Delta
Also written Option delta · δ · Hedge ratio of an option
The change in an option's premium for a one-rupee change in the underlying — the first and most used Greek, and the hedge ratio that says how much underlying to hold against an option position.
In plain language
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:
| Position | Delta | Behaves like |
|---|---|---|
| Call buyer | Positive | A long or bull position in the underlying |
| Call seller | Negative | A short or bear position |
| Put buyer | Negative | A short or bear position |
| Put seller | Positive | A long or bull position |
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.
How it works
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.
The formula
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 worked example
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.
Why NISM asks about it
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.
Common exam traps
- 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.
Where this is taught
- Series V-D · Chapter 16: Introduction to Optionsintroduced here
- Series XVI · Chapter 4: Commodity Optionsintroduced here
- Series I · Chapter 4: Exchange Traded Currency Optionsintroduced here
- Series IV · Chapter 4: Exchange Traded Interest Rate Optionsintroduced here
- Series VIII · Chapter 4: Introduction to Optionsintroduced here
- Series VIII · Chapter 5: Strategies using Equity Futures and Equity Options
- Series V-D · Chapter 21: Exchange Traded Interest Rate Options
Related terms
- Hedge ratioThe number of futures (or options) needed to hedge a position. For one stock with its own futures it is 1:1; for a portfolio against index futures it is portfolio value × beta ÷ contract value.
- GammaThe rate at which an option's delta changes for a one-unit change in the underlying — the second-order Greek, and the reason a delta hedge stops working as soon as the market moves.
- At-the-moneyAn option whose strike price is closest to the spot price, so exercising it immediately would produce neither a gain nor a loss — the strike where the whole premium is time value and uncertainty peaks.
- ThetaThe option Greek that measures time decay — the change in an option's premium for a one-day decrease in time to expiry. It is negative for a long option, call or put alike.
- Option premiumThe price an option buyer pays the seller for the right the contract carries — non-refundable, and made up of intrinsic value plus time value.
- Wasting assetAn option, whose time value shrinks towards zero every day it is held and is worth nothing at expiry — so an option buyer loses money simply from the passage of time.
- Delta hedgingOffsetting an option position's delta with the underlying or futures so the combined delta is zero, then buying or selling as delta changes to keep it there — the option trader's way of managing risk.
- Delta neutral positionA combined options-and-futures position whose net delta is zero, so a small move in the underlying causes an equal and offsetting gain and loss across the two legs — built and maintained by delta hedging.
- Call optionA contract giving its buyer the right, but never the obligation, to buy the underlying at a fixed strike price — so the loss is capped at the premium and the gain is not.
- Price bandThe highest and lowest price at which a contract may trade on a given day, set as a percentage of its base price to block erroneous and manipulative orders.
- Market riskThe risk of loss from movements in market prices — one named category in a manager's risk framework, alongside credit, liquidity and operational risk, and the one measured with VaR and stress tests.
- Open interestThe total number of derivative contracts outstanding and not yet settled in an underlying — counted on one side only, because every long is matched by a short.
- VegaThe option Greek that measures sensitivity to volatility — how much an option premium changes for a one per cent change in the implied volatility of the underlying. It is positive for a long call and a long put alike.
- RhoThe option Greek that measures interest rate sensitivity — the change in an option premium for a one percentage point change in the risk-free rate. It is positive for calls and negative for puts.
- Put-Call RatioPut open interest (or volume) divided by call open interest on the same underlying, read as a contrarian sentiment gauge: below 1 is taken as bearish, above 1 as bullish.
- MoneynessWhether exercising an option right now would give the buyer a positive, zero or negative cash flow — classifying it as in the money, at the money or out of the money.
- Black-Scholes modelA 1973 closed-form option pricing model using spot, strike, volatility, time to expiry and the risk-free rate; Indian exchanges use it for the theoretical price of exchange-traded currency options.
- Future Equivalent Open InterestOpen interest measured as the net delta-adjusted position across futures and options on an underlying, instead of adding up notional contracts — the basis for SEBI's index derivatives position limits.