Private Wealth Management puzzles, solved step by step
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063A one-year call and a one-year put, both struck at Rs 1,000 on a stock trading at Rs 1,000, cost Rs 60 and Rs 40. The one-year interest rate is 7%. Does put-call parity hold, and if not, what trade locks in the gap?Private banking
Try it first
Which side of parity is cheap?
Show the worked solution
Parity fails by about Rs 45.42: the call is cheap relative to the put. Parity says call minus put equals the stock minus the strike's present value, 1,000 minus 1,000 / 1.07, which is Rs 65.42. The market prices it at Rs 20. Buy the call, sell the put, short the stock and lend Rs 934.58: that collects Rs 45.42 today and nets to zero at expiry at any stock price, assuming no dividends.
Why must a call, a put and a bond be tied together?
Two routes to the same destination must cost the same, or everyone takes the cheaper one. Owning a call and lending the strike's present value gives you, in a year, the stock if it ends above Rs 1,000 and Rs 1,000 in cash if it ends below. Owning the stock plus a put gives exactly the same thing. Two portfolios with identical payoffs in every state must cost the same today, and that equality is put-call parity. Rearranged, call minus put must equal stock minus the present value of the strike.
The relationshipC, P call and put prices, Rs 60 and Rs 40 S the stock price, Rs 1,000 K/(1+r) the strike discounted one year at 7%, Rs 934.58 What it says in wordsCall minus put should equal the stock less the strike's present value; here it falls about Rs 45 short.Call minus put is Rs 20 while stock minus the present value of the strike is Rs 65.42; buying the call, selling the put, shorting the stock and lending Rs 934.58 collects the Rs 45.42 gap today and nets to zero at expiry whatever the stock does. How do you prove the trade has no risk left?
Check both ends. If the stock ends at Rs 1,300, the call pays Rs 300, the put expires, you buy back the stock for Rs 1,300 and the loan returns Rs 1,000: 300 minus 1,300 plus 1,000 is zero. If it ends at Rs 700, the call expires, the put costs you Rs 300, the stock buyback costs Rs 700 and the loan returns Rs 1,000: zero again. Every payoff cancels at expiry, so the Rs 45.42 collected today is kept, worth about Rs 48.60 a year later.
Then say what would dissolve the gap in real life. An expected dividend with a present value of about Rs 45.42 would make these prices consistent, because the short seller must pay it. Costly stock borrowing, early exercise of American-style options and wide dealing spreads also eat into it. For a private banking client, parity matters because structured notes are built from exactly these pieces, and it is how you check whether a note is fairly priced.
Where candidates lose it
The first trap is assuming an at-the-money call and put should cost the same. With a positive interest rate the call is worth more, by the stock less the strike's present value.
The second is naming the direction but not the full trade. The interviewer wants all four legs and the proof that the expiry payoffs cancel; saying buy the cheap call alone leaves you holding stock market risk.
What the interviewer asks next
- What dividend, paid before expiry, would make these prices consistent with parity?
- If the put were the cheap side instead, what would the four legs be?
- How does parity help you check the price of a capital-protected note offered to a client?
075A one-year reverse convertible, bought for Rs 100 on a share trading at Rs 100, pays a 12% coupon. At maturity it returns Rs 100 unless the share ends below Rs 80, in which case the investor gets shares worth the final price instead. What does he receive in total if the share ends at Rs 81, and at Rs 79?Private banking
Try it first
How far apart are the two outcomes, Rs 81 and Rs 79?
Show the worked solution
Rs 112 at Rs 81 and Rs 91 at Rs 79: a Rs 2 move in the share costs him Rs 21. Above the Rs 80 barrier he gets his Rs 100 back plus the Rs 12 coupon. Below it, he receives shares worth the final price plus the coupon, so at Rs 79 he takes the full fall from Rs 100. The coupon is mostly payment for selling a put on the share that only switches on below Rs 80.
Why is there a cliff at Rs 80?
Think of travel insurance that pays nothing unless your flight is delayed four hours, and then pays in full: a delay of three hours fifty-nine and four hours one are worth completely different amounts. The barrier switches the note from paying Rs 100 to paying the share's value, so crossing it moves the payout by the whole distance from Rs 100 to the share price, not by the Rs 2 the share moved. At Rs 81 he gets Rs 100 plus Rs 12; at Rs 79 he gets Rs 79 plus Rs 12.
The note pays a flat Rs 112 whenever the share ends at or above Rs 80, but just below the barrier it pays shares worth the price plus the coupon, so at Rs 79 the investor gets Rs 91 and the payout drops Rs 21 for a Rs 2 move. What is the investor actually selling for the 12%?
Split the note into pieces. He lends Rs 100 for a year, which on its own earns an ordinary interest rate. He also sells the bank a put on the share struck at Rs 100 that only comes alive if the share ends below Rs 80, a knock-in putA put option that exists only if the underlying price crosses a barrier level; once knocked in, it pays like an ordinary put.. The part of the 12% above the interest rate is the premium for that put, and the put is why the note's downside looks like owning the share while its upside is capped at Rs 112. Above Rs 112 a shareholder does better; below Rs 80 the investor does no better than a shareholder, apart from the coupon.
What to say about suitability, without making a call on the product. The note swaps an uncertain equity return for a fixed Rs 12 in most outcomes and a large loss in the others, and the loss arrives as a jump. A client should see the Rs 79 row, not only the Rs 112 headline, and understand that the chance of breaching Rs 80 depends on how volatile the share is.
Where candidates lose it
The trap is treating the barrier as a buffer: a 20% protection, so a fall to Rs 79 costs only Rs 1 below the barrier. It costs Rs 21 against Rs 100, because below the barrier the protection disappears entirely.
The second miss is calling the coupon income. Most of the excess over an ordinary interest rate is option premium, paid for taking the share's downside below Rs 80.
What the interviewer asks next
- At what final share price is the investor exactly back to his Rs 100?
- Why do reverse convertibles on more volatile shares offer higher coupons?
- How would a barrier that is watched every day, instead of only at maturity, change the risk?