Gamma: How Fast the Exposure Changes

Convexity, priced

Delta changes, and gamma is how fast. On the worked position — two $100 calls with a 0.52 delta and a 0.045 gamma — the share-equivalent exposure is 104 shares, so a dollar in the stock is $104. After a $2 rise the delta is 0.52 + 2 × 0.045 = 0.61, and the exposure is 122 shares. The extra gain that came from the change in delta rather than from the delta itself is about ½ × 0.045 × 4 × 100 × 2 contracts = $18, and three properties of that expression are worth noticing. It scales with the square of the move, so four times the move is sixteen times the benefit. It is small for small moves, which is why gamma feels like nothing until it is everything. And its sign is always favourable for a long option: whatever direction the market moves, the delta moves with it, so a long option gets longer when it is right and shorter when it is wrong. That asymmetry is what “long convexity” means, and it is why the round trip in the prediction costs money. After the $2 rise the position behaves like 122 shares; when the stock falls the $2 back, the loss is taken on an exposure that is shrinking — genuinely shrinking, since the delta falls as the stock declines — so the round trip does not return to its starting value. In practice the long-gamma position pays for this on the other side, in theta, which is the next lesson: the same ½γΔS² that makes a move profitable is what the position forfeits daily for the privilege of being long it. A cleaner way to hold the whole idea is that gamma and theta are the same trade: a position is either long gamma and short theta, or short gamma and long theta, and neither state can be entered for free. Gamma is not a constant either, and where it concentrates explains a great deal of behaviour. It is largest at the money, because that is where the probability of the option finishing on either side of the strike is most balanced and therefore where a small move most changes the exercise decision. It grows as expiry approaches, because with little time left the same price change does proportionally more to that decision — so an at-the-money option in its final week has enormous gamma and correspondingly enormous theta. And it shrinks as the option moves deep in or out of the money, because there the answer to “will this be exercised” has largely stopped changing. Those three facts are the whole explanation for why short-dated at-the-money options are discussed as gamma instruments: the payoff is dominated by the curvature rather than by the level of the delta. The same position, before and after a $2 move — Two contracts, delta 0.52, gamma 0.045: Exposure 104 shares, so $104 a dollar of stock · After a $2 rise: Delta 0.61, exposure 122 shares — the position got longer ← · Gamma’s contribution to that move: ½ × 0.045 × 2² × 100 × 2 = $18 · After a $10 decline: First-order delta 0.07, exposure 14 shares — it gets shorter as it loses ← Note where gamma is largest in a chain: at the money, and more so near expiry. That is why the same option moves from being a direction trade to being a curvature trade as the clock runs down.

What long and short gamma imply in practice

A trader who is long gamma is short time. Their position makes money on movement and loses money on stillness, and the amount of movement required to break even is computable in advance: it is the move at which ½γΔS² equals the theta being paid. That break-even is the honest statement of what a long option position needs, and it is a better answer than “the stock has to go up” — a long call that rises less than the gamma-theta break-even still loses, which is the mechanism behind the good-news loss that the next lesson treats properly. A trader who is short gamma is in the mirror position: paid every day the market stands still, and exposed to a large loss on a large move, with the loss growing faster than the credit because the same ½γΔS² term now works against them. Delta hedging is what turns that description into a business. A market maker who sells an option is short gamma and must buy the underlying as it rises and sell as it falls — buying high and selling low, which is a loss — and the daily theta they collect is the compensation for that mechanical behaviour. The frequency of rebalancing is therefore a choice with a price: hedge too often and the transaction costs eat the theta, hedge too rarely and the hedging error grows with the square root of the interval. That is the whole economics of market making in one sentence, and it is also why a market maker widens a spread when volatility is high: the hedging error they are accepting for a given rebalancing interval has grown. Two structural risks follow from gamma rather than from direction. The first is pin risk: when a stock settles very close to a strike, an option’s exercise decision is genuinely uncertain to the last minute, and a short position can end up assigned on some contracts and not others, leaving an unplanned position in the shares over the weekend. The second is the concentrated case — an at-the-money option in its final days, where gamma and theta are both extreme — which is the subject of the short-dated lesson later in this rung, and the reason those instruments are not simply “cheap options” but a bet on realised movement measured against an enormous decay. • Long gamma is short time: the break-even is the move at which ½γΔS² pays the theta. • Short gamma is paid to stand still and punished for moving — the same term with the sign reversed. • Delta hedging is buying high and selling low mechanically; theta is the fee for doing it. • Rebalancing frequency is a cost trade-off: too often pays the spread, too rarely builds error. • Pin risk and extreme short-dated gamma are structural risks of the curvature, not of the direction. The most expensive gamma mistake available to a retail trader is selling short-dated at-the-money options because the premium looks large. The premium is large *because* gamma is large: the position collects a fixed amount and is exposed to a loss that grows with the square of the move, so a 4% move in the underlying can cost several times the credit. A short-gamma position does not fail gradually; it fails on the day the market moves, and the fact that it was profitable every day before is the shape of the payoff rather than evidence of an edge.

Dollar gamma: the number that decides the day

Gamma is described as the rate at which delta changes, which is precise and hard to feel. The version that can be used is **dollar gamma**: the change in the position’s value for a one-percent move in the underlying, expressed in money. It converts the convexity into the unit a P&L statement uses, and it makes the difference between a position that is exposed and one that is fragile visible at a glance. The construction is a chain of multiplications. Gamma per contract is the change in delta for a one-point move in the underlying; multiply by the number of contracts, by the multiplier of one hundred, and by the size of the move you care about, and the result is the amount by which the position’s delta exposure changes over that move. Divide the P&L consequence by the capital at risk and the position has a number that can be compared with any other position in the book. The reason it matters more than gamma itself is that gamma has no units a trader can trade; dollar gamma is the same quantity denominated in the currency of the account. The number explains a set of behaviours that otherwise look like market mysteries. Positions held near a strike into expiry have a very large dollar gamma, because gamma rises sharply as time value collapses — which is why a small underlying move in the final week can produce a P&L swing that dwarfs anything the position did in the preceding month. It is also the arithmetic behind the observation that a large short-gamma book forces its holder to trade in the direction of the move: the hedging requirement grows with the move, so a dealer who is short gamma must chase, and a dealer who is long gamma leans against it. That is the mechanism behind pinning at large open-interest strikes and behind the amplified intraday ranges that appear when positioning is concentrated. The comparison that makes it practical is dollar gamma against the daily decay that pays for it. A long-gamma position collects convexity and pays theta, and the break-even is the move computed in the previous read; a short-gamma position does the reverse, collecting time and paying for every large move. Expressed as dollar gamma divided by daily theta, the ratio tells you how many standard deviations of movement are needed per day to keep the long position solvent, and it is the single most useful pair of numbers on a multi-leg ticket. One caution about precision, which is also a caution about platforms. Gamma is a second derivative computed inside a model, so its value is more sensitive to the model’s assumptions and to numerical method than delta is — which is why the gamma displayed on a far out-of-the-money strike can be mostly numerical noise, and why different systems disagree on it more than they disagree on the first derivative. Treat small dollar-gamma figures on remote strikes as approximate, and trust the number on positions near the money and near expiry, where it is large and therefore meaningful. • Dollar gamma is the P&L change for a one-percent move, in the account’s currency. • It is largest near the strike into expiry, which is why the last week is different. • Short-gamma books must trade with the move; long-gamma books lean against it. • Pair it with daily theta to see the movement the position needs per day. A useful habit after any multi-leg order is filled: write the position’s dollar gamma per percent next to its daily theta. Two numbers that cost a minute to compute describe the entire risk the structure will take to expiry.

Sources

• Delta, gamma and the convexity of an option positionNatenberg, “Option Volatility and Pricing”
• Delta hedging, hedging error and the cost of rebalancingTaleb, “Dynamic Hedging”
• Gamma near expiry, pin risk and settlement mechanicsOption Clearing Corporation (OCC) expiration and exercise documentation