Case 008Option pricing and arbitrage checksWarm up

A client wants a price for the three-month 21,500 index put but only the 21,500 call is quoted, at 1,050. The index is 22,000, the rate 6.8% and the dividend yield 1.2%. Price the put.

The situation

A client of Indravati Wealth holds a large Satpura 50 portfolio and asks for an indicative price on a three-month put struck at 21,500 to protect it. The desk's screen shows only the three-month 21,500 call, quoted at 1,050. The index is at 22,000, the three-month rate is 6.8% a year and the index dividend yield is 1.2% a year, both continuously compounded.

Both options are European and expire on the same day. The client wants the number and one sentence on how you got it.

Your task

Price the put from the call using put-call parity, show each term, and say what the price would be if you dropped the dividend.

Quick check

Before the formula: the put is 500 points out of the money and the call is 500 in. Should the put cost more or less than the call less 500?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The put is worth about 253.5. By parity, put equals call less the index discounted at the dividend yield plus the strike discounted at the rate: 1,050 less 22,000 times e to the minus 0.003, 21,934.1, plus 21,500 times e to the minus 0.017, 21,137.6. Dropping the dividend term gives 187.6, about 66 too low, and dropping all discounting gives 550, far too high. The put is priced off the call with no view on volatility.

Step 1Why does the call tell you the put's price?

If you own a shop and a friend owns an identical shop, the price one of you gets fixes what the other is worth. A long call plus a short put at the same strike is a forward on the index at 21,500, so once the call price and the forward are known the put has nowhere else to be. The forward side is the index you will hold, less the dividends it pays before expiry, less what it costs to carry the strike: 22,000 times e to the minus 0.012 times 0.25 is 21,934.1, and 21,500 times e to the minus 0.068 times 0.25 is 21,137.6. That is put-call parityThe relationship that a call less a put at the same strike and expiry equals the asset, net of its income before expiry, less the discounted strike., and it holds whatever volatility is.

The relationship
P=C−Se−qT+Ke−rT=1,050−21,934.1+21,137.6=253.5P = C - S e^{-qT} + K e^{-rT} = 1{,}050 - 21,934.1 + 21,137.6 = 253.5
C, Pthe 21,500 call at 1,050 and the put being priced
S e^{-qT}the index 22,000 discounted at the 1.2% dividend yield for a quarter
K e^{-rT}the strike 21,500 discounted at 6.8% for a quarter
What it says in wordsThe put equals the call, less the index net of the dividends it will pay before expiry, plus the present value of the strike.
Parity with each term filled in, and the two shortcuts that misprice the putPut253.5Call1,050S e^(-qT)21,934.1K e^(-rT)21,137.6=-+22,000 x e^(-0.012 x 0.25): the index you will hold, less the dividends it pays first21,500 x e^(-0.068 x 0.25)Drop the dividend term: 1,050 - 22,000 + 21,137.6187.666 too lowDrop all discounting: 1,050 - 22,000 + 21,500550297 too highThe dividend term is worth about 66 points on this put and is the one most often dropped.
Filling parity in gives the put as 1,050 less 21,934.1 plus 21,137.6, about 253.5; dropping the dividend term gives 187.6, about 66 too low, and dropping all discounting gives 550, the error that treats a three-month forward as if it were spot.
Step 2What does each adjustment do to the number?

Take the terms one at a time, because the interviewer will. The strike discount, 21,500 less 21,137.6, is about 362 points: the put holder receives 21,500 in three months, not today, and that money is worth less. The dividend discount, 22,000 less 21,934.1, is about 66 points: the index will shed those dividends before expiry, which lowers the forward and raises the put. Dropping it gives 187.6, and on a protection trade for a client that is a 26% underpricing of what the put is worth. Using neither discount gives 1,050 less 500, 550, which would have the client paying more than double the fair price.

VersionPut priceError against parity
Full parity, dividends and discounting253.50
No dividend term187.6-65.9
No discounting at all550+296.5
Against the parity price of 253.5, leaving out dividends underprices the put by about 66 points and leaving out discounting overprices it by about 297.
Step 3What do you tell the client, and what is the limit?

One sentence: the put is priced off the quoted call through a no-arbitrage relationship, so it carries the same volatility the market is charging for the call, and no view of ours. The quoted 1,050 is presumably a mid; a dealer would show the client a put at parity off the call's offer, and the bid-offer on the call passes straight into the put. Two more limits: the formula is for European options, and index options on Indian exchanges are European, but confirm it for the contract in hand; and the 1.2% yield is an estimate of dividends over the next three months, so a large special dividend from a heavyweight index member would move the put by itself.

Where candidates lose it

The common loss is to apply parity in its textbook form without the dividend yield, which is the version most candidates memorise. On an index with a 1.2% yield that is a 66 point error on a 253 point option.

The second is to argue about volatility or skew. The put and call here share a strike and an expiry, so they share one implied volatility by construction, and parity needs no volatility input at all. Reaching for a model is a signal you have not understood what parity is for.

What the interviewer asks next

  • The client wants the 21,000 put instead. Can you still price it from the 21,500 call?
  • The quoted call is an American option on a single stock. What changes?
  • How much does the put move if the dividend yield estimate rises to 2%?
  • Build the arbitrage if a dealer quoted the put at 300.

Company names and figures are illustrative.