Delta

An option is not a share, and it does not move rupee for rupee with one. Delta is the exchange rate between the two.

A call with a delta of 0.60 gains about 60 paise when the underlying gains one rupee. A delta of 0.25 gains 25 paise. Delta is the speed at which an option tracks its underlying.

The signs are worth memorising as a block, because they are examined directly:

Buyer and seller of the same option have deltas of the same magnitude and opposite sign — as they must, since one of them gains exactly what the other loses.

Delta is also called the hedge ratio, and that name is the practical use. If you hold n shares of a stock, then n ÷ delta is the number of calls you would need to write to hedge them. The resulting portfolio is delta neutral: a rise in the shares is offset exactly by the loss on the written calls, and a fall the other way.

Two facts make the whole thing work, and both are examinable:

• A futures contract has a delta of roughly 1, because the futures price moves hand in hand with the spot price. That is why futures are the natural hedging instrument.
• Delta itself changes as the underlying moves. The rate at which it changes is gamma — "the acceleration of the delta". A call with a delta of 0.50 and a gamma of 0.08 will have a delta of 0.58 after a one-rupee rise in the stock.

Because delta moves, a delta-neutral position does not stay neutral. The trader must keep buying or selling futures to drag the portfolio delta back to zero, and that continuous process is delta hedging. Note the key word in every statement about delta neutrality: the position is unaffected by small changes in the underlying price.

Delta = Change in option premium ÷ Unit change in price of the underlying

Position delta = Delta × Lot size × Number of lots   (negative if short)

Gamma = Change in option delta ÷ Unit change in price of the underlying

Delta of a futures contract ≈ 1. Delta of the underlying share itself = 1.

A stock trades at Rs 1,000. A trader sells 10 lots of the at-the-money 1,000-strike call, lot size 50, at a premium of Rs 38.

Premium received = 38 × 50 × 10 = Rs 19,000
Underlying units shorted (in delta terms), at delta 0.50:
Position delta   = −0.50 × 50 × 10 = −250

He is short 250 "share equivalents". A one-rupee rise in the stock costs him:

0.50 × 500 units = Rs 250

The hedge. A futures contract has a delta of 1, so each lot of 50 contributes +50 of delta. To cancel −250 he needs:

250 ÷ (1 × 50) = 5 futures lots
Position delta of futures = +1 × 50 × 5 = +250
Combined delta = −250 + 250 = 0     ← delta neutral

Now a one-rupee rise loses Rs 250 on the calls and gains Rs 250 on the futures. Net zero.

Why it does not last. The stock runs from Rs 1,000 to Rs 1,010. With a gamma of 0.008 per rupee, the call delta rises from 0.50 to 0.58:

New option delta position = −0.58 × 500 = −290
Futures position          = +250
Net delta                 = −40      ← no longer neutral

Over that Rs 10 move the calls lost roughly 0.54 (average delta) × 10 × 500 = Rs 2,700 while the futures made 10 × 250 = Rs 2,500. A net loss of about Rs 200 on a supposedly hedged book — the cost of gamma. He buys a sixth futures lot (+50), taking the futures delta to +300 against −290, and the position is neutral again until the next move.

That rebalancing, repeated, is delta hedging.

Chapter 4 (Introduction to Options) introduces the Greeks — Delta, Gamma, Theta, Vega and Rho — and calls delta the most important of them. Chapter 5, section 5.4 (Delta-hedging), works the neutralisation example above in full. Expect: the sign table (call buyer positive, put buyer negative, and so on) as a direct question; a one-line computation of the change in premium from a given delta; the identification of delta as the hedge ratio; and the number of futures lots needed to make a short option position delta neutral. Delta also underpins the newer delta-adjusted open interest measure in section 5.5.1.

• Put deltas are negative for the buyer. Candidates apply the call signs to puts. A long put gains when the underlying falls, so its delta must be negative.
• Delta is a rate, not a probability. A delta of 0.60 means 60 paise per rupee. It is loosely used as a rough chance of finishing in the money, but that is not the workbook definition and not the exam answer.
• Delta is valid only for small moves. The whole delta-hedging section turns on this: over a large move, gamma makes the hedge wrong.
• Delta neutral is not risk free. The position still loses to gamma, to time decay and to a change in volatility. It is neutral to small price moves only.
• A futures contract has a delta of about 1, not of the option it hedges. The hedge is sized by the option's delta and delivered in futures.
• Gamma adds to delta; it does not replace it. Delta 0.50 plus gamma 0.08 on a one-rupee move gives a new delta of 0.58, not 0.08.
• The magnitudes for buyer and seller of the same option are identical; only the signs flip.