Part VI · Dealers · Chapter 6C

Delta hedging

Neutralizing direction

1 · The first hedge is one multiplication

The dealer is short your one call, so its delta exposure is $-\Delta$, where $\Delta$ is the call's delta. Scale it up: a dealer that has sold $n$ contracts, each on a multiplier of $m = 100$ shares, is short $n\,m\,\Delta$ share-equivalents, and its hedge is that many shares:

This is delta hedging: holding the underlying in the amount that offsets a position's delta. The calculator below does the multiplication for a dealer that has sold SPY-like calls struck at $600, with 30 days to expiry and priced at 18% implied volatility. Its two sliders set the contracts the dealer sold (1 to 5,000, starting at 100) and the spot price ($500 to $700, starting at $600). The table walks the two steps: the delta per contract and the net exposure, then the shares the dealer buys and their dollar value. The curve shows the hedge the same position needs at every spot on the slider.

contracts dealer sold

100

spot price

$600

Step	This position
1 · delta per contract	0.5357
1 · net delta exposure	−5,357 shares
2 · dealer buys	5,357 shares
2 · notional	$3,214,200

At the opening settings the table reads a delta of 0.5357, a net exposure of −5,357 shares, a purchase of 5,357 shares and a notional of $3,214,200. The delta is a little above the round 0.50 of the worked example, because an at-the-money call's delta sits above one half (Chapter 6B). Now move spot to $606, 1% higher: the delta rises to 0.6112 and the hedge to 6,112 shares. Move it to $590 and the hedge falls to 4,066 shares: the shares bought at one spot are the wrong hedge at the next.

2 · Why the hedge is local, not global

Here is the catch that every dealer has known since 1973, when the Black-Scholes model was published. The hedge is correct only at the current spot. The moment spot moves, the option's delta changes by its gamma, which leaves the existing share position the wrong hedge. The dealer must re-hedge, and after the next move it must re-hedge again. This catch drives everything from Chapter 6D through Part VIII.

In idealized continuous time this re-hedging never stops, and that continuous re-hedging is exactly what makes Black-Scholes pricing work mathematically (Chapter 7A). In practice dealers re-hedge at discrete intervals instead: when the residual delta exceeds some threshold, on a fixed time grid, or when market events demand it.

3 · The noise of hedging at intervals

The choice of when to re-hedge leaves a residual risk. Suppose the dealer re-hedges at intervals $\Delta t$ apart, measured in years, where $\Delta$ in front of a letter means "a change in", as in Chapter 5C. Between hedges the option's value moves with its full Greek profile while the share hedge sits still. The hedging error $\varepsilon$ over one interval has variance approximately

where $\Gamma$ is the option's gamma, $S$ the price of the underlying and $\sigma$ its volatility. The expression encodes a key insight: the more frequently you re-hedge, the smaller the noise, but the more transaction costs you pay.

Where the variance comes from

Step 1: the error over one interval. By Chapter 5C, a delta-hedged option gains or loses about $\tfrac12\Gamma S^2 \left[(\Delta S/S)^2 - \sigma^2 \Delta t\right]$ over a short interval. The sign flips for the dealer, which is short, but the variance does not.

Step 2: the size of the move. Over a short interval, model the return as $\Delta S/S = \sigma\sqrt{\Delta t}\,Z$, where $Z$ is a standard normal variable and $\sigma$ is the volatility the option was priced at, so that $(\Delta S/S)^2 = \sigma^2 \Delta t\,Z^2$ and the error becomes

$$\varepsilon = \tfrac12\,\Gamma S^2 \sigma^2 \Delta t\,(Z^2 - 1)$$ (6C.3)

Step 3: no bias. Since $\mathbb{E}[Z^2] = 1$, the error averages zero: hedging at intervals adds noise, not a systematic loss. If realized volatility differs from the priced one, the average becomes the realized-against-implied P&L of Chapter 5C, whatever the interval.

Step 4: the variance. $\operatorname{Var}(Z^2 - 1) = \mathbb{E}[Z^4] - 1$, and the fourth moment of a standard normal variable is 3 (Chapter 1B). So $\operatorname{Var}(Z^2 - 1) = 2$, and squaring the constant in front gives

$$\begin{aligned} \operatorname{Var}(\varepsilon) &= \left(\tfrac12\Gamma S^2\sigma^2\Delta t\right)^2 \times 2 \\ &= \tfrac12\,\Gamma^2 S^4 \sigma^4 (\Delta t)^2 \end{aligned}$$ (6C.4)

Step 5: over a whole horizon. Over a horizon $\tau$ there are $\tau/\Delta t$ intervals. If their errors are independent, and gamma and spot are held fixed for simplicity, the variances add:

$$\operatorname{Var}(\text{total}) \approx \tfrac12\,\Gamma^2 S^4 \sigma^4\, \tau\, \Delta t$$ (6C.5)

The standard deviation of the total error is therefore proportional to $\sqrt{\Delta t}$. Halving the interval cuts it by a factor of $1/\sqrt{2} = 0.71$, and quartering the interval halves it.

There is an optimal frequency, and it depends on the relative sizes of gamma, volatility and the cost of crossing the spread. For institutional dealers with very low transaction costs, it is high: every few seconds or even faster. For retail-scale operations with higher costs it is lower, a trade-off that Chapter 6E works with numbers.

For this book's purposes, the key point is that dealers re-hedge continuously enough that the aggregate hedge flow is observable in the market. The continuous-time idealization is good enough to predict where flow appears. When spot moves through strikes with high open interest (OI), the dealer book's net delta shifts, and re-hedging activity appears in the underlying market.

4 · How often, and at what threshold

Different dealer business models re-hedge at different frequencies:

• Equity wholesalers, such as Citadel Securities, re-hedge essentially continuously through aggregated order flow. Their hedging is interleaved with their market making and is largely invisible as discrete flow.
• Options market makers on Cboe re-hedge when a threshold is breached. Common thresholds are a residual delta above 5% to 10% of the original position size, or a move of 0.25% to 0.50% in spot since the last hedge.
• Structured product desks re-hedge less often, hourly or even less. They run larger residual deltas and accept the variance, because their books are larger and their cost per trade is higher.

Across the dealer ecosystem, hedge flow scales with realized volatility: on calm days hedging is small, and on volatile days it is large. This is the structural reason why volatility itself becomes self-reinforcing during stress regimes, and Chapter 8D returns to it with a formal feedback model.

5 · The last 30 minutes

The single most important empirical paper on dealer hedging at intraday horizons is Baltussen, Da, Lammers & Martens (2021), "Hedging Demand and Market Intraday Momentum", Journal of Financial Economics, Volume 142, Issue 1, October 2021, pages 377 to 403.

The authors used intraday returns on over 60 futures contracts around the world, on equities, bonds, commodities and currencies, from 1974 to 2020. They document a robust intraday momentum pattern: the return over the rest of the day, from the previous close to the last 30 minutes, positively predicts the return over those last 30 minutes. The predictive power is economically and statistically highly significant, and the move it predicts reverts over the next several days.

The authors' explanation of the mechanism is that dealers rebalance toward the close to limit their overnight residual exposure. When dealers are net short gamma in aggregate, their re-hedging trades go in the direction of price moves. They sell more as the price falls and buy more as it rises, as the 1% move above showed. This creates intraday momentum that concentrates in the final 30 minutes.

The Dealer Mechanics Lab adds that dealers were typically net short gamma in the past. In the paper, the sign of dealer gamma is an assumption, not an observation. Its measure for the S&P 500, net gamma exposure computed from index option open interest from 1996 to May 2020, assumes that market makers are long calls and short puts. By that measure dealer gamma was negative on about 2,930 of 6,088 days, slightly under half. The momentum appeared when the measure was negative, growing stronger the more negative it was. It was weak or absent when the measure was positive.

The effect sizes are economically meaningful. The Dealer Mechanics Lab puts the Sharpe ratios of simple intraday momentum strategies, built on the rest-of-day return's power to predict the close, in the 0.5 to 1.0 range. Summaries of the paper report higher figures: annualized Sharpe ratios of about 0.87 to 1.73 for its timing strategy at the level of an asset class, before costs. Either range is strong by intraday standards, though insufficient to support running the strategy after transaction costs.

The mechanism extends beyond options, because leveraged ETFs (LETFs) and inverse ETFs also generate short-gamma hedging needs as their leverage drifts from its target during the day. They rebalance toward the close in the same direction as the day's move.

Other rule-based players trade the same way at the end of the day. The hedging of variance swaps, and the rebalancing of risk parity portfolios and of volatility-targeting funds, all go in the same direction as the recent move. A volatility-targeting fund is one that scales its exposure to hold its volatility near a target. All of these hedging activities contribute to the documented intraday momentum.

This is why the close is often the most mechanical part of the day, and why moves into 4 pm ET frequently extend rather than reverse. The last 30 minutes are different, and Baltussen and his coauthors document the difference quantitatively across decades and markets.

Three findings limit that picture. First, it describes whole markets, index and other futures. Among individual stocks, Baltussen, Da and Soebhag (2024) find the opposite in the last 30 minutes: the day's losers bounce against its winners. They find that neither liquidity nor gamma hedging explains the reversal. They point instead to attention-driven retail buying and to short sellers managing their risk at the end of the day.

Second, the rebalancing rule above overstates what leveraged funds actually trade. Ivanov and Lenkey (2018) find that investors' money moving into and out of leveraged and inverse equity ETFs, from 2006 to 2014, offset much of the rebalancing. Its effect on late-day returns was then economically insignificant.

Third, the effect shrinks where it is predictable. Barbon, Beckmeyer, Buraschi and Moerke (2022) find that leveraged-fund rebalancing moved stocks' closing prices more than option hedging did. It moved them less over time, though, as liquidity providers learned to anticipate it. Both effects reversed at the next day's open.

6 · Check yourself