Risk Management puzzles, solved step by step

Puzzles: 100
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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?
Show the worked solution
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?
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?
Show the worked solution
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?
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?
Show the worked solution
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?

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

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

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

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

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