Case 033Option strategies and trade ideasCore

A fund with no spare cash wants exposure to Purandar Motors at Rs 700. The three-month 700 call is 42 and the 700 put 30, rates are 7%, and the future trades at 712. Build a synthetic long, compare it with the future, and say whether anything is mispriced.

The situation

Chembra Opportunities Fund has used its cash but its manager wants three months of exposure to Purandar Motors, trading at Rs 700. The three-month 700 call is offered at Rs 42 and the three-month 700 put bid at Rs 30. The three-month future trades at Rs 712. The lot size is 500 shares and three-month rates are 7% continuously compounded; Purandar pays no dividend before expiry.

The manager asks the dealing desk to build the position with options, and to check it against simply buying the future.

Your task

Construct the synthetic long, work out the forward price it locks in, compare it with the future and with the fair forward, and say whether any leg is mispriced enough to trade.

Quick check

Long the 700 call at 42 and short the 700 put at 30: at what share price does the position break even at expiry?

Worked solution

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

30-second answerThe answer to give first

Long the 700 call and short the 700 put locks in a purchase at 712, the same price as the future at 712 and within 40 paise of the fair forward of 712.36. Nothing is mispriced enough to trade: the gaps are Rs 72 and Rs 179 a lot before costs. The synthetic needs Rs 12 a share of premium plus margin on the short put; the future needs margin alone, so for a fund with no cash the future is the simpler route.

Step 1How does a call and a put make a forward?

Think of agreeing to buy a flat at a fixed price in three months. If the market rises you gain, if it falls you lose, and either way you end up owning the flat at that price. A long call gives you the gain above the strike; a short put hands you the loss below it. Hold both at the same strike and you have promised to buy at 700 whatever happens, which is a forward, and the net premium of 42 less 30, Rs 12, is the price of moving that promise from 700 to an effective 712. This is put-call parityA call less a put at the same strike and expiry equals the stock less the discounted strike. It holds because both sides pay the same at expiry. written as a trade rather than a formula.

Long call plus short put is a forward; check it against the future-100-50+50+100600650700750800zero at 712long call, paid 42short put, received 30both: a long forwardPayoff per share at expiryThree prices for the same thing711.0711.5712.0712.5713.0712.00 future, quoted712.00 synthetic, 700 + 42 - 30712.36 spot carried at 7%gaps of 0.36 and 0.14: inside costs
At expiry the long 700 call bought for 42 and the short 700 put sold for 30 add up to a straight line through zero at 712, a long forward, and that synthetic price sits alongside the quoted future at 712 and the fair forward of 712.36 within 40 paise.
Step 2Is the synthetic, the future or the theory out of line?

Three prices should agree. The synthetic says 712. The future says 712. The fair forward is spot carried at 7% for three months, 700 times e to the 0.0175, which is 712.36. Parity says call less put should equal spot less the discounted strike, 700 less 687.86, that is 12.14; the market's 12 is 0.14 a share cheap, and the future is 0.36 a share cheap against theory. Strictly, the Rs 12 of premium is paid today and is worth Rs 12.21 by expiry, so the synthetic locks in 712.21 rather than 712; the shortcut of adding the premium to the strike is fine for a three-month quote and is how a desk talks. On a 500-share lot the gaps are Rs 72 and Rs 179, well inside bid-offer, brokerage and the margin financing a real arbitrage would need. The honest answer is that the three prices agree and there is nothing to pick off.

The relationship
C−P=S−Ke−rT⇒K+C−P=SerT=F,700+42−30=712≈700 e0.0175=712.36C - P = S - K e^{-rT} \quad\Rightarrow\quad K + C - P = S e^{rT} = F, \qquad 700 + 42 - 30 = 712 \approx 700\,e^{0.0175} = 712.36
C, Pcall and put prices at the same strike, 42 and 30
Kthe common strike, 700
Sspot, 700
Fthe forward price
r, T7% continuous rate, a quarter of a year
What it says in wordsThe strike plus the net premium paid is the forward price the options lock in, and it should match spot grown at the rate.
Step 3Which route should the fund take?

The manager's constraint is cash, so compare what each route needs. The synthetic needs Rs 12 a share of net premium paid today, Rs 6,000 a lot, plus initial margin on the short put, and it carries pin risk at 700 on expiry day, when one leg may be assigned and the other not; the future needs initial margin alone and settles cleanly. Both carry daily variation margin, which is the cash the fund does not have, so the real question for the manager is how a margin call would be met on a down day. The synthetic earns its place when the options are mispriced against the future by more than costs, or when the fund wants to shade the exposure later by lifting one leg. Here, with everything in line, the future is the cheaper and simpler trade.

State the limits. The parity check assumes no dividend before expiry; a dividend would lower the fair forward and the put-call difference together, and a candidate who forgets it can mistake a dividend for a mispricing. It also assumes the quoted 42 and 30 are prices the fund can actually deal at, in size, at the same moment. And the fund has not solved its cash problem: both routes are leverage, and the manager should say so.

Where candidates lose it

The common loss is seeing 42 against 30 and declaring the calls expensive. The gap of 12 is not a mispricing; it is the cost of carry on Rs 700 for three months at 7%, 12.14, which is what parity predicts.

The second is comparing the synthetic with spot rather than with the forward. The synthetic locks in 712, and 712 against 700 looks like a bad price until you remember the fund keeps its Rs 700 for three months. Compare like with like: 712 against the future at 712 and the fair forward of 712.36.

What the interviewer asks next

  • Purandar announces a Rs 10 dividend payable before expiry. Which of the three prices move, and by how much?
  • The put is bid at 26 instead of 30. Is there a trade now, and what does it need to work?
  • The fund wants to limit its downside below 650. How would you adapt the structure, and what does it cost?
  • On expiry day the stock closes at 700.50. What happens to each leg of the synthetic?

Company names and figures are illustrative.