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075

Case 075Option pricing and arbitrage checksCore

A stock is at Rs 900 and pays a Rs 10 dividend in one month. The three-month 900 call is 48 and the 900 put 30, rate 7% continuous. Check put-call parity, and if it fails, build the arbitrage.

1The situation

Ambivli Chemicals trades at Rs 900 and has declared a dividend of Rs 10 a share, payable in one month. European options on the stock expiring in three months show the 900 call at Rs 48 and the 900 put at Rs 30. The desk can borrow and lend at 7% a year, continuously compounded.

A junior trader notices the call looks expensive against the put and asks whether there is a trade.

2Your task

Check put-call parity with the dividend, measure any mispricing, construct the arbitrage with its cash flows today, at the dividend date and at expiry, and say what could stop it working.

Quick check

Before working it: if parity fails with the call too expensive, the arbitrage is to:

Worked solution

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

30-second answerThe answer to give first

Parity fails: call minus put is 18, but spot less the present value of the dividend and the strike is only about 5.67, so the call is about 12.33 too expensive relative to the put. Sell the call at 48, buy the put at 30 and buy the stock at 900, borrowing 882. At expiry the stock, put and short call are worth exactly 900; with the dividend reinvested and the loan repaid, about 12.55 a share is left, the 12.33 grown at 7%.

Step 1What should call minus put be worth?

A call and a short put at the same strike together oblige you to buy the stock at 900 at expiry, whichever way it goes: that is a forward contract. A forward to buy at 900 is worth today's share price, less the dividend you will not receive because you do not own the share yet, less the present value of the 900 you will pay. It is like agreeing today to buy a flat in three months at a fixed price: the agreement is worth the flat's value now, less the rent you will miss in the meantime, less what the purchase price is worth today. With a 7% rate, the dividend in one month is worth 9.94 today and the strike in three months 884.39, so call minus put should be 900 less 9.94 less 884.39, about 5.67.

The relationship
C−P=S−De−rtD−Ke−rT=900−10e−0.07/12−900e−0.0175≈5.67C - P = S - D e^{-r t_D} - K e^{-rT} = 900 - 10e^{-0.07/12} - 900e^{-0.0175} \approx 5.67
C, Pthe call and put prices at the same strike and expiry
D, t_Dthe Rs 10 dividend and its timing, one month
K, T, rthe 900 strike, three months, and the 7% continuous rate
What it says in wordsCall minus put must equal the share price after removing the present value of the dividend and of the strike; anything else leaves a riskless profit.
Parity says the gap should be 5.67; the market shows 1818.00Market: call - put48 - 305.67Parity: S - PV(D) - PV(K)900 - 9.94 - 884.3912.33The tradeSell the call +48Buy the put -30Buy the stock -900Borrow +882Locks 12.33 today,12.55 at expiry
The market's call minus put is 18 while parity, 900 less the dividend's present value of 9.94 and the strike's 884.39, says 5.67, so the call is about 12.33 too expensive relative to the put and the trade sells the call, buys the put and buys the stock.
Step 2How is the arbitrage built, and what does it lock in?

Sell the rich side and hedge it with the cheap side. Sell the call for 48, buy the put for 30 and buy the stock for 900, borrowing the net 882; at expiry the stock plus the put minus the call is worth exactly 900 whatever the stock does, because above 900 the call is exercised against you and below 900 you exercise the put. Along the way the stock pays the Rs 10 dividend in one month, which you invest to expiry. At expiry you receive 900, add the dividend grown to 10.12 and repay the loan grown to 897.57, leaving 12.55 a share: the 12.33 mispricing carried forward at 7%.

DateCash flow, per shareAmount
TodaySell call +48, buy put -30, buy stock -900, borrow +8820.00
One monthDividend received, invested at 7% to expiry+10.00
ExpiryStock + put - call delivers exactly+900.00
ExpiryDividend with interest+10.12
ExpiryRepay 882 with three months' interest-897.57
Net at expiry+12.55
The conversion costs nothing today and leaves 12.55 a share at expiry, which is the 12.33 mispricing grown at 7% for three months.
Step 3What could stop the trade from working?

Four things, each with a number. The dividend: the position is long the stock, so it relies on receiving Rs 10; if the dividend were cancelled, parity would say 15.61 and the edge would fall from 12.33 to 2.39. Funding: the prices imply a financing rate of about 12.6% a year, so a desk that can only borrow at a rate near that has no trade; the market may simply be telling you money is dearer than 7% for this stock. Execution: 48 and 30 may be mid prices, and selling the call at the bid and buying the put at the offer can easily eat several rupees of the 12.33. And exercise style: the analysis assumes European options; if the call could be exercised early, just before the dividend, the short call could be taken away and the stock with it. Check each before calling the edge free money.

The view to give is measured. The gap is large enough to be worth checking, and parity mispricings this size in a liquid name are usually explained by one of the four items above rather than by an oversight in the market. If the dividend is confirmed, the options are European, the desk funds near 7% and the executable prices are close to these, the conversion locks in the difference; if not, the apparent edge is the market's price for one of those risks.

Where candidates lose it

Candidates check parity without the dividend, get 900 less 884.39, 15.61, and conclude the call is only 2.39 rich. The dividend is paid before expiry to whoever holds the stock, so its present value comes off the share price.

The second miss is building the reverse trade, buying the call and selling the put, because the call price is the higher number. Sell the side that is rich relative to parity, not the one with the bigger price.

What the interviewer asks next

  • The call is 40 and the put 30 instead. Is there still a trade, and in which direction?
  • The company may raise the dividend to Rs 20. Does that help or hurt the conversion?
  • How would a hard-to-borrow stock change the parity relationship, and which side would look rich?
← Case 074A stock at Rs 1,200 reports results tomorrow, and the at-the-money straddle expiring in a week costs Rs 84. Its last eight results-day moves were 5%, 3%, 9%, 4%, 6%, 2%, 7% and 4%. Buy the straddle, sell it, or sell an iron fly with wings at 1,100 and 1,300 for Rs 60?Case 076 →A client is long 50 lots of calls with delta 0.30, short 20 lots of puts with delta minus 0.40 and long 10 lots of futures. What is the net delta, and how many futures make the book flat?

Company names and figures are illustrative.

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