Case 076Greeks and managing an options bookWarm up

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?

The situation

A client of Dudhganga Securities, a retail and HNI broker, holds three positions in Panchganga Cement, where one lot is 250 shares. He is long 50 lots of out-of-the-money calls with a delta of 0.30, short 20 lots of out-of-the-money puts with a delta of minus 0.40, and long 10 lots of the stock future.

The risk desk wants the book's net delta in lots and in shares, and the number of futures that would take it to zero.

Your task

What is the net delta of the book, and what single futures trade makes it flat?

Quick check

Before adding anything: does the short put position add to or subtract from the book's long exposure?

Worked solution

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

30-second answerThe answer to give first

The book is long 33 lots of delta, 8,250 shares, and selling 33 futures makes it flat. The calls contribute 50 times 0.30, 15 lots. The short puts contribute minus 20 times minus 0.40, plus 8 lots, because selling a negative delta leaves you long. The futures add 10. Fifteen plus eight plus ten is 33, so 33 lots of futures sold brings the delta to zero, in shares and in rupees of exposure.

Step 1Why does the short put add to the long side?

Think of a shopkeeper who has promised to buy a neighbour's scooter at Rs 50,000 if the neighbour asks. If scooter prices rise, the neighbour never asks and the shopkeeper keeps the fee; if prices crash, he is forced to buy at Rs 50,000 and loses. He is exposed like an owner without owning anything. A short put is a long exposure to the stock, so its delta contribution is positive. In symbols: position times deltaHow much an option price moves for a one rupee move in the stock, between 0 and 1 for a call and between minus 1 and 0 for a put., minus 20 lots times minus 0.40, is plus 8 lots. The two minus signs are where candidates lose the question.

Add each leg's delta in lots: the short put adds, it does not subtract50 long calls, delta 0.30+1520 short puts, delta -0.40+8 (short x negative = long)10 long futures, delta 1+10Net delta of the book+33 lots = 8,250 sharesSell 33 futures-33 lotsHedged book0scale: 1 lot of delta = 250 shares
The calls add 15 lots of delta, the short puts add 8 and the futures add 10, so the book is long 33 lots, or 8,250 shares, and selling 33 futures takes it to zero.
Step 2How do you add the legs without mixing units?

Keep one unit through the whole sum. Lots of delta is the cleanest: each leg's delta times its lot count, with the sign of the position. The calls give plus 15, the short puts plus 8, and a future has a delta of 1, so ten long futures give plus 10. The net is plus 33 lots, which is 33 times 250, or 8,250 shares of exposure. If the stock is at Rs 400, that is roughly Rs 33 lakh of directional risk, which is the number the risk desk will quote back to the client.

LegLotsDelta per lotDelta in lotsShares
Long calls+500.30+15.03,750
Short puts-20-0.40+8.02,000
Long futures+101.00+10.02,500
Net+33.08,250
Each leg's delta in lots is the signed lot count times the option delta; the book nets to 33 lots long, which is 8,250 shares of Panchganga Cement.
Step 3What does the hedge fix, and what does it leave behind?

Selling 33 futures removes the first-order exposure, and only that. The calls and puts still carry gamma, so the delta drifts as the stock moves: a rally lifts the call delta above 0.30 and the book turns long again; a fall deepens the put delta and the short puts turn the book long on the way down, which is the uncomfortable side. A delta hedge is a snapshot, true at today's price and stale by tomorrow's. Say that out loud and the interviewer knows you have managed a book rather than read about one. A fractional answer, such as 32.6 lots, is rounded to whole futures and the residual stated.

Where candidates lose it

The common loss is subtracting the put leg: 15 minus 8 plus 10 gives 17 lots, and the hedge is wrong by nearly half. The put's delta is negative and the position is short; the product is positive.

The second is reporting the answer in lots and forgetting that the desk thinks in shares and rupees. State both, and say the hedge is only good until the stock moves.

What the interviewer asks next

  • The stock rallies 5% and the call delta moves to 0.42. Which way is the book now, and by how much?
  • If futures are not available, how would you flatten the delta using the 400 call and 400 put?
  • Why might the client prefer to stay long some delta rather than hedge to zero?

Company names and figures are illustrative.