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009A stock trades at 100 and in one period will be either 120 or 80. Interest rates are zero. Price a call option struck at 100 by building a portfolio of shares and borrowing that copies it, and explain why the real-world probability of the up move does not appear in the price.Pricing, options and index mathsCoreOptions market makingQuant trading
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If you believe the stock goes up with probability 90%, what is the call worth?
Show the worked solution
The call is worth 10. It pays 20 if the stock goes to 120 and 0 at 80. Half a share pays 60 or 40, so half a share with a loan of 40 pays 20 or 0, exactly the call. That portfolio costs 50 - 40 = 10 today. If the call traded at any other price, you could buy the cheap one and sell the dear one for a riskless profit, so no probability is needed.
How do you build the copy?
Match the swing first. The call's payoff moves by 20 between the two states while the stock moves by 40, so the copy needs 20/40 = 0.5 of a share: that ratio is the option's deltaHow much an option's value changes for a one-unit change in the underlying price; here, the number of shares that copies the option.. Half a share is worth 60 or 40 at the end, which is 40 more than the call in both states. Borrow 40 today, repay 40 at the end with zero interest, and the copy pays exactly 20 or 0.
A call struck at 100 on a stock that moves to 120 or 80 is copied by half a share and a loan of 40, which pays 20 or 0 exactly as the call does and costs 10 today, so the call is worth 10 with no probability used.
Why does the chance of the up move not matter?
Think of a shop selling a bundle of two items that you can also buy separately. The bundle's price is pinned by the parts, whatever you think about how useful the items are. The call is a bundle of half a share and a loan; the share price already reflects everyone's views about the up move, so the option inherits them and adds none of its own. A 90% view is a reason to hold the stock itself, not a reason to pay more for the call than its parts cost.
The relationship
$\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{20-0}{120-80} = 0.5 \qquad C_0 = \Delta S_0 - B = 50 - 40 = 10 = q\,C_u + (1-q)\,C_d,\; q = \frac{S_0 - S_d}{S_u - S_d} = 0.5$
\Delta shares held in the copy
B the amount borrowed, 0.5 x 80 - 0 = 40
q the risk-neutral weight on the up state, fixed by the prices, not by beliefs
What it says in wordsThe copy's cost gives the price, and the same price is an average of the payoffs using weights set by today's stock price.
What would you do if the call traded at 12?
Sell the dear thing and buy the cheap one. Sell the call for 12, buy half a share for 50 and borrow 40, a net cash inflow of 2 today; at the end the portfolio pays exactly what you owe on the call in either state. The 2 is kept whatever happens. The weight q = 0.5 that reproduces the price is called the risk-neutral probability, but it is a pricing weight backed out of the stock price, not a forecast. Say that distinction; interviewers listen for it.
Where candidates lose it
The trap is pricing the call as an expected payoff under your own view: 90% of 20 is 18. That price can be arbitraged against the stock, so nobody could trade it for long, and the interviewer wants to hear that the copying portfolio pins the price.
The second loss is getting 10 by assuming a 50% chance. The number is right by coincidence of the symmetric tree; ask yourself what happens with an up move to 130, and the risk-neutral weight changes to 1/2.5 = 0.4.
What the interviewer asks next
Price the put struck at 100 and check put-call parity.
What changes if interest rates are 5% for the period?
The stock can go to 130 or 80 instead. Price the call again.
036A stock pays a growing dividend and is valued with the Gordon model at a discount rate of 10% and growth of 6%. What is its duration, and roughly how much does its price change if the discount rate rises by one point?Pricing, options and index mathsCoreBlackRockNew York · 2026
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What is the stock's duration, its percentage price sensitivity to the discount rate?
Show the worked solution
Duration is 1/(r - g) = 25 years, so a one-point rise cuts the value by about a fifth. The Gordon price is D1/(r - g), and its percentage sensitivity to r is 1/(r - g) = 1/0.04 = 25. With a Rs 4 dividend the price moves from Rs 100 at 10% to Rs 80 at 11%, a 20% fall. The 25% duration estimate overshoots because the price curve is convex.
Why does a stock have a duration at all?
A promise of money in one year hardly changes in value when rates move; a promise of money in twenty five years changes a lot, because the rate is compounded over every one of those years. A stock is a stream of dividends stretching forever, and when the dividends grow, most of its value sits in cash flows far in the future, so it behaves like a very long bond. Duration measures exactly that: the percentage price change for a change in the discount rate.
The relationship
$P = \frac{D_1}{r-g} \qquad -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g} = \frac{1}{0.10-0.06} = 25$
D1 next year's dividend, Rs 4 in the illustration
r discount rate, 10%
g dividend growth rate, 6%
1/(r - g) percentage price change per unit change in r
What it says in wordsDifferentiate the Gordon price and divide by price: the sensitivity is one over the gap between the discount rate and growth.
At a 10% discount rate the Rs 4 dividend stock is worth Rs 100 and its duration is 25; at 11% the price is Rs 80, a 20% fall, while the tangent line predicts Rs 75, and at 12% the gap widens to Rs 66.7 against Rs 50 because the price curve is convex.
Why does the estimate say 25% when the price falls 20%?
Duration is the slope at one point, and the price curve bends. For a one-point rise the tangent predicts a 25% fall, but the exact move from Rs 100 to Rs 80 is 20%, because the curve is convexCurving upward, so it always sits above any of its tangent lines. and flattens as r rises. The same bend makes a one-point fall worth more than 25%: at 9% the price is Rs 133.3, up 33.3%. For a big rate move, reprice exactly instead of trusting the slope. Duration here also equals price over dividend, 100/4, which is a quick way to say it: one over the dividend yield.
For precision, the Macaulay durationThe present-value weighted average time at which cash flows arrive. is (1 + r)/(r - g) = 27.5 years, and dividing by 1 + r gives the modified duration of 25. A 10-year bond paying 10% at a 10% yield has a modified duration of about 6.1. The gap r - g is what matters, which is why high-growth stocks carry the most duration: the same stock with 2% growth would have a duration of 12.5 years. The limitation is that the Gordon model holds growth fixed while rates move; in practice both shift together.
Where candidates lose it
The usual loss is saying a stock has no duration because it has no maturity, or that it is infinite because it pays forever. Both skip the one line of calculus that gives 1/(r - g).
The second is quoting 25% as the exact price change. It is the slope at 10%; the exact fall to 11% is 20%, and saying why, convexity, is what separates a strong answer.
What the interviewer asks next
What happens to duration as growth approaches the discount rate?
Why might a stock's measured sensitivity to bond yields be much lower than 25?
What is the price change for a one-point fall in the discount rate?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks MSE Forecasting equation
048Calls on the same stock and expiry are quoted: the 95 strike at 9.80 bid, 10.20 offered, and the 100 strike at 4.30 bid, 4.50 offered. Is there an arbitrage, and exactly how would you trade it?Pricing, options and index mathsCoreWolverine Trading, Chicago, ILUSA · 2019
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What can you lock in, per spread, at these quotes?
Show the worked solution
Yes: sell the 95 call at 9.80 and buy the 100 call at 4.50, collecting 5.30 for a position that can never cost more than 5. A 95/100 call spread pays between 0 and the strike gap of 5 at expiry, so its price must sit between 0 and 5. The market lets you sell it for 5.30, which locks in at least 0.30 per spread, more if the stock ends below 100.
What is a 95/100 call spread worth at most?
Think of two coupons for the same shirt: one lets you buy it for Rs 950, the other for Rs 1,000. The first is worth more, but never by more than Rs 50, because the most it can save you over the second is the Rs 50 difference in price. Long the 95 call and short the 100 call pays the stock's rise above 95, capped once it reaches 100, so at expiry it is worth between 0 and the strike gap of 5. Anything that is certain to pay no more than 5 cannot be worth more than 5 today; with interest it is worth at most 5 discounted, slightly less.
Selling the 95 call at its 9.80 bid and buying the 100 call at its 4.50 offer collects 5.30 for a spread whose payoff is zero below 95 and capped at 5 above 100, so at least 0.30 is kept whatever the stock does at expiry.
Which side of each quote do you trade at?
This is where the question is really won or lost. You sell at the bid and buy at the offer, so the spread you can sell is worth 9.80 - 4.50 = 5.30 to you, not the mid of 5.60. 5.30 is still above 5, so the bound is broken at prices you can actually deal at. Buying the spread would cost 10.20 - 4.30 = 5.90 for something worth at most 5, a certain loss, so only one direction works. Check the stock price cases: below 95 both calls expire worthless and you keep 5.30; at 97 you owe 2 on the short call and keep 3.30; at 100 or above you owe exactly 5 net and keep 0.30.
Stock at expiry Short 95 call pays Long 100 call receives Net owed You keep
90 0.00 0.00 0.00 5.30
95 0.00 0.00 0.00 5.30
97 -2.00 0.00 2.00 3.30
100 -5.00 0.00 5.00 0.30
110 -15.00 +10.00 5.00 0.30
The 5.30 collected less what the spread owes at expiry is never below 0.30, because the short 95 call and the long 100 call together never owe more than 5.
The relationship
$0 \le C(95) - C(100) \le (100 - 95)\,e^{-rT} \qquad 9.80 - 4.50 = 5.30 > 5$
C(K) price of the call with strike K, same stock and expiry
e^{-rT} discount factor to expiry; it makes the upper bound slightly below 5
What it says in wordsA call spread is worth between zero and the discounted strike gap; selling it for more than the gap is free money.
What could stop the arbitrage from paying?
If the calls are American and the short 95 is exercised early, exercise the 100 call too: you pay the stock price minus 95 and receive the stock price minus 100, a net 5, and you already hold 5.30. The real frictions are fees, the margin the short call ties up, and the risk that the quote vanishes after you trade one leg. Trade both legs together as a spread order. On a real screen a 0.30 bound violation lasts seconds, which is why the interviewer is testing whether you can see it fast and name the side, not whether such quotes are common.
Where candidates lose it
The common loss is reasoning with mid prices: 10.00 - 4.40 = 5.60 and a claimed profit of 0.60. Nobody deals at mids; you sell at the bid and buy at the offer, and the honest edge is 0.30.
The second is getting the direction backwards and buying the spread because the 95 call looks cheap next to its payoff. Say the bound first, the spread is worth at most 5, and the direction follows: sell it.
What the interviewer asks next
The 105 call is quoted 1.10 bid, 1.30 offered. Is there a butterfly arbitrage across 95, 100 and 105?
What is the lower bound on the 95/100 call spread, and what quotes would break it?
How does a dividend before expiry change the early-exercise argument?
Asked at Wolverine Trading, Prop Trading, Chicago, IL, USA, 2019 (Wall Street Oasis): pricing options given an ask and a bid price for options with different strikes if you were to short one and long another
066A 2-year bond pays a 6% annual coupon and trades at par. What are its Macaulay duration and modified duration?Pricing, options and index mathsCorePIMCOLos Angeles · 2024
Try it first
Which is closest to the Macaulay duration?
Show the worked solution
Macaulay duration is 1.943 years and modified duration is 1.833. At par the yield equals the coupon, 6%. The year 1 coupon is worth 6 / 1.06 = 5.660 and the final payment 106 / 1.06^2 = 94.340, adding to 100. Weight each time by its share: 0.0566 x 1 + 0.9434 x 2 = 1.943. Divide by 1.06 for modified duration: a 1 point rise in yield cuts the price by about 1.83%.
What is Macaulay duration actually measuring?
Think of a seesaw with weights placed along it at the dates money arrives, each weight equal to what that payment is worth today. Macaulay duration is the point where that seesaw balances: the average time to your money, weighted by present value. A zero-coupon bond has all its weight at maturity, so its duration equals its maturity. Any coupon puts a little weight earlier and pulls the balance point forward. Here the coupon is small and the bond is short, so the pull is small, and that is the whole shape of the answer before any arithmetic.
The year 1 coupon is worth 5.66 today and the year 2 payment 94.34, so the present-value weighted average time is 1.943 years, just 0.057 years before maturity, and dividing by 1.06 gives a modified duration of 1.833.
How do you get the numbers fast in your head?
At par, the yield is the coupon, so discount at 6%. The coupon's present value, 6 / 1.06, is about 5.66, and the rest of the price, 94.34, sits at year 2, so duration is 2 minus the coupon's weight: 2 - 0.0566 = 1.943. That shortcut works for any two-period bond: start at the maturity and subtract the early weight times the time it saves. For a par bond there is also a closed form, (1 + y)/y times (1 - 1/(1 + y)^n), which gives the same 1.9434.
The relationship
$D_{Mac} = \sum_t t\,\frac{PV_t}{P} = 1\times\frac{5.660}{100} + 2\times\frac{94.340}{100} = 1.943 \qquad D_{mod} = \frac{D_{Mac}}{1+y} = \frac{1.943}{1.06} = 1.833$
PV_t the present value of the cash flow at time t, discounted at the yield
P the price, 100 at par
y the yield per period, 6%
What it says in wordsDuration is the average payment date weighted by value today; modified duration converts it into a price sensitivity.
Why are there two durations, and how good is the estimate?
Macaulay duration is a time; modified duration is a slope. Modified duration is the percentage price change for a one point change in yield, so this bond loses about 1.83% if yields rise from 6% to 7%, about Rs 0.018 per Rs 100 for each basis point. Repricing exactly gives 98.192 at 7%, a fall of 1.808%, and 101.859 at 5%, a rise of 1.859%. The rise is bigger than the fall, which is convexityThe curvature of the price against yield: prices rise more when yields fall than they drop when yields rise by the same amount.; for a two-year bond it is a small correction, for a thirty-year bond it is not.
Where candidates lose it
The trap answers are 2 years, from forgetting the coupon, and 1.83 given as the Macaulay figure, from mixing the two definitions. Say which duration you are giving and in what units: years for Macaulay, percent per point of yield for modified.
The second slip is discounting at the wrong rate. A bond at par yields its coupon, so there is nothing to solve for; say that first and the arithmetic is two divisions.
What the interviewer asks next
What is the duration of a 2-year zero-coupon bond, and of a 2-year bond with a 20% coupon?
Estimate the price of this bond if yields rise 50 basis points.
Why does duration fall as the coupon rises, holding maturity fixed?
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par

\Delta	shares held in the copy
B	the amount borrowed, 0.5 x 80 - 0 = 40
q	the risk-neutral weight on the up state, fixed by the prices, not by beliefs

D1	next year's dividend, Rs 4 in the illustration
r	discount rate, 10%
g	dividend growth rate, 6%
1/(r - g)	percentage price change per unit change in r

Stock at expiry	Short 95 call pays	Long 100 call receives	Net owed	You keep
90	0.00	0.00	0.00	5.30
95	0.00	0.00	0.00	5.30
97	-2.00	0.00	2.00	3.30
100	-5.00	0.00	5.00	0.30
110	-15.00	+10.00	5.00	0.30

C(K)	price of the call with strike K, same stock and expiry
e^{-rT}	discount factor to expiry; it makes the upper bound slightly below 5

Try it first

How do you build the copy?

Why does the chance of the up move not matter?

What would you do if the call traded at 12?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a stock have a duration at all?

Why does the estimate say 25% when the price falls 20%?

Where candidates lose it

What the interviewer asks next

Try it first

What is a 95/100 call spread worth at most?

Which side of each quote do you trade at?

What could stop the arbitrage from paying?

Where candidates lose it

What the interviewer asks next

Try it first

What is Macaulay duration actually measuring?

How do you get the numbers fast in your head?

Why are there two durations, and how good is the estimate?

Where candidates lose it

What the interviewer asks next