the move as a percentage change: 4 over 22 is about 18%, and some candidates multiply vega by that. Vega is quoted per point of volatility, so the move is four points, not 18%.
The second slip is the sign. Positive vega loses when volatility falls; say the direction before the number.
What the interviewer asks next
What position would have made money on this move, and what would its vega be?
Short-dated volatility fell 6 points and long-dated only 2. How would you estimate the P&L now?
How would you hedge the book's vega without changing its delta?
021A desk is short 10,000 put options on a Rs 500 stock. Each put has a delta of minus 0.3 and a gamma of 0.004 per rupee. The stock falls 10%. Compare the delta-only loss estimate with the delta-gamma estimate.Options and GreeksHardBank market riskModel validation
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How much larger is the delta-gamma loss than the delta-only loss?
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Delta alone says a Rs 1.5 lakh loss; delta and gamma say Rs 2.0 lakh, a third more. The stock falls Rs 50. Each put gains 0.3 times 50, Rs 15, from delta, plus half of 0.004 times 50 squared, Rs 5, from gamma. Short 10,000 puts, the desk loses Rs 20 a put. A full revaluation in this example gives about Rs 2.06 lakh, so the gamma term captures most of the gap.
Why does the delta estimate fall short?
Think of driving downhill with the brakes a little weak. The first hundred metres feel manageable, but every metre adds speed, so the second hundred covers more ground than the first. A put gains value faster the further the stock falls, because its delta grows more negative as it moves towards being in the money; a short put position therefore loses at an accelerating rate. Delta is the slope at today's price, so a straight line from it misses the curve. GammaHow much an option delta changes for a one rupee move in the underlying price. measures that bend.
The relationship
$\Delta V \approx \delta\,\Delta S + \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 = (-0.3)(-50) + \tfrac{1}{2}(0.004)(2{,}500) = 15 + 5 = 20$
delta the put delta, minus 0.3
Gamma the put gamma, 0.004 per rupee
Delta S the stock move, minus Rs 50
Delta V the change in each put's value, Rs
What it says in wordsEach put gains Rs 15 from the slope and Rs 5 from the bend; the desk is short, so it loses both.
For a 10% fall in the stock, the straight delta line predicts a Rs 1.5 lakh loss on 10,000 short puts, but the true P&L curve bends away to about Rs 2.06 lakh, and the delta-gamma estimate of Rs 2.0 lakh lands close to it.
Why does this matter for VaR?
Because a VaR built on delta alone treats every option book as a straight line. For a book that is short options, the straight line understates the loss in exactly the large moves that VaR is meant to capture, and the understatement grows with the square of the move. On a 5% fall the gamma term adds Rs 1.25 a put against Rs 7.50 from delta, one sixth; on a 10% fall it adds a third; on a 20% fall it would add two thirds. A risk team would use delta-gamma at minimum and full revaluation for large scenarios.
Say the limit of the gamma fix too. Delta-gamma is still an approximation: gamma itself changes as the stock moves, and volatility usually jumps when a stock falls 10%, which adds a vega loss for a short option book. In this example, built with an assumed 30% volatility and a strike of about Rs 463, full revaluation gives Rs 2.06 lakh, a little above the delta-gamma figure, before any volatility move.
Where candidates lose it
The trap is stopping at the delta answer, Rs 1.5 lakh, and treating gamma as a rounding detail. On a 10% move it adds a third to the loss, and for a short option book it always adds, never subtracts.
The second slip is the sign. Gamma is positive for the option holder; the desk is short, so the gamma term increases its loss whichever way the stock moves.
What the interviewer asks next
What would the P&L be if the stock rose 10% instead?
How many shares would you trade to delta hedge the book, and does that remove the gamma loss?
Implied volatility rises 5 points as the stock falls. What else would you need to estimate the full loss?
035A stock trades at Rs 500. An investor who owns it buys a 450 put for Rs 12 and sells a 560 call for Rs 10, both expiring on the same date. What is the range of outcomes at expiry?Options and GreeksWarm upAsset manager risk
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What is the worst loss per share at expiry, including the premiums?
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Between a loss of Rs 52 and a gain of Rs 58 a share. The put guarantees a sale at 450 at worst; the sold call hands over anything above 560. Between the strikes the investor simply holds the stock. The pair costs Rs 12 minus Rs 10, a net Rs 2, so the outcome runs from 450 less 500 less 2, minus 52, to 560 less 500 less 2, plus 58.
What does each leg of the collar do?
A farmer worried about a price crash agrees with a trader: if prices fall below a floor, the trader pays the floor; in return, if prices soar above a ceiling, the farmer sells at the ceiling. The farmer gives up the dream harvest to remove the nightmare one. The bought put is the floor, the sold call is the ceiling, and the premium from the call pays for most of the put. That structure is a collarA position that holds a stock, buys a put below the current price and sells a call above it, locking the outcome between two strikes.: the investor still owns the stock between 450 and 560 and nothing outside it.
The collar follows the stock between the 450 and 560 strikes and is flat outside them, so the outcome runs from a loss of Rs 52 to a gain of Rs 58 a share, with breakeven at Rs 502 after the net premium of Rs 2.
How do you check the two ends quickly?
Take one price in each region and walk through it. At 400, the stock is worth 400 and the put pays 50, so the holding is worth 450; the call expires worthless. At 620, the stock is worth 620 and the call costs 60, so the holding is worth 560. Whatever happens, the holding ends between 450 and 560, and subtracting the Rs 500 cost and the Rs 2 net premium gives the range of minus 52 to plus 58. Breakeven is Rs 502, the starting price plus the net premium.
Say what the collar does not do. It removes the tails; it does nothing for moves inside the band. And the cheap net premium is not free protection: the investor paid by selling every rupee of gain above 560. Whether that trade is sensible depends on what the investor needs, a floor for a known liability, say, rather than on the Rs 2.
Where candidates lose it
The frequent slip is to forget the premiums and quote minus 50 to plus 60. The interviewer gave you two premium numbers for a reason; the net Rs 2 moves both ends.
The opposite slip is reading the sold call as an unlimited risk. The investor owns the shares, so the call is covered: its cost is the lost upside above 560, not an open-ended loss.
What the interviewer asks next
Which strikes would make the collar cost exactly zero, and what do you give up?
The stock is at 440 a month before expiry. How has the collar's delta changed?
Why might a promoter holding a large stake use a collar rather than simply selling shares?
046A trader who has sold a call hedges it by buying the stock whenever it trades above the strike and selling whenever it falls below. The hedge looks costless: you only hold the stock when the option is in the money. Why does it lose money over time?Options and GreeksHardQuant riskBank market risk
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If the trader checks the price more often, what happens to the expected cost of the hedge?
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Because you always buy a little above the strike and sell a little below it, and those gaps add up to the option's time value. Prices do not stop at the strike; by the time you trade they have moved through it. Watching more closely shrinks each gap but multiplies the crossings, so the cost never vanishes. Here, with a Rs 100 strike, 1.5 a day of volatility and 40 days, the average cost is about Rs 3.78 a share at any monitoring frequency.
Where does the money leak out?
Think of a thermostat set to switch the heater on at 20 degrees and off at 20 degrees. It cannot switch at exactly 20; it reacts at 20.3 on the way up and 19.7 on the way down, and it keeps flicking all evening. A stop-loss hedge buys when the price is already above the strike and sells when it is already below, so every round trip loses the gap between the two. On the path in the figure the trader traded 5 times and the gaps added to Rs 3.86 a share.
Each buy in the stop-loss hedge happens above the Rs 100 strike and each sell below it, so this path's 5 trades lose Rs 3.86 a share; averaged over many paths the loss is about Rs 3.78, the option's time value, whether the price is checked daily or sixteen times a day.
Why can faster monitoring not fix it?
Because of how random paths behave near a level. Check four times as often and each gap roughly halves, but the path crosses the strike roughly twice as often, so the total barely changes. In a simulation of 2,000 paths, the average cost is 3.84 with daily checks (3.9 crossings), 3.79 at four a day (7.9 crossings) and 3.79 at sixteen a day (16.0 crossings). Checking continuously would mean infinitely many crossings of zero size each, and the cost still would not go away.
The relationship
$E[\text{crossing costs}] = E[(S_T - K)^+] - (S_0 - K)^+ \approx \frac{\sigma\sqrt{T}}{\sqrt{2\pi}} = \frac{1.5\sqrt{40}}{2.507} = 3.78$
\sigma\sqrt{T} price volatility over the option's life, 1.5 a day for 40 days, about 9.5
(S_T - K)^+ the call's payoff at expiry
What it says in wordsThe expected leakage from crossing the strike equals the part of the option's value that comes from time and volatility.
That identity is the real lesson. The stop-loss hedge only replicates the option's intrinsic value; the crossing costs are the time valueThe part of an option price above what it would pay if exercised now, which reflects the chance of favourable moves before expiry. the trader collected as premium and is now paying back. Higher volatility means more and bigger crossings, which is why volatile underlyings command bigger premiums. Proper delta hedging spreads the same cost smoothly as gamma losses rather than lumpy crossing losses; neither makes the premium free money.
Where candidates lose it
Candidates accept the premise that the hedge is costless and look for a trading cost or bid-offer spread to explain the loss. Transaction costs make it worse, but the loss is there even with none: it is built into how prices cross a level.
The second trap is saying monitor more often and the problem goes away. It does not, and the reason, crossings multiply as gaps shrink, is what separates a memorised answer from an understood one.
What the interviewer asks next
How does delta hedging spread this same cost differently?
What happens to the stop-loss hedge's cost if volatility doubles?
The stock starts well above the strike. What does the hedge cost now, and why?
060A one-year European call and put on a non-dividend stock, both struck at 100, trade at 12 and 7. The stock is at 100. What interest rate does put-call parity imply?Options and GreeksCoreBank market risk
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Which relationship do you use?
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About 5.26% a year with annual compounding, or 5.13% continuously compounded. Parity says call minus put equals stock minus the present value of the strike. Here 12 minus 7 is 5, so the present value of 100 must be 95. A one-year discount factor of 0.95 means 100 over 95 minus 1, about 5.26%.
Why must call minus put equal the stock minus the discounted strike?
Agreeing today to buy a house next year at a fixed price is the same as buying it now with money borrowed until then: either way you own the house next year and pay the fixed price. A long call plus a short put at the same strike is that agreement. At expiry the pair pays the stock minus the strike in every state, so today it must cost the same as owning the stock and owing the strike in a year, which is S minus the present value of K.
A long call and a short put struck at 100 together pay the stock price minus 100 in every outcome. Call 12 minus put 7 is 5, which must equal the stock at 100 minus the present value of the strike, so that present value is 95 and the implied rate is 5.26% a year.
Why would a risk manager care about the rate hidden in option prices?
Because it is a check that comes free. If the rate implied by parity sits far from the funding rate the desk actually pays, either the marks are stale, a dividend has been missed, or the options are American and parity no longer holds exactly. Model validation teams run exactly this test on option books to catch mispriced marks before they show up as a loss.
The relationship
$C - P = S - \frac{K}{1+r} \;\Rightarrow\; 5 = 100 - \frac{100}{1+r} \;\Rightarrow\; r = \frac{100}{95} - 1$
C, P the call and put prices, 12 and 7
S the stock price, 100
K the common strike, 100
r the one-year interest rate implied by the prices
What it says in wordsThe gap between an at-the-money call and put is the interest on the strike, so the prices reveal the rate.
The limitation: parity holds exactly only for European options on a stock paying no dividend before expiry. A dividend would lower the stock's forward and push the implied rate the other way, so state the assumption before you give the number.
Where candidates lose it
The common error is to write 5 over 100 and answer 5%. That treats 5 as the interest on 100, but 5 is what you save today, and the rate is measured on the 95 you actually pay. 100 over 95 minus 1 is 5.26%.
The other trap is forgetting the conditions. Say European, no dividends, and one year, then give the number; an interviewer will often follow up with a dividend to see if you adjust.
What the interviewer asks next
The stock pays a dividend of 2 in six months. What rate is implied now?
The call trades at 13 with the put unchanged. What trade locks in a profit?
Why does parity not hold exactly for American options?
085A rule of thumb says an at-the-money option is worth about 0.4 times volatility times the square root of time times the price. What is a three-month at-the-money call on a Rs 1,000 stock with 20% volatility worth?Options and GreeksWarm upBank market riskQuant risk
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Roughly what is the premium?
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About Rs 40. The typical one-year move is 20% of Rs 1,000, Rs 200. Over three months it is Rs 200 times the square root of 0.25, Rs 100. The call is worth about 0.4 of that typical move, Rs 40. A full Black-Scholes calculation at zero rates gives Rs 39.88, so the rule is within a rupee.
Why is the premium a slice of the typical move?
Imagine insuring a shop's daily takings against a bad day. The premium depends on how much the takings usually swing, not on how large the takings are. An at-the-money option pays out on the upside half of the stock's moves, so its value is proportional to the size of a typical move over the option's life, not to the stock price itself. The price enters only through converting volatility into rupees.
Rs 1,000 times 20% volatility is a Rs 200 one-year move, times the square root of 0.25 is a Rs 100 three-month move, and 0.4 of that is a Rs 40 premium, within 0.12 of the Black-Scholes value of Rs 39.88.
The relationship
$C_{ATM} \approx 0.4\, \sigma \sqrt{T}\, S = 0.4 \times 0.20 \times \sqrt{0.25} \times 1000 = 40$
sigma annual volatility, 20%
T time to expiry in years, 0.25
S share price, Rs 1,000
0.4 about 1 over the square root of 2 pi
What it says in wordsMultiply the price by volatility and the square root of time to get a typical move, then take 0.4 of it.
Where does 0.4 come from, and when does the rule fail?
If the stock's move is roughly normal with standard deviation of one typical move, the average payoff from the upside half is that deviation divided by the square root of 2 pi, 0.399. The 0.4 is not a fudge factor; it is the average size of the positive half of a normal move. The rule is built for at-the-money options with short expiries and low rates. It fails for options far in or out of the money, where the payoff is mostly intrinsic value or mostly zero, and over long horizons, where interest rates and the skew of returns start to matter.
A risk manager uses this to sanity-check a trader's mark in ten seconds. If a three-month at-the-money call on a Rs 1,000 stock is marked at Rs 70, the mark implies volatility of about 35%, which is a question worth asking.
Where candidates lose it
The common error is scaling time linearly: a quarter of a year, so a quarter of Rs 200, then 0.4 of Rs 50 gives Rs 20. Volatility scales with the square root of time, so three months is half a year's move.
The other slip is applying the rule to a deep out-of-the-money option. Say it is an at-the-money shortcut and name where it breaks.
What the interviewer asks next
What is the matching at-the-money put worth at zero rates?
How much does the premium rise if volatility doubles, and if time to expiry doubles?
A trader marks the call at Rs 70. What implied volatility is that?
096An options book on a Rs 1,000 stock has a gamma of 500 shares per rupee and theta of minus Rs 40,000 a day. How large a daily move does the book need to break even, and what annual volatility does that imply?Options and GreeksHardBank market riskQuant risk
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Roughly what daily move breaks even?
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About Rs 12.65 a day, a 1.26% move, which is roughly 20% annual volatility. A delta-hedged long gamma book earns half of gamma times the move squared each day and pays theta. Setting 250 times the move squared equal to Rs 40,000 gives a move of the square root of 160. Times the square root of 250 trading days, 1.26% a day is about 20% a year.
What does a long gamma book earn, and what does it pay?
Picture a shopkeeper who pays a fixed daily rent for a stall that earns money only when a crowd passes, and earns much more from a big crowd than a small one. A quiet day loses the rent; a busy day covers it many times. A delta-hedged long option book pays theta every day and earns from gamma, which pays in proportion to the square of the stock's move, so small moves lose and large moves win. Rebalancing the hedge after each move locks in that gamma profit: buying low after a fall and selling high after a rise.
The relationship
$\tfrac{1}{2}\,\Gamma\,(\Delta S)^2 = |\Theta| \quad\Rightarrow\quad \Delta S = \sqrt{\frac{2 \times 40{,}000}{500}} = \sqrt{160} \approx 12.65$
Gamma change in delta per rupee move, 500 shares
Delta S the stock's daily move in rupees
Theta time decay paid each day, Rs 40,000
What it says in wordsThe book breaks even on the day when the gamma profit from the move equals the theta paid.
Gamma profit of 250 times the move squared crosses the Rs 40,000 daily theta at moves of plus and minus Rs 12.65, so smaller moves lose money, larger moves make it, and the breakeven corresponds to about 20% annual volatility.
Why does the breakeven turn into a volatility?
Because that is how the market priced the options. A Rs 12.65 move on a Rs 1,000 stock is 1.26% a day, and 1.26% times the square root of 250 is 20.0% a year. For a delta-hedged option book, the breakeven daily move is the implied volatility in disguise: the book profits if realised volatility beats about 20% and loses if it falls short. That is the one-line view a market risk manager wants: this book is long volatility at about 20%.
State the limits. Gamma and theta change as the stock moves and time passes, so the breakeven drifts; the relation holds day by day, not for a month in one go. And the average of squared moves is what counts, so one large day can pay for several quiet ones, which is why realised volatility, not the number of up days, decides the result.
Where candidates lose it
The common slip is dropping the half, setting 500 times the move squared equal to 40,000 and getting Rs 8.94. Write the Taylor term, one half gamma times the move squared, before any numbers.
The second miss is stopping at Rs 12.65. The interviewer wants the translation into volatility, because that is how a risk manager describes the position: long volatility at about 20%.
What the interviewer asks next
Realised volatility comes in at 15% for a month. Roughly what does the book lose?
How do gamma and theta change as expiry approaches?
Why is a short gamma book said to pick up coins in front of a steamroller?

nu	vega, Rs lakh per volatility point
Delta sigma	the change in implied volatility, in points

delta	the put delta, minus 0.3
Gamma	the put gamma, 0.004 per rupee
Delta S	the stock move, minus Rs 50
Delta V	the change in each put's value, Rs

\sigma\sqrt{T}	price volatility over the option's life, 1.5 a day for 40 days, about 9.5
(S_T - K)^+	the call's payoff at expiry

C, P	the call and put prices, 12 and 7
S	the stock price, 100
K	the common strike, 100
r	the one-year interest rate implied by the prices

Try it first

How do you turn a Greek into rupees?

When is the Rs 20 lakh estimate wrong?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the delta estimate fall short?

Why does this matter for VaR?

Where candidates lose it

What the interviewer asks next

Try it first

What does each leg of the collar do?

How do you check the two ends quickly?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the money leak out?

Why can faster monitoring not fix it?

Where candidates lose it

What the interviewer asks next

Try it first

Why must call minus put equal the stock minus the discounted strike?

Why would a risk manager care about the rate hidden in option prices?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the premium a slice of the typical move?

Where does 0.4 come from, and when does the rule fail?

Where candidates lose it

What the interviewer asks next

Try it first

What does a long gamma book earn, and what does it pay?

Why does the breakeven turn into a volatility?

Where candidates lose it

What the interviewer asks next