Delta, and what hedging involves

The Greeks are the partial derivatives of the pricing formula. That sounds like a technicality and is not: each one answers a question an option holder actually has, and delta answers the first one.

Three definitions of the same number

As a derivative. Delta is the rate of change of the option's price with respect to the underlying:

$\Delta = \frac{\partial C}{\partial S}$

Differentiating chapter 6's formula gives, for a call and a put on the same strike and expiry,

$\Delta_{\text{call}} = N(d_1), \qquad \Delta_{\text{put}} = N(d_1) - 1$

As the replicating holding. It is chapter 3's $\Delta$ — the units of the underlying in the portfolio that manufactures the option. In the limit of small steps, the ratio of spreads becomes the derivative.

As an equivalent position. A call with a delta of 0.56 behaves, for small moves, like holding 0.56 units of the index. This is the most useful reading in practice: it converts an options book into an equivalent exposure to the underlying.

The three are the same number for one reason — Derman and Kani's: an option can be hedged with the underlying to create "an instantaneously riskless portfolio." The quantity that cancels the risk is the quantity that replicates the option is the quantity that measures its sensitivity.

And note the put relation. $\Delta_{\text{call}} - \Delta_{\text{put}} = 1$, always. Differentiate chapter 2's parity, where the only term involving $S$ is $S$ itself, and that falls out immediately. A put's delta is negative, between −1 and 0, which is the formal statement that a put gains when the underlying falls.

What delta does across strikes

Same index at 24,000, thirty days, 14% volatility, 6.5% interest.

Strike	Call price	Call delta	Put delta	$N(d_2)$
22,000	₹2,120.58	0.9899	−0.0101	0.9887
23,000	₹1,176.02	0.8875	−0.1125	0.8797
23,500	₹771.04	0.7510	−0.2490	0.7381
24,000	₹450.57	0.5609	−0.4391	0.5450
24,500	₹230.21	0.3592	−0.6408	0.3443
25,000	₹101.39	0.1938	−0.8062	0.1830
26,000	₹12.22	0.0328	−0.9672	0.0300

Deep in the money, delta approaches 1. The option is nearly certain to be exercised, so it tracks the index almost one for one. It has become a forward contract with a small amount of optionality left.

Far out of the money, delta approaches 0. The index moving 100 points barely changes a 26,000 call worth ₹12.

At the money it is a little above a half, for chapter 6's reason: the risk-neutral drift tilts things slightly.

The error worth correcting

Delta is widely described as the probability the option expires in the money. It is not, and the last column of that table is the correction.

$N(d_2)$ is the risk-neutral probability of finishing in the money. $N(d_1) = \Delta$ is not a probability at all. At the 24,000 strike they are 0.5450 and 0.5609 — close, which is why the confusion survives. At 23,500 they are 0.7381 and 0.7510; at 26,000, 0.0300 and 0.0328. The gap is small for short-dated options and widens with time and volatility.

And chapter 3 applies to both. Even $N(d_2)$ is a risk-neutral weight, not a forecast. So "delta is the probability of expiring in the money" is wrong twice over: wrong quantity, and the right quantity would not be a real-world probability either.

What delta does as expiry approaches

This is the behaviour that surprises people, and it is worth tabulating.

Time left	23,500 call (in the money)	24,000 call (at the money)	24,500 call (out of the money)
90 days	0.7150	0.6046	0.4875
30 days	0.7510	0.5609	0.3592
7 days	0.8770	0.5295	0.1612
2 days	0.9808	0.5158	0.0256
6 hours	1.0000	0.5056	0.0000

Deltas polarise. With the index unchanged, the in-the-money call's delta climbs to 1 and the out-of-the-money call's collapses to 0. Nothing happened to the index; only the time changed.

Why. Delta reflects the chance of ending on the exercising side. With ninety days left, a 500-point move is unremarkable, so all three options have a real chance either way. With six hours left, 500 points is nearly impossible, so each option's fate is essentially settled.

And the at-the-money column barely moves at all — it stays near 0.51 to 0.60 throughout, because a strike exactly at the index is a coin flip whatever the horizon.

The practical consequence is the one that catches people. Near expiry, the at-the-money delta is stable while the deltas just either side of it are not — so a small index move flips an option from a 0.03 delta to a 0.98 delta within hours. Chapter 9 names that effect: it is gamma, and it is largest exactly here.

Delta in Indian contract units

The textbook delta of 0.5609 is in index units. A real position is in lots.

SEBI fixes lot sizes so that contract value falls between ₹15 lakh and ₹20 lakh, as a multiple of 5 and not less than 10, reviewed every six months with at least two weeks' notice. At an index of 24,000, that band implies a lot of roughly 60 to 80 units:

Lot size	Contract value at 24,000
65	₹15,60,000
70	₹16,80,000
75	₹18,00,000

So converting a delta into a hedge is two multiplications. One option contract with a delta of 0.5609 and a lot size of 75 carries the exposure of $0.5609 \times 75 = 42.07$ index units, or about ₹10.1 lakh of index at 24,000.

And the coarseness is the point. You cannot hold 42.07 units; the available hedging instrument comes in lots of 75 too. One future over-hedges this position by nearly a factor of two. Chapter 7's divisibility assumption fails most severely for the smallest positions, which is to say for individuals.

Working the problem

Short one lot of the 24,000 call. Delta 0.5609, lot size 75.

Step 1 — the exposure. A short call has negative delta: $-0.5609 \times 75 = -42.07$ index units. A 100-point rise in the index costs about $42.07 \times 100 = ₹4{,}207$.

Step 2 — the hedge. Buy 42.07 index units — which means buying index futures, since the cash index is not tradeable. At a futures lot of 75, that is 0.56 of a contract. You can buy one, leaving you long 32.93 units net, or none, leaving you short 42.07. Neither is delta neutral, and this is the ordinary situation for a position of this size.

Step 3 — what the hedge does and does not neutralise.

Neutralised: the first-order effect of a small index move. That is all delta is.

Not neutralised — the change in delta itself. Delta is 0.5609 at an index of 24,000 and something else at 24,100. The hedge is correct at one point and stale everywhere else. That is gamma, chapter 9.

Not neutralised — volatility. If implied volatility rises, the call you are short gets dearer and the future does nothing. That is vega, chapter 10.

Not neutralised — time. As a seller, time passing is in your favour, and no futures position affects it. That is theta.

Not neutralised — financing and margin. The short option attracts initial margin at a 99% one-day VaR, extreme loss margin, and an additional 2% on expiry day. The futures leg attracts its own. Both must be funded.

Step 4 — the 400-point overnight fall. Two things happen, and the second matters more.

You lose on the hedge and gain on the option — but not equally. The long futures position loses about $42.07 \times 400 = ₹16{,}828$. The short call gains, because the call falls from ₹450.57 to ₹251.65, which is ₹198.93 per index unit, or ₹14,920 per lot. The two do not cancel:

$+₹14{,}920 - ₹16{,}828 = -₹1{,}908$

The delta-hedged short position lost about ₹1,900 on a move it was hedged against. The reason is that delta fell as the index fell, so the call's gain was smaller than the linear estimate — and the hedge was sized on the old delta. Had the index risen 400 instead, the same arithmetic gives a loss of about ₹1,716. Either direction loses; only a quiet day pays, and a quiet day pays about ₹653. That asymmetry is gamma, and chapter 9 is about it.

And the hedge is now wrong. At 23,600 with 29 days left the call's delta is 0.3915, so the correct hedge is about 29 units rather than the 42 you hold. You are over-hedged by 13 units and must sell.

Step 5 — the uncomfortable observation. The index fell overnight. There was no moment at which to adjust, because the move happened while the market was closed. The replication argument of chapters 3 to 5 assumed continuous trading precisely to rule this out, and chapter 7 listed it as an assumption that fails. An overnight gap is the cleanest example of a hedge that was correct at the close and irrelevant by the open.

Which is why a short option position is not made safe by delta hedging. It is made less directional. The Derivatives chapter on selling options and tail risk is about what remains, and what remains is everything in chapters 9 and 10.

The point

Delta is $\partial C/\partial S$, equal to $N(d_1)$ for a call and $N(d_1) - 1$ for a put, and it is simultaneously the replicating holding and the position's equivalent exposure to the underlying. It runs from 0 to 1 across strikes, polarises towards 0 or 1 as expiry approaches while the at-the-money delta stays near a half, and it is not the probability of expiring in the money — that is $N(d_2)$, and even that is a risk-neutral weight. Converting it to an Indian contract means multiplying by a lot size SEBI pegs to a ₹15–20 lakh contract value, which makes exact hedges unavailable to small positions. And a delta hedge neutralises only the first-order effect of a small move: the change in delta, volatility, time and financing are all still there.