Delta & Gamma Up Close

In the options-pricing course you met the Greeks as a polite roll-call: delta, gamma, theta, vega, rho — five gauges, each a partial derivative, each measuring “if this input twitches, how much does my premium move?” That was the tourist version. Now you’re running a live book, and a live book doesn’t care about all five equally. Two of them — delta and gamma — eat almost all your attention, your hedging budget, and (on a bad day) your P&L. This lesson zooms all the way in on those two.

Quick reminder of the framing, because it explains why delta and gamma dominate. A Greek is a term in the Taylor expansion of your position value $V$ around the current spot $S$:

$\Delta V \approx \underbrace{\Delta\,\Delta S}_{\text{delta term}} + \underbrace{\tfrac{1}{2}\Gamma\,(\Delta S)^2}_{\text{gamma term}} + \underbrace{\Theta\,\Delta t}_{\text{theta term}} + \dots$

Delta is the first-order response to the stock; gamma is the second-order one. Once you hedge away delta (which every desk does, constantly), the gamma term is the leading thing left — the curvature that delta-hedging can’t kill. Everything practitioners agonise over — re-hedge frequency, pin risk, convexity P&L — lives in those first two terms. So let’s earn our keep there.

Delta: the hedge ratio that doubles as a probability

Analogy. Delta is your option’s exchange rate into the underlying stock. A delta of 0.45 says: for every $1 the stock moves, your option moves about $0.45 — as if you were holding 0.45 of a share per option. It’s the conversion factor that turns “I own a derivative” into “I own this much stock-equivalent.”

Definition. Delta is the first derivative of value with respect to spot:

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

From Black–Scholes, a call’s delta is $N(d_1)$, living in $[0, 1]$, and a put’s delta is $N(d_1) - 1$, living in $[-1, 0]$. Deep out-of-the-money options barely react ($\Delta \to 0$); deep in-the-money options move nearly one-for-one with the stock (call $\Delta \to 1$, put $\Delta \to -1$). At the money, delta sits near 0.5 for a call and about $-0.5$ for a put.

Delta has two readings, and a good trader holds both in their head at once:

• Reading one — the hedge ratio. Delta is literally shares of stock per option. It tells you how much stock to trade against the option to cancel first-order moves. This is the reading you live by on a hedged book.
• Reading two — a rough probability of finishing in-the-money. People love to say “a 0.30-delta call is about 30% likely to expire ITM.” It’s a serviceable back-of-envelope read — but notice the trap below.

Delta-hedging, worked

You’re long 10 call contracts, each on 100 shares, with $\Delta = 0.45$ per option. Your position delta is the delta per option, scaled by the contract multiplier and the number of contracts:

$\text{position delta} = 0.45 \times 100 \times 10 = +450$

So your book behaves exactly like being long 450 shares of stock — a $1 up-tick adds roughly $450, a $1 drop costs roughly $450. To neutralise that first-order exposure you short 450 shares. Now a small wobble nets out:

• Stock rises $1. Calls gain about $0.45 \times 100 \times 10 = 450$, i.e. $450. Short 450 shares loses $450. Net $\approx 0$.
• Stock falls $1. Calls lose about $450. Short stock gains $450. Net $\approx 0$.

That’s delta-neutral: the position barely flinches on a small move. The whole catch lives in the word small — delta is just the slope at this instant, and the slope itself drifts as the stock moves. Which gauge governs that drift? Gamma. But first, lock in the delta arithmetic.

When it matters

Delta is the Greek you re-check most often, because it’s the exposure you’re paid to not have if you run a market-neutral book. The hedge-ratio reading drives every share you trade; the probability reading is a fast sanity check on moneyness (“this 0.05-delta wing is basically a lottery ticket”). The danger is mistaking the slope for the whole story — which is exactly what gamma punishes.

Position delta & dollar delta: speaking the desk’s language

Analogy. Position delta tells you “how many shares I effectively own”; dollar delta translates that into “how many dollars of stock I’m long.” A risk manager doesn’t care that you hold 450 share-equivalents of a $30 stock the same way they care about 450 share-equivalents of a $3,000 stock — one is $13,500 of exposure, the other is $1.35 million. Dollars are the common currency across a book full of different-priced names.

Definition. Aggregate the signed deltas of every position to get the book’s net position delta, then multiply by spot to get dollar delta:

$\text{dollar delta} = (\text{position delta}) \times S$

Worked example. Your book holds the +450 share-equivalents from the calls above, plus a separate leg that’s $-120$ share-equivalents. Net position delta $= 450 - 120 = +330$. If the stock trades at $80:

$\text{dollar delta} = 330 \times 80 = 26{,}400 \text{ dollars}$

You’re carrying $26,400 of long stock exposure — a 1% move in the stock ($0.80) is worth about $264 (that’s $330 \times 0.80$). Dollar delta is what lets a desk compare and aggregate exposure across names trading at wildly different prices.

Gamma: the curvature that makes delta move

Analogy. If delta is your speed, gamma is your acceleration — how fast the speedometer needle itself is swinging. Cruising at a steady 50 mph (high delta, zero gamma) is predictable. Flooring it (high gamma) means your read of “how fast am I going” goes stale in seconds. Gamma is why the hedge you set five minutes ago is already wrong.

Definition. Gamma is the rate of change of delta — equivalently the second derivative of value:

$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2} = \frac{n(d_1)}{S\,\sigma\,\sqrt{T}}$

where $n(d_1)$ is the standard-normal density (the bell curve’s height) at $d_1$. Three things fall straight out of that formula. Gamma is the same for a call and its same-strike put (they share $d_1$). It’s bell-shaped, peaking at the money and decaying toward 0 deep ITM or deep OTM (where delta has flattened at 1 or 0 and stops changing). And it’s always positive for a long option — call or put — because owning optionality is owning curvature.

Gamma, worked

You hold a position with gamma 0.08 (per share, per option). The stock moves up $2. Your delta changes by approximately:

$\Delta\Delta \approx \Gamma \times \Delta S = 0.08 \times 2 = 0.16$

So a call that was sitting at delta 0.45 is now around $0.45 + 0.16 = 0.61$ after a $2 rally — without you touching anything. Your hedge, set for a 0.45 delta, is now under-hedged by 0.16 delta per option. Re-balancing back to neutral is the chore gamma forces on you, over and over. The bigger the gamma, the more often the stock knocks your hedge out of true.

When it matters

High gamma means your delta-hedge goes stale fast — you re-hedge more often, and each re-hedge trade costs spread and commission. Gamma is the difference between a hedge you can set and forget for an hour and one you’re babysitting tick-by-tick. The sign of your gamma also decides whether that babysitting pays you or bleeds you — which is the convexity story.

Dollar gamma & convexity: the curvature you pay for

Analogy. Long gamma is like a seatbelt that also hands you cash in a crash: the bigger the jolt — up or down — the more you collect. Short gamma is selling those seatbelts; you pocket a steady premium right up until the crash that wipes it out. Gamma is convexity, and convexity is never free.

Definition. From the Taylor expansion, the gamma contribution to P&L per option, for a move of $\Delta S$, is the curvature term:

$\text{gamma P\&L} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2$

Notice the $(\Delta S)^2$: it’s squared, so it’s positive whichever way the stock moves and it grows with the size of the move, not its direction. Long gamma ($\Gamma > 0$) earns this term on any big move — a convex smile that curves up on both sides. Short gamma ($\Gamma < 0$) pays it out on any big move — the frown. “Dollar gamma” scales this to your full position so you can read the convexity in money terms.

Worked dollar-gamma example. You’re long, delta-neutral, with position gamma 0.08 per share across 1,000 share-equivalents of contracts (so position $\Gamma = 0.08 \times 1{,}000 = 80$ deltas per $1). The stock makes a $5 move ($\Delta S = 5$). The curvature P&L, beyond what delta already accounted for, is:

$\tfrac{1}{2}\,\Gamma\,(\Delta S)^2 = \tfrac{1}{2} \times 80 \times (5)^2 = \tfrac{1}{2} \times 80 \times 25 = 1{,}000 \text{ dollars}$

A $5 move in either direction hands the long-gamma book about $1,000 of convexity profit — on top of whatever the (re-hedged) delta did. Flip to short gamma and that same $5 move is a $1,000 loss. The asymmetry is the whole game: you either own the curvature and profit from chaos, or you’ve sold it and dread chaos.

Gamma explodes near expiry: the hedging nightmare

Analogy. A near-expiry at-the-money option is a coin balanced on its edge with seconds left to fall. Its delta has to resolve from “anyone’s guess” (≈0.5) to a hard 0 or 1 over a razor-thin price range — so the rate delta changes, gamma, goes through the roof. The hedge that was fine an hour ago is hopeless now.

Why. Look again at $\Gamma = n(d_1)/(S\sigma\sqrt{T})$. As $T \to 0$, the denominator’s $\sqrt{T} \to 0$, so for an at-the-money option gamma blows up toward infinity. The bell curve of gamma-vs-spot gets taller and narrower as expiry approaches: enormous gamma in a sliver around the strike, almost none elsewhere.

The practical horror: pin risk. If the stock is sitting right at the strike into the close on expiry day, your delta can flip wildly between roughly 0 and 1 on tiny moves — you genuinely don’t know if you’ll be assigned, and any hedge you put on is instantly wrong. That’s pin risk: the stock “pinned” to the strike, your delta hedge thrashing, transaction costs piling up as you chase it. Short-dated ATM gamma is where well-run desks get hurt.

A quick name-drop of the next level down. Gamma itself has derivatives: speed is $\partial\Gamma/\partial S$ (how gamma changes as spot moves — it tells you how fast your gamma profile shifts), and charm is $\partial\Delta/\partial t$ (how delta decays just from time passing — it warns you your hedge drifts overnight even if the stock doesn’t move). You won’t hand-compute these, but desks running big short-dated books absolutely watch them, because near expiry they stop being rounding errors.

The sign table: who’s long and short what

For a long position the signs of delta and gamma are completely fixed; writing (selling) flips every sign. Burn this grid in:

The pattern worth memorising: any long option is long gamma (call or put — you own curvature, you love big moves), and any short option is short gamma (you’ve sold curvature, big moves hurt you). Delta is the only one of the two whose sign depends on call-vs-put. So “long gamma” and “short gamma” cut across the call/put split entirely — what matters for gamma is whether you own the option or wrote it.

Putting it together

Delta is your stock exposure and your hedge ratio — $N(d_1)$ for a call, $N(d_1)-1$ for a put — and, loosely (overstated), the chance of finishing ITM. Aggregate it into position delta, scale by spot for dollar delta, and you can read the book’s linear risk in one number. Gamma is the curvature that makes delta move: same for call and put, positive when you’re long, bell-shaped and peaking ATM, and the source of the convexity term $\tfrac{1}{2}\Gamma(\Delta S)^2$ — convexity you profit from when long and dread when short. Near expiry, ATM gamma explodes and hands you pin risk. Hedge delta, and gamma is the thing that’s left to manage.

Next up — Theta, Vega, and the Cost of Convexity — we put a price tag on the gamma you just learned to love. Owning curvature isn’t free: every quiet day, theta nibbles at the premium you paid, and a shift in implied volatility moves your whole book through vega. The gamma–theta trade-off is where running an options book stops being a math exercise and starts being a daily judgement call.