Chapter 4: Delta and gamma · US Options Course

In this chapter

Opening: the position grew on its own
4.1 Three ways to read delta
4.2 How delta changes
4.3 Portfolio delta and beta weighting
4.4 Gamma: the speed of delta
4.5 The delta–gamma P&L approximation
4.6 The gamma–theta trade-off
4.7 A gamma scalping walkthrough
4.8 Market-maker gamma and the market
Three cases
Checklist and self-test

After this chapter you can

Convert any options position into “how many shares it equals”
Estimate your P&L for ±3% and ±5% moves before placing an order
Work out how far the stock must “move” each day for a long option to break even

Prerequisites

Chapter 1: the four basic positions and payoff diagrams at expiry
Chapter 3: the six inputs to premium

Conventions

Examples use Black-Scholes, a 4% rate, no dividends
Days are calendar days; a year is 365 days
One contract = 100 shares

OpeningThe position grew on its own

You buy 10 at-the-money calls: stock at $100, strike 100, 5 days to expiry, implied volatility 30%. Over the next few days you don't add a single contract, and the stock rises only 2%.

On the day you buy, each option's Delta is 0.513, so 10 contracts are equivalent to holding 513 shares. By the last day, with the stock at $102 and 1 day to expiry, Delta has become 0.899, equivalent to 899 shares. You did nothing, yet your directional exposure grew by 75%.

Table 4-0 Share equivalent of 10 contracts of the 100 call over the last five days before expiry
Time	Stock price	Days left	Delta per contract	Equivalent shares	Option price
Day 1	100.0	5	0.513	513	1.43
Day 2	100.8	4	0.612	612	1.72
Day 3	101.2	3	0.679	679	1.82
Day 4	101.6	2	0.769	769	1.93
Day 5	102.0	1	0.899	899	2.09

This is Gamma at work: it makes Delta change automatically with price and time. Stock has no such property; 100 shares are always 100 shares. Option risk can “grow on its own,” and this chapter teaches you to see it coming.

4.1Three ways to read delta

The same Delta number can be used in three ways. All three are valid, but each has its limits.

Reading 1: price sensitivity

When the underlying rises $1, the option's price changes by about Delta dollars. For a call with a Delta of 0.53, a move in the stock from 100 to 101 takes the option from about 3.59 to 4.12. Call Deltas lie between 0 and 1; put Deltas lie between −1 and 0.

Reading 2: hedge ratio and “share equivalent”

Delta × 100 is the number of shares one contract is equivalent to. This is the most practical reading, because it lets you add up stock and different options directly.

Position Delta = Σ (Delta per contract × 100 × contracts × direction)

Direction: +1 for long, −1 for short. Each share of stock always has a Delta of 1.

Buy 2 calls with Delta 0.40: +0.40 × 100 × 2 = +80 shares
Sell 3 puts with Delta −0.30: −0.30 × 100 × 3 × (−1) = +90 shares
Total +170: directionally the same as holding 170 shares. Selling a put is bullish, just like buying a call

Reading 3: probability of expiring in the money (approximate)

Delta is often used as the “probability of expiring in the money.” That is only an approximation: in Black-Scholes, Delta is N(d₁), while the risk-neutral probability of finishing in the money is N(d₂), and the former is always slightly larger. Also, this is the probability implied by market prices, not the real-world probability.

Table 4-1 Delta vs. probability, stock $100, 30 days, IV 30%
Call strike	Delta N(d₁)	Probability ITM at expiry N(d₂)	Probability of touching the strike before expiry ≈ 2 × N(d₂)
105	0.313	0.284	≈ 0.57
110	0.152	0.133	≈ 0.27

Key point

“The probability of touching is about twice the probability of finishing in the money” matters especially to sellers. A call with a Delta of 0.15 has only a little over a 10% chance of expiring in the money, but nearly a 30% chance that the stock touches 110 during those 30 days. A seller's stop-loss usually gets hit on the touch, not at expiry.

4.2How delta changes

Three things push Delta around: price, time and volatility. The effect of price on Delta is Gamma (Section 4.4); the effects of time and volatility have their own names, Charm and Vanna. Get a feel for them first with the chart below.

Delta's S-curve against the stock priceFigure 4-1 · Strike 100, interactive

Days left 30 days Implied volatility 30%

Current settingsReference: 30 days, IV 30%

—Stock at 95

—Stock at 100

—Stock at 105

Drag the days left down to 1–3 and the curve becomes an almost vertical step: near the money, a tiny price change makes Delta jump from 0.2 to 0.8. Raise implied volatility and the curve flattens: out-of-the-money Deltas are pushed up and in-the-money Deltas are pulled down.

Price: the closer to the money, the faster delta changes

Table 4-2 Call Delta by strike, stock $100, IV 30%
Strike	85	90	95	100	105	110	115
30-day call	0.98	0.90	0.75	0.53	0.31	0.15	0.06
5-day call	1.00	1.00	0.93	0.51	0.09	0.00	0.00
30-day put	−0.02	−0.10	−0.25	−0.47	−0.69	−0.85	−0.94

Put Delta equals call Delta minus 1 (same strike, same expiration). So the at-the-money call is about 0.53 and the at-the-money put about −0.47, not an even split: d₁ contains a drift term from the interest rate and volatility, which puts the at-the-money call's Delta slightly above 0.5.

Time: charm pushes delta toward 0 or 1

Even if nothing else changes, Delta changes as a day passes. This rate of change is called Charm. Out-of-the-money Deltas drift toward 0 each day, and in-the-money Deltas drift toward 1. Going from 30 days to 29, the 110 call's Delta drops from 0.152 to 0.148; with 10 days left it is only 0.031. The 90 call, meanwhile, rises from 0.904 to 0.985 at 10 days.

Watch out

Before the Friday close, market makers adjust their hedges on large numbers of options about to expire. Out-of-the-money Deltas collapse to zero and the stock held against them is unwound; that flow comes from Charm. A lot of “unexplained moves on expiration-day afternoons” are related to this.

Volatility: vanna changes delta

The change in Delta for each 1-point change in implied volatility is called Vanna. The 110 call (30 days) has a Delta of 0.152 at IV 30%; if IV rises to 31%, it becomes 0.161; at 40%, 0.228; if IV falls to 20%, only 0.057.

This means that when IV collapses after earnings, both the price and the Delta of your out-of-the-money options shrink at the same time: even if you're right on direction, you capture less of the move.

4.3Portfolio delta and beta weighting

When you hold several stocks and options, the Delta of each position on its own can't answer “how much am I betting on the market overall?” The solution is to convert every position into “how many shares of SPY it equals.”

SPY-equivalent shares = Position Delta × Stock priceSPY price × β

Position Delta is expressed in shares; β is the stock's Beta to the S&P 500. The price ratio converts “shares” into “dollars” and then back into SPY shares.

Example: SPY is at $780. 200 shares of Stock A at $150 with β 1.3 equal 200 × 150 / 780 × 1.3 = 50 shares of SPY. The calculator below comes filled in with three positions; you can replace them with your own.

Beta-weighted delta calculatorTool 4-A · Editable

SPY priceDelta of the SPY put used to hedge

Position	Type	Quantity	Delta per unit	Underlying price	β	SPY-equivalent shares

—Portfolio total: SPY-equivalent shares

—Equivalent SPY market value

—SPY puts to buy for a full hedge

When the type is “Option,” enter the number of contracts as the quantity (negative for short) and the per-contract Delta as Delta per unit; when it is “Stock,” enter the number of shares, and Delta per unit is fixed at 1.

Mind the units

The portfolio above totals about 300 shares of SPY. To hedge it, you short about 300 shares of SPY, or buy about 7–8 contracts of an SPY put with Delta −0.40 (−40 shares each). Mistake “300” for a number of contracts and your hedge is off by a factor of 100.

Beta weighting often exposes “hidden concentration”: three tech stocks that look unrelated plus a QQQ call may, once converted, all be the same directional market exposure. Positions in tech, high-Beta stocks, small caps and calls are often one and the same directional risk; you can't count tickers and conclude you are diversified.

4.4Gamma: the speed of delta

Gamma is the change in Delta for each $1 rise in the underlying. If Delta is your speed, Gamma is the gas pedal.

Sign: long options (calls or puts) have positive Gamma; short options have negative Gamma; stock has zero Gamma.
Positive Gamma: Delta grows as the stock rises (you get longer as it goes up) and shrinks as it falls (you get less long as it goes down). The P&L curve bends upward.
Negative Gamma: the exact opposite. You get shorter as the stock rises and longer as it falls, and the P&L curve bends downward.

Gamma peaks at the money and spikes near expiryFigure 4-2 · Strike 100, IV 30%

The same at-the-money option has a Gamma of about 0.033 with 60 days left and about 0.18 with 2 days left, more than five times larger. The out-of-the-money 110, by contrast, has almost no Gamma near expiry: it is too far away to move Delta much anymore.

Table 4-3 Gamma and Theta of an at-the-money option (stock and strike both 100, IV 30%)
Days left	Gamma	Delta gain per contract after a $1 rise (shares)	Daily Theta (per share, $)
365	0.0128	1.3	−0.021
90	0.0265	2.7	−0.038
30	0.0462	4.6	−0.062
7	0.0960	9.6	−0.124
2	0.1796	18.0	−0.227
1	0.2540	25.4	−0.319

Notice that Gamma and Theta grow together. This is no coincidence; as Section 4.6 explains, Theta is the price of Gamma.

Comparing across underlyings: use “dollar gamma”

Gamma's unit is “change in Delta per $1.” A $1 move is 10% for a $10 stock but only 0.1% for an $800 stock, so you can't compare Gamma directly across underlyings at different prices. In practice people use 1% dollar Gamma: how many dollars' worth of stock your Delta changes by when the stock moves 1%.

1% dollar Gamma = Γ × 100 × contracts × S × (1% × S)

This is the quantity Section 4.8 uses to estimate market makers' Gamma exposure.

4.5The delta–gamma P&L approximation

The single most useful formula before you place an order: use the current Delta and Gamma to estimate the P&L after a stock move of ΔS.

P&L ≈ Δ · ΔS + 12 Γ · (ΔS)² + Vega · Δσ + Θ · Δt

The first two terms are the familiar Delta–Gamma approximation. The last two are the effects of volatility and time; they are often ignored, but around earnings they are usually the main event (see Case 2).

Take a 30-day at-the-money call: Delta 0.532, Gamma 0.0462, price 3.591. If the stock rises $5, the approximation gives 0.532 × 5 + ½ × 0.0462 × 25 = 3.24; repricing with Black-Scholes gives 3.207. If it falls $5, the approximation gives −2.084 and the actual result is −2.082. For the same $5, the gain on the way up beats the loss on the way down: that is the asymmetry of positive Gamma.

The approximation is nearly perfect for small moves and drifts off for large onesFigure 4-3 · 30-day ATM call, P&L per share

IV change 0 points Days passed 0 days

Actual P&L (repriced)Delta onlyDelta + Gamma

With both sliders at 0, the pink and black lines nearly coincide within ±5. Drag IV down 20 points and the whole black line sinks while the approximation doesn't budge: this is where “right on direction but still losing money” comes from after earnings. Drag the days passed to the right and you can watch Theta eat the premium day by day.

Table 4-4 Approximation vs. full repricing (per share, IV and time unchanged)
Stock move	Actual P&L	Delta only	Delta + Gamma	Error
−10	−3.122	−5.324	−3.012	+0.110
−5	−2.082	−2.662	−2.084	−0.002
−3	−1.387	−1.597	−1.389	−0.002
+3	+1.799	+1.597	+1.805	+0.006
+5	+3.207	+2.662	+3.240	+0.033
+10	+7.300	+5.324	+7.635	+0.335

When to trust it

Move within 1 daily standard deviation (about S × IV ÷ 19): the error is negligible
Move beyond 2–3 standard deviations, or less than a week to expiry: Gamma itself is changing fast, and the error grows quickly
An event ahead (earnings, FOMC): you must add the Vega term, and it is better to reprice outright

4.6The gamma–theta trade-off

Buying options gets you positive Gamma at the cost of paying Theta every day; selling options collects Theta at the cost of carrying negative Gamma. The two aren't independent: the Black-Scholes equation ties them together.

Θ + 12 σ²S²Γ + rSΔ − rV = 0

Drop the interest-rate terms and you get Θ ≈ −½ σ² S² Γ: the size of Theta is proportional to Gamma. The bigger an option's Gamma, the bigger its daily time decay.

How far the stock must move each day to break even

If you buy an option and hedge its Delta to zero with stock, a day's P&L has only two terms left: ½Γ(ΔS)² from Gamma, minus Theta. Set them equal and you get the daily move needed to break even:

|ΔS|_breakeven = √(2|Θ| / Γ) ≈ S × σ√365 ≈ S × σ19

Counting calendar days, you divide by √365 ≈ 19. Traders often say “divide by 16”; that counts trading days (√252 ≈ 15.9, rounded to 16): the weekend's two days of time decay are folded into Friday, so each trading day requires a bigger move.

30-day at-the-money call: Theta −0.0624, Gamma 0.0462, breakeven move √(2 × 0.0624 / 0.0462) = $1.64. The shortcut gives 100 × 0.30 / 19.1 = $1.57; the small gap comes from the interest-rate term. Under the trading-day convention it is 100 × 0.30 / 16 ≈ $1.88.

Hedged daily P&L is a parabolaFigure 4-4 · ATM call + stock hedge, per share

Days left 30 days Implied volatility 30%

—Daily move to break even

—Shortcut S·σ/√365

—Daily loss if the stock stays put

Between the parabola's two crossings of the horizontal axis (the shaded area) is the loss zone: the stock didn't move enough for Gamma's gains to cover Theta. Reduce the days left and the parabola gets narrower and deeper: more Gamma, but more expensive Theta.

Bottom line

Buying options is essentially a bet that “realized volatility will exceed implied volatility”; selling options is the opposite bet. Direction is only part of the trade.

4.7A gamma scalping walkthrough

Gamma scalping turns the conclusion of Section 4.6 into a procedure: buy options to get positive Gamma, then every time the stock moves, use stock to bring Delta back to zero. Positive Gamma makes the process “sell high, buy low” automatically: when the stock rises, Delta grows and you sell stock; when it falls, Delta shrinks and you buy it back.

Here is a concrete position over five days: buy 10 at-the-money 30-day straddles (10 calls + 10 puts, strike 100, IV 30%), hedge to zero at each close, and assume slippage of $0.02 per share. The starting position has Gamma of about 92.5 shares per dollar and Theta of about −$114 a day.

One position, two scenariosFigure 4-5 · 10 straddles, hedged at each close

Close▼ Sell stock▲ Buy stock

Day	Close	Move	Option P&L	Hedge P&L	Rebalance (shares)	Day total	Cumulative

—Cumulative P&L after five hedged days

—Same period, straddle unhedged

—Realized volatility of this path

In Scenario A the stock closes day 5 at 101.9, nearly back where it started; holding the straddle unhedged would lose $304. Hedged, you sell high and buy low every day and make $797 over five days. In Scenario B the moves are too small: Gamma earns only a dozen or so dollars a day while Theta takes $114, and you lose $542 over five days. The dividing line between the two outcomes is whether realized volatility exceeds the 30% implied volatility you bought at.

Extra costs in the real world

Hedging frequency: the more often you hedge, the closer you get to the theoretical value, but the higher your commissions and slippage. In practice, traders often hedge when Delta drifts past a threshold (say, beyond ±50 shares) rather than on a fixed schedule.
Gaps: an overnight gap leaves you no time to rebalance. Positive Gamma benefits from gaps; negative Gamma is most dangerous in them.
IV changes: the walkthrough assumes IV stays constant. If IV falls after you buy, the straddle loses extra value, and the scalping profit may be swallowed by the Vega loss.

For individual traders

This is routine work for market makers, but for individuals it is costly and demanding. The more realistic use is to read it the other way around: when you sell options to collect Theta, you are on the other side of this table, profiting from “Scenario B” and fearing “Scenario A.”

4.8Market-maker gamma and the market

Market makers usually hedge the Delta of the options they sell, and the Gamma they hold determines whether those hedges push prices along or hold them back.

Market makers' gamma sign sets the direction of their hedgingFigure 4-6 · Schematic

In a positive-Gamma environment, market makers' hedging goes “against the trend,” so the market tends to trade sideways and get pinned near strikes with large open positions. In a negative-Gamma environment, hedging goes “with the trend,” and moves in both directions get amplified. The price level that divides the two environments is what people call the Gamma Flip.

How to estimate market makers' gamma exposure

No one can see market makers' positions directly. All the GEX (Gamma exposure) data out there is estimated, and the most common method has three steps:

For each strike, multiply open interest (OI) by the Gamma of the options at that strike, then convert to 1% dollar Gamma as in Section 4.4.
Assume market makers are net long calls and net short puts (customers mostly sell covered calls and buy protective puts). Calls then contribute a positive sign and puts a negative one.
Sum across all strikes to get net GEX; repeat the calculation at different hypothetical stock prices, and the level where the net value flips from positive to negative is the Gamma Flip.

Table 4-5 Two hypothetical strikes (SPY $780, 7 days to expiry, IV 16%)
Contract	Open interest	Gamma per contract	1% dollar Gamma	Sign for market makers
780 call	60,000	0.0231	$842M	+
770 put	80,000	0.0190	$922M	−
Net			−$80M	Slightly negative Gamma

How to read it: if market makers really held these positions, every 1% drop in SPY would force them to sell about $80 million more stock to stay neutral. The numbers are hypothetical, to illustrate the method.

Limits of the estimate

“Market makers long calls, short puts” is only an average assumption; at individual strikes it may be the exact opposite
OI updates only once a day, so it misses same-day 0DTE flow
Vanna and Charm drive hedging too; looking at Gamma alone misses the flows caused by IV and time

From the field

Many traders' pre-market notes use this framework. Before one triple-witching day, for example, a note judged that a break below SPY 760 favored buying puts while holding above 760 favored buying calls, on the grounds that positive Gamma dominated above 760. Another attributed a sharp MU rally to market-maker hedging driven by a large volume of calls expiring the next day, not to improving fundamentals. Another warned that for chip stocks on a winning streak, you should wait until after options expiration to draw conclusions: if they pull back after expiry, the rally was pushed up mainly by hedging flows. Barchart's Gamma page is a commonly used free place to check.

CasesThree cases

Case 1

The same ±3% is a completely different position in the final days

An at-the-money 100 call, IV 30%. Compare its Delta at stock prices of 97, 100 and 103 with different numbers of days left:

Days left	Stock 97	Stock 100	Stock 103	Delta gap, 97 to 103
30	0.392	0.532	0.665	0.27
10	0.286	0.519	0.740	0.45
5	0.202	0.513	0.809	0.61
3	0.137	0.510	0.867	0.73
1	0.027	0.506	0.971	0.94

With 30 days left, as the stock wobbles between 97 and 103, Delta moves only between 0.39 and 0.67, and the position behaves like a steady stock. With 1 day left, the same wobble makes Delta jump back and forth between 0.03 and 0.97: one moment you have almost no position, the next you are effectively fully long.

For buyers: an at-the-money option in its last few days is a lottery ticket. Your chance lies in a jump, but Theta rises in step with Gamma (Table 4-3).
For sellers: the last few days collect the most premium, but one big up day can take you from “almost no risk” to “fully short.” This is why so many sellers' accounts get into trouble in expiration week.

Case 2

A delta–gamma stress test before earnings that missed the biggest term

Before earnings you buy 20 contracts of the 30-day at-the-money 100 call. IV has been bid up to 60% ahead of the report; the position has Delta of +1,084 shares, Gamma of 46.1, and Vega of about $227 per IV point. The day after earnings, IV falls back to 35%.

Table 4-6 P&L on 20 calls the day after earnings ($)
Stock move	Delta–Gamma approximation	Repriced, IV unchanged	Repriced, IV down to 35%
−8	−7,194	−7,360	−11,798
−5	−4,842	−5,053	−10,147
0	0	−240	−5,839
+5	+5,995	+5,739	+445
+8	+10,146	+9,835	+5,025

The first two columns are close, so the Delta–Gamma estimate of the price effect is fine. The whole gap is in the third column: IV drops 25 points, times $227 per point, is a Vega loss of about $5,700. The stock rose 5% as expected, and you made only $445.

Key point

Before an event, run at least two sets of stress tests: price × IV unchanged, and price × IV falling back to its normal pre-event level. If even “right on direction” doesn't make money in the second set, you picked the wrong structure; consider a spread (the two legs' Vega partly cancels) or a calendar spread.

Case 3

When options “push” a stock

GameStop in January 2021 is the most famous example of negative-Gamma amplification. Retail traders bought short-dated out-of-the-money calls in huge numbers, and the market makers who sold them had to buy stock to hedge. The rising stock made those out-of-the-money Deltas grow quickly (Gamma), while soaring IV pushed Delta up further (Vanna). Market makers had to buy more and more stock, the buying drove the price higher, and the loop fed on itself.

You don't need an extreme case like this to use the logic. Day to day, the small-scale version is more common: a large pile of calls builds up at a strike near expiry, market makers buy stock to hedge, and the stock gets pushed toward that strike. A rally like this comes from an “options effect,” not from better fundamentals. The test is the same: after expiration, when the hedging demand disappears, see whether the price falls back.

Delta and gamma