Your Hedge Is Worth More and Does Less
A single call spread bought against a ladder of capped liabilities creates two mismatches at once. They have opposite signs. At expiry they cancel; on every day before it they do not — and the standard hedge effectiveness report shows you both without ever saying they are related.
The complaint is always the same shape. The hedge was matched at inception — delta-neutral, signed off, documented. The index went up all year. At the November review somebody notices the book is under-hedged and has been for months.
Nobody rebalanced wrong. Nobody missed a trade. Nothing was mismarked. The book drifted because of a decision made a year earlier about how to group the liabilities, and the price of that decision was never written down.
It could have been. That is the whole point of this piece.
The intuition, and why it is only half true
Start where most people start, because the intuition is a good one.
A cohort has policies capped at 5% and policies capped at 10%. You hedge it with one call spread at the 7.5% average. The index rallies past 7.5%, your hedge is pinned at its cap and stops earning — but the 10%-cap policies are still crediting, all the way to 10%. You are short the difference.
Every part of that sentence is true, and the conclusion is still wrong.
It is wrong because there is a mirror region below, and it comes first. Between 5% and 7.5%, the hedge is still accruing while half the liability is already pinned at its 5% cap. The hedge over-earns there by exactly as much as it under-earns above.
Laid out across the range:
| Index up | Liability credits | Hedge pays | Hedge − liability |
|---|---|---|---|
| +5.00% | 5.000% | 5.00% | 0.000% |
| +6.25% | 5.625% | 6.25% | +0.625% |
| +7.50% | 6.250% | 7.50% | +1.250% |
| +8.75% | 6.875% | 7.50% | +0.625% |
| +10.00% | 7.500% | 7.50% | 0.000% |
| +15.00% | 7.500% | 7.50% | 0.000% |
The hedge is never behind. It is ahead through the entire middle of the range, peaking at the average cap itself, and it converges back to level once the index clears the highest liability cap.
This is not an accident of the 50/50 split. A capped credit, min(index, cap), is a concave function of the cap. Jensen's inequality then says the average of the capped payoffs can never exceed the payoff at the average cap:
average of min(X, capᵢ) ≤ min(X, average cap)
The left side is the liability. The right side is the hedge. A notional-matched hedge at the weighted-average cap is never short at expiry, for any distribution of caps. I ran two thousand randomly generated ladders looking for a counterexample and there isn't one, because there cannot be.
So the terminal problem is real but it is the opposite of the one people describe. It is not a shortfall. It is waste — you are buying up to 1.25 points of coverage, in this cohort, that no policyholder can ever claim. That is money spent on the option budget for a payoff region the liabilities do not reach.
Which raises the obvious question. If the endpoint is fine — over-covered, even — why does an aggregated book so reliably show up under-hedged on the daily report?
Because the endpoint is not where you live.
So why does the path misbehave? Gamma
The received version goes like this. Early in the term, when the index is near the lower strike, a call spread is modestly long gamma. As expiry approaches and the index climbs toward the cap, the short leg takes over and gamma turns sharply negative.
The second half is right. The first half is wrong, and it is wrong at exactly the moment you are sizing the hedge.
Where gamma actually peaks
Gamma is not largest at spot. It is largest at the strike where d₁ = 0, which is
K* = S · exp((r − q + σ²⁄2)T)
For an index at 100, one year out, at 20% volatility, with rates at 4.5% and a dividend yield of 1.5%, that is 105.13.
That number sits above spot. Which means a call spread struck at the money has its short leg nearer the gamma peak than its long leg. The short leg carries more gamma than the long leg, and the net is negative.
Priced out, at spot 100 with a year to run:
| Structure | Gamma at inception |
|---|---|
| 100 / 105 | −0.000604 |
| 100 / 110 | −0.000107 |
| 100 / 115 | +0.001278 |
Every structure a real indexed annuity program would use — caps between 3% and 10% — is short gamma from the first day. The sign only turns over once the cap is far enough out that the short leg falls away from the peak, which is well beyond where anyone sets a cap.
The 100/105 is the most negative of the set, because 105 is almost exactly the gamma peak. The structure that looks most conventional is the one most exposed.
Note also what drives K*. It is the forward, not volatility. Cut rates and K* falls toward spot; raise them and it moves further out. The sign of your gamma at inception is a rates question before it is a volatility question.
This also follows from the general result for where a call spread's gamma changes sign, S* = √(K₁K₂) · exp(−(r − q + σ²⁄2)T), which for the 100/105 gives 97.47. Below that level the spread is long gamma. Above it — which is where you are on day one — it is short.
But that is not what makes the hedge drift
Here is the part that matters, and it is the part I got wrong the first time I wrote about this.
Being short gamma does not, on its own, cause any drift at all. Take the liability and hedge each segment with its own matching spread — the textbook one-to-one approach. Every position in the book is short gamma. Net gamma is exactly zero. The hedge and the liability move together forever, because they are the same instrument.
Drift requires a mismatch in gamma, not a level of it.
Where the mismatch comes from
Consider a cohort split evenly between policies capped at 5% and policies capped at 10%. Hedging that with one spread at the average cap of 7.5% is standard practice, and it is a reasonable thing to want to do — one trade instead of two, one position to reconcile, one line in the file.
Size it so the delta matches at inception and you get:
| Delta | Gamma | |
|---|---|---|
| Liability (50% at 5% cap, 50% at 10% cap) | 0.1402 | −0.000355 |
| Hedge (single 7.5% cap, delta-matched) | 0.1402 | −0.000483 |
| Net | 0.0000 | +0.000124 |
Both sides are short gamma. The hedge is more short gamma than the liability. So when the index rises, the hedge sheds delta faster than the liability does, and the book slides under-hedged.
The reason is Jensen's inequality again — the same tool as the payoff section, pointed at a different quantity. There it was applied to the capped credit and told us the hedge over-covers. Here it is applied to gamma, which is also curved in the cap, so the average of the gammas is not the gamma of the average cap:
- average of the two gammas: −0.000355
- gamma of a single spread at the average cap: −0.000483
- gap: +0.000127
That gap is the net gamma, almost to the digit. The mismatch is not an approximation error or a modelling artifact. It is a specific, computable consequence of replacing a distribution of caps with its mean.
And it disappears when the distribution does. A homogeneous cohort — every policy capped at 7.5%, hedged with one 7.5% spread — has net gamma of exactly zero, no matter how short gamma the position is. The drift is caused by the spread of caps inside the cohort, not by the aggregation itself.
That is an actionable distinction. It says the question is not "how few trades can we get away with," but "how wide a cap distribution are we willing to put inside one hedge."
What it costs over a year
Run the index up 1% a month for a year and watch the two sides separate:
| Month | Index | Liability delta | Hedge delta | Net delta |
|---|---|---|---|---|
| 0 | 100 | 0.1402 | 0.1402 | 0.0000 |
| 3 | 103 | 0.1603 | 0.1600 | 0.0002 |
| 6 | 106 | 0.1894 | 0.1886 | 0.0008 |
| 9 | 109 | 0.2350 | 0.2319 | 0.0030 |
| 11 | 111 | 0.2496 | 0.2336 | 0.0160 |
Matched at inception, under-hedged by 0.016 of delta eleven months later, having done nothing wrong.
A steadily rising market is the bad case specifically because the mismatch is one-directional. In a choppy market the drift accumulates and unwinds; the hedge is under-hedged for a while, then over-hedged, and the errors partly cancel. In a persistent rally every month pushes the same way, and the gap compounds — note that most of it arrives in the last two months, as gamma steepens into expiry.
On a $1 billion book, a net delta of 0.016 means a 1% index move costs about $160,000 in unhedged P&L. That is a real number and it is worth avoiding, but it is worth being precise about the size: it is not a catastrophic hole, and describing it as one is a good way to lose an audience that can do the arithmetic. What makes it worth managing is that it is free to avoid if you see it coming, and expensive to fix if you do not — because you fix it by buying options in exactly the market that created the problem.
The final payoff, incidentally, nets out fine. At year end the index is up 12%, the 5% cap policies credit 5%, the 10% cap policies credit 10%, the average is 7.5% and the hedge pays 7.5%. If you only ever look at settlement, none of this shows up. It shows up in the path — in the P&L volatility between here and there, and in what the hedge is worth if you have to move it.
What this looks like on a real effectiveness report
The simulation above is synthetic. The signature is not.
Take a live indexed annuity hedge book — index call spreads and vanilla calls against index-linked liabilities — and run the standard effectiveness tests over seventeen months of daily data. 362 observations. Three measures, all of them conventional, all of them on the same report.
The value ratio. Asset market value over liability value. It runs from 0.949 to 1.227 and averages 1.106. The hedge assets are worth about eleven percent more than the liabilities they cover, and they are worth more on 359 of the 362 days. Read that column alone and the book looks comfortably over-covered.
The dollar-offset ratio. Asset dollar delta over liability dollar delta, against the conventional 80%–125% band. It runs from 0.906 to 1.204 and averages 0.985. Every single day is inside the band — 362 out of 362, not one exception. By this test the hedge is effective every day of the period.
Regression R². Trailing twelve months of daily asset and liability deltas, against the conventional 0.80 threshold. The six rolling windows report 0.71, 0.75, 0.71, 0.83, 0.89 and 0.84. Three of the six fall below 0.80.
Same book, same days, same two columns of underlying data. One measure says over-covered, one says effective, one says marginal.
Now put the first two side by side, day by day.
On 270 of the 362 days — 74.6% — the hedge was worth more than the liability and responded less than the liability. More value. Less sensitivity. And the correlation between the two ratios across the period is +0.105, which is to say almost none: watching the value ratio tells you essentially nothing about the delta ratio.
That is what being short gamma against your own liability looks like once you cash it out. The hedge holds its worth and stops moving. It has bought intrinsic value and given up responsiveness — which is exactly the trade the earlier sections describe, showing up in a production report as two columns that appear to be measuring the same thing and are not.
Neither ratio has a sign term or a memory, so the thing that actually happened appears on neither. The book was under-hedged on 271 of 362 days, 74.9% of the time, with a median shortfall of 2.33% of liability delta. If the sign of the net delta were tracking noise you would expect about half the days on each side; 271 of 362 is roughly nine and a half standard deviations from a coin flip. That is not noise.
And it is not a constant policy haircut either, which is the tell:
| Quarter | Days under-hedged | Median net delta |
|---|---|---|
| Q2 2025 | 47% | +1.06% |
| Q3 2025 | 88% | −2.65% |
| Q4 2025 | 100% | −4.51% |
| Q1 2026 | 78% | −2.07% |
| Q2 2026 | 61% | −1.49% |
| Q3 2026 | 74% | −3.05% |
One full quarter — sixty-six consecutive trading days — without a single day on the other side of the line.
Every one of those days passed the limit test. The limits are absolute bands: green inside a fixed net-delta tolerance, amber at twice it, red at three times. A book pinned just inside the green band on the same side every day for a year is green every day for a year. The control measures magnitude. It is structurally blind to direction and to persistence.
I want to be careful about what this does and does not establish. A book can run systematically under-hedged for reasons that have nothing to do with gamma: an option budget that buys slightly less than full delta, a deliberate policy of carrying a small net long, timing between when liabilities are struck and hedges placed, or plain valuation differences between the asset and liability models. Any of those produces a negative net delta column. The quarterly pattern argues against a fixed haircut, since a deliberate 1.5% under-hedge does not swing from 47% of days to 100% and back to 61%. But the honest claim is that this book carries the signature the gamma mechanism predicts — persistent, one-sided, worse in some quarters than others — not that the mechanism is proven to be the cause.
What it does establish is narrower and more useful. The tests everyone runs would not have told you. One passed every day. One suggested the book was over-covered. The third flagged something ambiguous in half its windows and gave no direction. A quarter spent entirely on one side of the line appears on no line of the report.
Adding it costs nothing. One row, counted over a column the report already contains:
Net delta negative on 271 of 362 days. Longest one-sided run: 66 days.
Notional or delta: which error did you choose?
There is a prior question underneath all of this, and different desks answer it differently without always knowing there was a question.
When you buy one call spread against a ladder of capped liabilities, how big do you buy it? Two conventions are in common use. Match the notional — buy one unit of hedge for one unit of liability. Or match the delta — size it so the hedge and the liability have the same sensitivity on the day you put it on.
They are not the same trade, and the difference is not a rounding decision.
You have one lever and two things you might want it to do. Match the endpoint, or match today. You cannot do both with a single instrument, so the sizing convention is a choice about which error you would rather carry.
Notional sizing guarantees the endpoint. From the concavity result above, a notional-matched hedge at the average cap is never short at expiry, whatever the index does. What it costs is a visible over-hedge on delta from the first day, and that over-hedge grows with the dispersion of the cap ladder.
Delta sizing guarantees today. The book is neutral at inception by construction — which is what the effectiveness report will show, and it is a comfortable number to report. What it costs is that the hedge is slightly under-sized on notional, so once the index clears the highest liability cap the hedge is short, permanently and by a fixed amount.
Priced across ladders that all carry the same average cap:
| Cap ladder | Delta-matched size | Notional-sized: delta gap at inception | Delta-sized: worst terminal shortfall |
|---|---|---|---|
| 7% / 8% | 0.9998 | +0.02% | −0.0018% |
| 5% / 10% | 0.9942 | +0.58% | −0.0435% |
| 3% / 12% | 0.9814 | +1.89% | −0.1394% |
| 2% / 15% | 0.9620 | +3.95% | −0.3226% |
The two right-hand columns are in different units — one is a percentage of liability delta, the other a percentage of notional — so they are not directly comparable, and nothing here says one is larger than the other. What the table does show is that both grow with the same driver. On a tight ladder the conventions agree to the fourth decimal and this whole argument is academic. The wider the spread of caps inside one hedge, the more the choice matters. It is the same lever again.
What makes the choice interesting is that the two properties are not equally durable.
The terminal guarantee from notional sizing holds for the full term. It cannot decay, because it is a statement about the payoff function rather than about current conditions.
The delta match decays immediately:
| Month | Index | Hedge delta vs liability delta |
|---|---|---|
| 0 | 100 | 0.00% |
| 4 | 104 | −0.20% |
| 6 | 106 | −0.40% |
| 8 | 108 | −0.83% |
| 11 | 111 | −6.41% |
By month eight, the delta-sized hedge is further from matched than the notional-sized hedge was on the day it was struck. The neutrality you bought has decayed past the error you were avoiding, and it did so inside three quarters.
That is not an argument that delta sizing is wrong. It is an argument that it is right for a different program. If you rebalance actively, the terminal property is close to irrelevant — you will re-size long before expiry, and starting neutral is worth more than starting over-hedged. If you strike the hedge and hold it to the crediting date, notional sizing is buying you a guarantee that delta sizing simply does not offer.
The part worth writing down is what each convention does to your own reporting.
A notional-sized book shows a dollar-offset ratio above 1.0 from day one, falling toward it. The report reads as a persistent over-hedge.
A delta-sized book shows a ratio starting at exactly 1.0 and falling below it. The report reads as a persistent under-hedge that deepens through the term.
Both stay inside the 80%–125% band throughout. The test cannot distinguish them. They are materially different positions, and the ratio that is supposed to tell you about hedge quality is in fact telling you, mostly, which sizing convention somebody chose — probably years ago, possibly by inheriting a spreadsheet.
So the useful question to put to a hedging program is not "is the offset ratio inside the band." It is: which convention are we sized on, was that a decision, and does the report we look at every day match the answer?
The fix, and the fix that does not work
Adding delta with futures is the obvious move and the wrong one. A futures overlay corrects the gap at a point in time and then immediately starts drifting again, because it has no gamma. You have hedged the symptom. You will be back next month, and the month after, transacting each time.
Adding a call spread overlay — struck near the current index level, maturing with the cohort — adds delta and gamma. It corrects the gap and keeps correcting it as the market moves, which is what the liability does. That is the difference between a hedge that tracks and a hedge you chase.
Better still, most of this is avoidable at the structuring stage. Split the cohort so the cap distribution inside each hedge is narrow. Ladder the maturities so the gamma steepening does not all land on one date. Both of those are decisions made once, in advance, in daylight.
What to write down
Five lines, before the cohort structure is set rather than after it drifts.
- The cap dispersion. What range of caps sits inside each hedge, and the net gamma that dispersion implies at inception. This is one calculation. It is the price of the convenience, and it is knowable on day one.
- The direction. Which way the book drifts if the index rises, and which way if it falls. An aggregated call spread hedge in a rising market goes under-hedged. Say so in writing, so that when it happens nobody has to work out whether it was expected.
- The trigger. The net delta or net gamma level at which somebody acts, and who that somebody is. Pre-committed, because the alternative is deciding under pressure in the market that caused the problem.
- The instrument. That true-ups are done with options rather than futures, and why. Otherwise the desk reaches for the liquid thing at the worst moment.
- The sizing convention. Notional or delta, stated once, with the reason. It determines which way your offset ratio leans for the life of the hedge, and it is the single cheapest thing on this list to write down and the most commonly inherited without a decision.
A program with those five lines has a drift that was chosen. A program without them has a drift that was discovered — and there is no way to tell, from the file afterwards, which one it had.
A correction
An earlier version of this analysis made two errors that this one corrects.
It stated that a call spread is modestly long gamma early in its life when the index sits near the lower strike. That is wrong, and its own simulation showed it was wrong. With positive carry the gamma peak sits above spot, so an at-the-money call spread is short gamma from inception.
It also implied that the weighted-average cap leaves the hedge short at expiry, because liabilities with caps above the hedge's cap keep crediting after the hedge is pinned. That region is real, but the earlier version counted only half the range: below the average cap the hedge over-earns against the policies already pinned at lower caps, by exactly as much. Notional-matched, the hedge is never short at settlement. The terminal exposure is over-coverage, not shortfall — the opposite sign to what was originally described.
The calculator
Every model figure here comes from a Black-Scholes-Merton workbook you can download and check — the terminal payoff ladder, the gamma peak and sign-change level, the cap-dispersion gap, the three sizing conventions side by side, and the eleven-month drift path, all on your own caps, volatility and rates.
It is on the Tools page.
Scarborough Road works with insurers and asset managers on derivatives governance, hedging operations, and the investment data infrastructure that demonstrates the governance was followed.
General commentary on hedging mechanics and governance practice. Not legal, actuarial or investment advice, and not a recommendation of any strategy or structure.
Modelled figures are outputs of a Black-Scholes-Merton implementation under stated assumptions — one-year term, index at 100, 20% volatility, r = 4.5%, q = 1.5%, European exercise, constant rates and volatility, no transaction costs — none of which hold exactly. The cap cohorts used in the modelled sections are illustrative and describe no actual book.
The effectiveness-report section draws on real client reporting, used with permission and de-identified. Entity names, counterparty names, position identifiers, notionals and limit values have been removed; every figure in that section is a ratio, a count or a percentage. Real programs face bid-offer, policyholder behaviour, surrender activity and a full volatility surface, all of which will move these numbers. Verify against your own positions and pricing systems.
