Part VI · Dealers · Chapter 6E

How often to hedge

Hedging frequency against cost

Where we are

Chapter 6D assumed continuous hedging, with the dealer delta-flat at every instant. Real dealers do not rebalance continuously, because each rebalance crosses the bid-ask spread and the friction is not free. This chapter weighs how often to re-hedge against what it costs, with a stylized SPX example, and says what the answer means for the gamma exposure of Part VIII.

What you need first

Dollar gamma per 1%, and the gamma gain or loss of a move, $\tfrac12 \times \$\Gamma_{1\%} \times p^2/100$ for a move of $p$ percent (Chapter 6D).
The noise of hedging at intervals, and the practitioners' thresholds (Chapter 6C).
The bid-ask spread (Chapter 1F) and basis points (Chapter 1E).
Mean reversion, random walks and the variance ratio (Chapter 3B).

The real optimization a dealer faces is how often to re-hedge. It must re-hedge often enough to keep its gamma exposure under control. It must not re-hedge so often that spread costs devour the variance it set out to capture, if it is long gamma, or to accept, if it is short.

1 · The rebalance threshold

Practitioners parameterize discrete hedging by a rebalance threshold: a delta-equivalent or a move in basis points from the last hedged level that triggers a fresh rebalance. Two examples are "rebalance when net delta drifts more than 5,000 share-equivalents" and "rebalance after every 25 bps move in SPX". A tighter threshold means more frequent rebalances, which give tighter gamma control and cost more in spread. The trade-off runs one way on each side, and the question is where the net optimum sits.

The two kinds of threshold are the same thing in different units, and dollar gamma converts between them. Write $\$\Gamma_{1\%}$ for a book's dollar gamma per 1%, the dollar value of the hedge trade that a 1% move forces (Chapter 6D). A move of $p$ percent then forces a rebalance worth $\$\Gamma_{1\%} \times p$ in notional, the dollar value of the underlying traded.

Worked example: Converting a threshold

Step 1: the book. Take the long-gamma dealer of this chapter's main example, whose calls carry $\$\Gamma_{1\%} = \$230$ million.

Step 2: from basis points to shares. A 25 bp move forces a rebalance of $230 \times 0.25 = \$57.5$ million, which at an index level of 6,000 is about 9,600 share-equivalents.

Step 3: from shares to basis points. A threshold of 5,000 share-equivalents is $5{,}000 \times 6{,}000 = \$30$ million of notional. That is $30/230 = 0.13$ of the dollar gamma per 1%, so for this book it fires after a move of about 13 bps.

2 · What a coarse threshold gives up

Chapter 6C found that hedging at intervals adds noise that averages zero. On a random walk, then, a coarse threshold changes the spread of the dealer's result, not its expected value. The expected variance captured is the same whether the dealer rebalances every 5 bps or every 50.

A real intraday path can double back on itself, and then the threshold matters for the average too. Each round trip that stays inside the threshold is a move the hedger never trades, so its gamma is never collected.

Worked example: An index that doubles back

Step 1: the path. In one day the index goes 6,000, 6,015, 6,000, 6,015, 6,000: four moves of 0.25% that end where they started.

Step 2: the 25 bp hedger. A long-gamma dealer with $\$\Gamma_{1\%} = \$230$ million rebalances after every move. By Chapter 6D, each move earns $\tfrac12 \times 230{,}000{,}000 \times 0.25^2/100 = \$71{,}875$ , so the four earn $287,500. Each rebalance trades $57.5 million and pays 0.83 bp of spread, $4,773, so the net is $287{,}500 - 4 \times 4{,}773 = \$268{,}408$ .

Step 3: the 50 bp hedger. The same dealer with a 50 bp threshold never trades, because no move reaches 50 bps. It collects none of the four moves. Both dealers pay the same theta, so the coarse hedger is about $268,000 behind.

Step 4: the other sign. For a short-gamma dealer every sign flips. The 25 bp hedger pays $287,500 plus the spread, while the 50 bp hedger pays nothing.

This is the mean reversion of Chapter 3B at intraday scale. On a path that doubles back, the variance of long steps is less than the sum of the variances of the short ones. The variance ratio test of Chapter 3B reads this as a ratio below 1. A hedger collects the variance of the steps it actually trades, so a coarse threshold collects less.

The figure below hedges one simulated day at the threshold you choose, and averages the same hedge over 600 days. Widen the threshold with mean reversion set to none, then with a strong pull back to the open: the average holds near 100% in the first case and falls in the second.

mean reversion t½ 15 min

threshold 25 bps

this day: 18 rebalances

this day collects 83%

average of 600 days = 86%

A coarse threshold misses the wiggles only when the path doubles back. A simulated day of the index in 780 half-minute steps, with the chapter's 1.26% daily volatility and a pull back toward the open whose half-life the mean-reversion slider sets. The hedger rebalances at the dots, whenever the price leaves the shaded corridor around its last hedge, and collects the squared moves between rebalances and to the close, as a share of the finest sampling. Below, that share at every threshold is averaged over 600 such days, with the middle half of the days shaded and this day as the small dot. With no mean reversion the average stays near 100% while the days spread out, and as the pull strengthens, wide thresholds collect less, down toward the day's variance ratio.

3 · Stylized arithmetic at SPX scale

The dealer-mechanics tradition of this part works the trade-off through for a long-gamma book at SPX scale. Take the long-gamma dealer of Chapter 6D, scaled from 1,000 calls to 5,000: long 5,000 SPX 6,000 calls at 18% IV with 30 days to expiry (DTE). Its daily theta is about $1.0 million, the per-contract theta of about $200 times 5,000. Its aggregate dollar gamma per 1% move is about $230 million, 5,000 times the $46,000 of one contract.

Suppose the day's realized volatility is 20% annualized, slightly above the 18% IV, or about $0.20/\sqrt{252} = 1.26\%$ a day. With continuous hedging, the expected gamma capture is

\mathbb{E}[\text{capture}] = \tfrac12 \times \$\Gamma_{1\%} \times \sigma_d^2 \times 100

(6E.1)

where $\sigma_d$ is the daily volatility as a decimal and the 100 converts the per-1% unit. This is Chapter 6D's gamma gain for a move of $p = 1.26$ percent, and with the numbers it is $\tfrac12 \times 230{,}000{,}000 \times 0.0126^2 \times 100 = \$1.83$ million. Net of the $1.0 million of theta, the frictionless P&L lands near +$830,000 a day, and that is the upper bound.

Now add friction, taking a spread on the hedge of about 1.7 bps, so that each rebalance pays about 0.83 bp, half the spread, as it crosses. A threshold of $h$ bps makes each rebalance trade $\$\Gamma_{1\%} \times h/100$ of notional: a 50 bp threshold on $230 million of dollar gamma rebalances about $115 million. The spread paid per rebalance is that notional times the half-spread, about $9,500 at 50 bps.

Threshold	Variance captured	Gamma gross	Rebalances a day	Spread cost a day	Net P&L
Continuous, frictionless	about 100%	+$1.83M	none	$0	+$830k
5 bps	about 95%	+$1.74M	about 40	−$80k	+$660k
15 bps	about 85%	+$1.56M	about 15	−$36k	+$520k
25 bps	about 70%	+$1.28M	about 6	−$22k	+$258k
50 bps	about 45%	+$820k	about 3	−$15k	−$195k
200 bps	about 12%	+$220k	about 0.5	−$3k	−$783k

Each row's gross is the captured share of $1.83 million, and its net is the gross less the $1.0 million of theta and the day's spread cost. For the 5 bp row, for example, the net is $1.74 - 1.00 - 0.08 = 0.66$ million.

The lesson of the example is a U-shape in the net column, in which a threshold of 5 to 15 bps is near optimal. Below 5 bps the spread eats every wiggle, and above 25 bps the dealer leaves so much variance uncollected that theta exceeds the gamma it captures. The 50 bp and 200 bp rows show the strategy flipping from positive to negative as the threshold widens: same option position, same realized volatility, same dealer.

The captured-variance column reflects the path's mean-reverting jaggedness. Across 1.26% of daily net motion, the cumulative path length, the sum of every absolute move, is typically 2.5 to 4 times the net move. The sum of squared moves then depends on how finely you slice the path. Fine thresholds catch nearly all of it, and coarse ones catch only the large swings.

Reading the table with care

Its left arm is not in the table. The continuous row is frictionless, so the claim that hedging finer than 5 bps loses to the spread is asserted rather than shown by a row.
Its clocks differ. The $1.0 million of theta is per calendar day, while the capture uses a trading day's move, $\sigma/\sqrt{252}$ for an annual volatility $\sigma$ . On one clock, realized 20% against implied 18% nets about $\tfrac12 \times 230{,}000{,}000 \times 100 \times (0.20^2 - 0.18^2)/252 = \$347{,}000$ a trading day before friction, not $830,000. Every net figure falls by about $480,000, and the net turns negative between 15 and 25 bps rather than between 25 and 50. The ranking of the thresholds does not change.
Its captured-variance column is an assumption about the path. On a random walk every threshold would capture about 100% on average, and a coarse one would add only noise (Chapter 6C).
Its spread is larger than the quoted SPX spread. The SPX spread quoted with this example, about $0.10 on 6,000, is only 0.17 bp. The table's costs use 1.7 bps, which on 6,000 is about $1.

4 · Why the spread matters

The spread cost of each rebalance is about the notional rebalanced times the half-spread. For SPX, at 1.7 bps of spread, that is about 0.83 bp per crossing. A 5 bp rebalance pays about 17% of the move's width in spread, and a 50 bp rebalance about 1.7%.

Set against what the rebalance earns, the tax is heavier still. A rebalance of notional $N$ after a move of $h$ earns about $\tfrac12 N h$ , the triangle of Chapter 6D, and pays $N s$ , where $s$ is the half-spread, so the ratio of cost to gain is

\frac{\text{spread cost}}{\text{gamma gain}} = \frac{2s}{h}

(6E.2)

Worked example: The spread tax at three thresholds

Step 1: SPX. With $s = 0.83$ bp, (6E.2) gives $2 \times 0.83/5 = 0.33$ at a 5 bp threshold, so a third of each rebalance's gain goes to the spread. At 50 bps it gives $2 \times 0.83/50 = 0.033$ , about 3%.

Step 2: a single stock. With a spread of 10 bps, $s = 5$ bps, a 5 bp threshold gives $2 \times 5/5 = 2$ : each rebalance pays twice what it earns. A 30 bp threshold brings the ratio down to 0.33, and 60 bps to 0.17.

For tight thresholds, then, friction is a tax of several percent of the trade. That is exactly why high-frequency trading (HFT) market makers, who provide the liquidity the dealer's rebalances consume, earn a steady fraction of the spread. The dealer's discrete hedging creates the order flow that the HFT layer is selling into.

For single-stock options, where spreads run 5 to 15 bps rather than 1.7, the optimum threshold widens to 30 to 60 bps. The dealer simply cannot afford fine-grained rebalancing, as Step 2 shows. This is also why dealers' realized-volatility estimates in illiquid products tend to exceed the underlying's true volatility: every hedge carries a spread-cost overhead that does not appear in the unhedged price path.

5 · What this means for gamma exposure

Chapter 8A treats gamma exposure (GEX) as if every contract's gamma were hedged instantly on every spot move, which it is not. A frequency-dependent friction layer sits between open interest and the dealer flow you actually observe. The idealization is approximately correct at daily and longer frequencies, because the path catches up over a day. It loosens visibly at intraday horizons, where discrete rebalancing shows up as latency in the hedge response and as residual delta drifting between rebalances.

So when you read "GEX is +$5B today, expect mean-reverting intraday character", add a mental supplement. Expect mean-reverting intraday character averaged over a rebalance cycle of a few hours. Expect also bursts of the opposite character at the moments a large dealer is sitting on hedge it has not yet delivered. Chapter 7A's hedging-friction figure lets you hedge one path on several schedules and compare what each costs.

In practice

If you hedge an option position yourself, price your spread first. Compute $2s/h$ with your own half-spread before choosing a threshold. A threshold that suits a dealer paying 0.83 bp can hand most of a retail hedger's gain to the spread.
Read gamma exposure over hours, not minutes. A positive reading implies damping averaged over the dealers' rebalance cycle, not at every tick.
Discount single-stock hedging results for cost, since wider thresholds and wider spreads make realized volatility measured through a hedge look higher than the stock's own.
Know where the two traditions part. This chapter's arithmetic is a dealer's, while the edge-by-edge case of Part X rejects a related retail practice, trading around published gamma-exposure levels, as never robust at retail (Chapter 10I).

When it fails

The optimum threshold depends on how much the day's path doubles back, and that changes from day to day and regime to regime. On a random walk the expected capture is the same at every threshold, and the choice trades cost against noise alone. The table's rebalance counts, spreads and captured shares are stylized, so read it for its shape, not its levels.

Common mistake: "hedge as often as you can"

Tighter is not always better: at 5 bps a third of each SPX rebalance's gain goes to the spread, while for a single stock the same threshold loses money on every trade. For a short-gamma book on a path that doubles back, tighter hedging also means paying for more of the wiggles.

Decision rule

Set the threshold where the spread tax $2s/h$ is a modest fraction of each rebalance's gain. For SPX at 1.7 bps of spread, the example puts the optimum at 5 to 15 bps, and for single stocks at 5 to 15 bps of spread, at 30 to 60 bps.
Widen the threshold as your spread widens, and never hedge on every tick.
Put theta and variance on one clock before trusting any hedging P&L arithmetic, your own included.

What to carry forward

A dealer hedges on a rebalance threshold, in share-equivalents or in basis points, and dollar gamma converts one into the other.
On a random walk the threshold changes only the noise of the result. On a path that doubles back, a coarse threshold also collects less of the variance.
The spread tax on each rebalance is $2s/h$ of its gamma gain, so thin spreads allow fine thresholds and wide spreads force coarse ones.
In the stylized SPX example, 5 to 15 bps is near optimal, and wide thresholds turn a profitable long-gamma book into a losing one.
Gamma exposure describes dealer flow averaged over a rebalance cycle of hours, not tick by tick.

6 · Check yourself

Question 1

A book has $230 million of dollar gamma per 1%. What notional does a 20 bp rebalance trade, and what does it pay at a half-spread of 0.83 bp?

Answer

It trades $230 \times 0.20 = \$46$ million, and pays $46{,}000{,}000 \times 0.000083 = \$3{,}818$ of spread.

Question 2

What share of a rebalance's gamma gain goes to the spread at a 10 bp threshold with a half-spread of 0.83 bp?

Answer

By (6E.2), $2 \times 0.83/10 = 0.17$ , about a sixth.

Question 3

Why does a coarse threshold capture less variance on the example's path but not, on average, on a random walk?

Answer

On a random walk the variance of long steps equals the sum of the variances of the short steps they contain, so the expected capture does not depend on the threshold. The example's path doubles back, so many small moves cancel before a coarse threshold is reached, and the hedger never trades them.

Question 4

In the doubling-back example, what does a short-gamma dealer pay with a 25 bp threshold, and with a 50 bp one?

Answer

With 25 bps it pays $4 \times 71{,}875 = \$287{,}500$ of gamma plus $4 \times 4{,}773 = \$19{,}092$ of spread, about $306,600. With 50 bps it never trades and pays nothing on those moves.

Question 5

A single-stock option has a spread of 12 bps. What threshold keeps the spread tax at a third of each rebalance's gain?

Answer

The half-spread is 6 bps, and $2 \times 6/h = 1/3$ gives $h = 36$ bps, inside the 30 to 60 bps given for single stocks.