Risk Management puzzles, solved step by step
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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.Bank 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 relationshipdelta 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?
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?Quant 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\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?
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?Bank 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 relationshipGamma 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?