Risk Management puzzles, solved step by step

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

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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?
Show the worked solution
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?
097A client offers you a game: a fair coin is flipped until the first tail, and you are paid Rs 2 raised to the number of flips. The client can pay at most Rs 1 crore. What is the fair price of the game?Probability and base ratesHardQuant riskCounterparty risk
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Roughly what is the capped game worth?
Show the worked solution
About Rs 24.19. A game that ends on flip n pays 2 to the n with probability one half to the n, so each flip count adds exactly Rs 1. Uncapped, that sum is infinite. The client can pay only Rs 1 crore, which 2 to the 23 stays below, so 23 flip counts add Rs 23 and every longer game pays Rs 1 crore, adding about Rs 1.19.
Why is the uncapped game worth an infinite amount?
Picture a prize that doubles every time you survive another round, while the chance of surviving halves. Each round's prize times its chance is always the same Rs 1. In this game every possible length contributes exactly Rs 1 to the expected value, one Rs 1 for each flip count, forever, so the sum never stops. This is the St Petersburg paradox: the arithmetic says pay anything, yet nobody would pay even Rs 100.
Each flip count from 1 to 23 adds exactly Rs 1 to expected value, but beyond the payer's Rs 1 crore cap the contributions halve each flip and add only Rs 1.19 in total, so the capped game is worth about Rs 24.19.
How does the cap turn infinity into Rs 24?
Find where the cap bites: 2 to the 23 is Rs 83,88,608 and 2 to the 24 is over Rs 1 crore, so the first 23 flip counts pay in full. Those contribute Rs 23; every longer game pays the capped Rs 1 crore, and the chance of lasting past 23 flips is one half to the 23, so the tail adds Rs 1 crore divided by 2 to the 23, about Rs 1.19. The fair price is about Rs 24.19.
The relationship
$EV = \sum_{n=1}^{23} 2^{-n} \cdot 2^{n} + 10^{7} \cdot 2^{-23} = 23 + 1.19 = 24.19$
2^{-n} the chance the first tail comes on flip n
2^n the payout if it does
10^7 the payer's cap, Rs 1 crore
What it says in wordsEvery flip count below the cap adds one rupee; the capped tail adds the cap times the chance of reaching it.
Why is this a counterparty risk question?
Because the value came entirely from payouts the client could never make. A promised payoff is worth only what the payer can actually pay, and most of this game's theoretical value sat in states where the payer would default. Raising the cap a thousandfold to Rs 1,000 crore only lifts the value to about Rs 34.16, because each doubling of capacity adds just one rupee. The same logic prices protection bought from a seller who could not survive the event it insures.
Where candidates lose it
The tempting answer is infinity, or a large number near the cap, because the uncapped maths is famous. The question gives the cap precisely to see whether you use it.
The other slip is stopping at Rs 23 and forgetting the capped tail, or adding Rs 1 crore for every long game rather than weighting it by the chance of getting there.
What the interviewer asks next
What would you pay if the client could pay at most Rs 1,000 crore?
Why might a risk-averse person pay far less than the expected value even with the cap?
Where do you see payoffs that are only as good as the payer's capacity in real markets?
100A firm's assets are worth Rs 100 crore with 25% volatility and 5% expected growth; Rs 70 crore of debt is due in one year. What is the distance to default, and what default probability does it imply?Credit risk arithmeticHardQuant riskBank credit risk
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Roughly what one-year default probability does the Merton model give?
Show the worked solution
A distance to default of about 1.50, implying a one-year default probability of about 6.7%. In the Merton model the firm defaults if assets end below the debt. The log of 100 over 70 is 0.357; add growth less half the variance, 0.019; divide by 25% volatility to get 1.50 standard deviations. The normal tail beyond that is 6.66%.
What is the model saying in plain words?
A homeowner with a Rs 70 lakh loan on a Rs 1 crore house is in trouble at repayment only if the house is then worth less than Rs 70 lakh. How likely that is depends on the cushion, Rs 30 lakh, and on how much house prices swing. The Merton model treats a firm the same way: default happens if the value of its assets at the debt's maturity falls below what it owes, so the default probability is the share of possible asset values that land below the debt. Distance to default is that cushion measured in standard deviations.
With assets of Rs 100 crore, 25% volatility and 5% growth, the one-year asset value centres near Rs 101.9 crore and only the tail below the Rs 70 crore of debt, 6.66% of outcomes, ends in default, a distance to default of 1.50.
The relationship
$DD = \frac{\ln(V/D) + (\mu - \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = \frac{0.357 + 0.019}{0.25} = 1.50 \qquad PD = N(-1.50) = 6.7\%$
V asset value today, Rs 100 crore
D debt due, Rs 70 crore
mu expected asset growth, 5%
sigma asset volatility, 25%
N the standard normal distribution
What it says in wordsDistance to default is the log cushion plus expected drift, in units of asset volatility; the default probability is the normal tail beyond it.
Why not just divide the cushion by volatility?
The simple version, Rs 30 crore of cushion over Rs 25 crore of one-year volatility, gives 1.2 standard deviations and a default probability of about 11.5%. It ignores two things that both favour the lender: assets are expected to grow 5%, and a lognormal asset value cannot fall as easily in rupees as it can rise, so the log cushion is wider than the simple ratio suggests. The simple form is still a useful first number in the room, as long as you say which way it errs.
Then give the model's limits. Asset value and asset volatility are not observed; in practice they are backed out from the equity price and equity volatility. The model assumes default happens only at maturity, a single debt payment, and normal log returns, and it tends to produce very low default probabilities for safe firms over short horizons. Commercial versions map distance to default onto historical default rates rather than trusting the normal tail.
Where candidates lose it
The common error is using the simple ratio, 1.2 standard deviations, and quoting 11.5% as the Merton answer. It is a quick estimate, not the model; show the log form or say that you are approximating.
The second is forgetting the minus half sigma squared term. It is small here, 0.031, but leaving it out, or adding it, moves the distance to default and shows the interviewer the formula is memorised rather than understood.
What the interviewer asks next
Asset volatility rises to 35%. What happens to the default probability?
How would you estimate asset value and asset volatility from the share price?
Why does the Merton model tend to understate short-term default risk?

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

2^{-n}	the chance the first tail comes on flip n
2^n	the payout if it does
10^7	the payer's cap, Rs 1 crore

V	asset value today, Rs 100 crore
D	debt due, Rs 70 crore
mu	expected asset growth, 5%
sigma	asset volatility, 25%
N	the standard normal distribution

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

Try it first

Why is the uncapped game worth an infinite amount?

How does the cap turn infinity into Rs 24?

Why is this a counterparty risk question?

Where candidates lose it

What the interviewer asks next

Try it first

What is the model saying in plain words?

Why not just divide the cushion by volatility?

Where candidates lose it

What the interviewer asks next