Delta hedging
Offsetting 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.
In plain language
A trader who has sold call options is hurt if the underlying rises. He could buy the stock or its futures to protect himself, but how much? Delta answers that. Delta measures how much the option's price moves for a small change in the underlying — a delta of 0.60 means a Rs 1 move in the stock moves the call by 60 paise.
If the trader buys just enough futures to exactly cancel the option position's delta, the combined position has a delta of zero. The workbook calls it a delta-neutral position: small moves in the stock price no longer affect it.
The catch is the word small. Delta itself changes as the stock moves, so the hedge that was right at Rs 100 is wrong at Rs 110. The trader must keep buying or selling futures to keep the combined delta near zero. That continuous adjustment is delta hedging.
How it works
Delta signs (Chapter 5, section 5.5.1):
| Position | Delta |
|---|---|
| Long futures | +1 |
| Short futures | −1 |
| Long call | 0 to +1 |
| Short call | 0 to −1 |
| Long put | 0 to −1 |
| Short put | 0 to +1 |
A stock futures contract has a delta of roughly one because the futures price goes hand in hand with the spot price.
The hedge ratio. Chapter 4 says delta "is often called the hedge ratio": for a holding of n shares, n divided by the delta gives the number of calls to write to create a hedge. In such a delta neutral portfolio, a gain on the shares from a price rise is exactly offset by a loss on the calls written, and vice versa.
Why it must be repeated. Gamma measures the change in delta for a unit change in the underlying. With a delta of 0.50 and a gamma of 0.08, a Rs 1 rise in the stock takes the delta to 0.58. The workbook's own example: when the stock moves from 100 to 110, the call delta may rise from 0.50 to 0.60, and the hedge must be enlarged.
The formula
Delta of option position = Option delta × Lot size × Number of lots (negative if short a call)
Delta of futures position = 1 × Lot size × Number of lots (negative if short)
Delta-neutral when the two add up to zero
Futures lots needed = |Delta of option position| ÷ Lot size
A worked example
The workbook's case (Chapter 5, section 5.4). A stock trades at Rs 100. A trader shorts 10 lots of ATM calls, lot size 50, delta 0.50.
Loss on short calls for a Re 1 rise = 0.50 × 50 × 10 = Rs 250
Delta of short calls = −0.50 × 10 × 50 = −250
Long futures needed = 250 ÷ 50 = 5 lots (delta 1 × 5 × 50 = +250)
Combined delta = −250 + 250 = 0 → delta-neutral
The stock rises to Rs 110 and the call delta rises to 0.60.
Delta of short calls = −0.60 × 10 × 50 = −300
Long futures needed = 300 ÷ 50 = 6 lots → buy 1 more lot
What happens if he does not rebalance (illustrative arithmetic on the workbook's numbers). Still long only 5 lots, each further Re 1 rise now costs 0.60 × 500 = Rs 300 on the calls but earns 1 × 250 = Rs 250 on the futures — a net loss of Rs 50 per rupee. The position is no longer neutral.
On the way back down (illustrative). If the stock falls back and the delta drops to 0.40, the calls' delta is −0.40 × 500 = −200, so only 4 lots of futures are needed: he sells 2 of his 6. A delta hedger buys futures as the market rises and sells them as it falls.
Why NISM asks about it
Chapter 4, section 4.7, defines delta, calls it the hedge ratio, and describes the "delta neutral" portfolio of shares and written calls; it also defines gamma. Chapter 5, section 5.4 (Delta-hedging), works the short-call-plus-futures example and names the process. Section 5.5.1 gives the delta of each futures and options position for the FutEq open interest calculation. Expect the number of futures lots needed, the sign of each position's delta, and why the hedge must be adjusted.
Common exam traps
- Delta-neutral holds only for small price changes. The workbook stresses "the key word here is small".
- Delta hedging is continuous, not a one-time trade; delta changes as the price moves (that is gamma).
- A short call has negative delta, so it is hedged by going long futures or stock.
- Futures delta is about 1; option delta is between 0 and 1 in magnitude.
- A delta 0.50 call on 10 lots of 50 needs 5 lots of futures, not 10.
- When the underlying rises and a short call's delta increases, the hedger must buy more futures, not sell.
Where this is taught
Free preparation for NISM Series V-DRelated 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.
- HedgingTaking a derivative position that moves opposite to an exposure you already have, so gains on one offset losses on the other and the future rate is locked in at a known level.
- DeltaThe 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.
- 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.
- 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.
- Option writerThe party who sells an option: receives the premium upfront and takes on the obligation to sell (call) or buy (put) the underlying if the buyer exercises. Gain is capped at the premium.
- 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.