The Overnight Drift After Sustained Selling: A Dose-Response Threshold, Its Option Expression, and the Cost of a Take-Profit

Abstract. Boyarchenko, Larsen and Whelan (2023) document that the S&P 500 equity premium accrues disproportionately in the overnight session, concentrated around the European open, and attribute it to market-makers being compensated for carrying inventory absorbed during the prior day. We replicate the headline result on an independent hourly ES futures series of 3,603 overnight sessions across eighteen years (April 2007 to November 2024), with 738 of those sessions following two or more consecutive down days, and extend it in three directions. First, conditioning on the number of consecutive down days into the close, the overnight premium is a step function rather than a gradient: statistically absent after zero or one down day, and roughly seven times the unconditional mean after two or more. Second, the premium is procyclical in a way that matters for a trader: the overnight leg was the larger loser inside 2008 and 2022. Third, when the exposure is taken through a one-day out-of-the-money call rather than the future, the return distribution's expectancy lives almost entirely in the right tail, so any take-profit rule tested turned a positive mean into a loss. We also report two null results: a scheduled-news catalyst lifts the dead zone but does not stack on the two-down-day drift, and no filter on the pit bar's close improved the set-up in three attempts. An out-of-sample window of 275 nights (November 2024 to November 2025, and May to October 2026) keeps the sign of the threshold effect and the dead zone, at about half the magnitude and without statistical significance; we show it is underpowered to decide either way. We are explicit about what this work does not do: it does not improve the Federal Reserve estimate, and the option results rest on 106 events.

1. Introduction and relation to prior work

The overnight drift is one of the better-documented regularities in U.S. equity index returns. Cooper, Cliff and Gulen (2008) showed that the close-to-open return carries the premium while the open-to-close return is near zero. Boyarchenko, Larsen and Whelan (2023; Review of Financial Studies, first circulated as Federal Reserve Bank of New York Staff Report 917, hereafter BLW) traced the timing within the overnight session to a window around the European open and proposed an inventory-risk mechanism: when the day session ends with net selling, dealers hold the inventory overnight and are compensated through a positive expected return until liquidity returns in Europe. BLW condition on the sign and size of the prior day's intraday return.

2. Data

• Futures. ES hourly bars, April 2007 to November 2024, U.S. Central time, one vendor file, continuous front-month. After alignment this is 3,603 complete overnight sessions with a 19:00 CT entry and 11:00 CT exit, 738 of them after two or more consecutive down days, and about 4,500 bars per clock hour for the intraday profile. Every cell in Tables 1–3 and 6 is drawn from this population, and the sample floor of n = 40 that we apply elsewhere is cleared by every cell in Tables 1 and 2 by an order of magnitude. Roll treatment is the vendor's; we did not back-adjust. Overnight-versus-day decomposition, hourly profile and all conditioning in Sections 3 and 4 use this series.
• Options. Our own logger's one-minute snapshots of the MES daily option chain and the XSP chain, 2025 to 2026, plus a five-year ES option study window beginning 2021, the first full year with daily expirations. Section 6 uses these.
• Event calendars. CPI and Employment Situation release dates from the FRED release-dates API (n = 233 and 221), FOMC announcement days from the Federal Reserve's historical calendars (n = 144). Nothing hand-typed.

The hourly-bar approximation is the main limitation of the futures work. The day leg is measured from the 08:00 bar close (a 09:00 CT price) to the 14:00 bar close (15:00 CT); the 08:30–09:00 opening half-hour therefore falls into the overnight leg. Entries are the 18:00 CT bar close and exits the 10:00 CT bar close.

3. Replication

Table 1 gives the two legs over the full sample. For a leg with nightly returns r, sample mean r̄, standard deviation s and n observations,

The hourly profile reproduces BLW's timing. The strongest hour of the 24-hour cycle is the 01:00 CT bar (02:00–03:00 ET), at +1.07 bps per hour with t = 4.12 on n = 4,537 bars, inside a cluster running 23:00 to 01:00 CT. A secondary bump appears at 05:00 CT (t = 2.76) and the 06:00 CT bar is mildly negative.

The premium is not crisis-proof. Within 2008 the overnight leg lost 38.6% cumulatively and within 2022 it lost 11.7%, in both years more than the day leg. It was positive through 2020. The premium behaves like insurance: paid on average, charged in the tail. This matters for Section 7.

4. A threshold in the conditioning variable

BLW condition on the prior day's intraday return. We instead count consecutive down days into the close, where a down day is a regular-session close below its open, and measure the overnight return from the 19:00 CT entry to the 11:00 CT exit.

The response is a step, not a slope. The two "on" cells hold 413 and 325 nights and the two "off" cells 1,982 and 883, so the contrast is between populations of hundreds and thousands of nights, not a handful. Solving (1) for the standard error of the two-down-day cell gives 9.3 / 2.12 ≈ 4.4 bps; the zero- and one-day cells sit inside their own noise. The third day adds nothing beyond the second.

4.1 A mechanism consistent with a step (theory, not tested here)

In the inventory models of Ho and Stoll (1981) and Grossman and Miller (1988), a dealer carrying inventory Q of an asset with return variance σ² and risk aversion γ requires an expected return of order

to hold it, and the overnight session is where the position cannot be laid off cheaply. If one day of selling is absorbed within normal capacity and a second day pushes Q past it, the required return jumps rather than rising smoothly. The data in Table 2 is consistent with that; it does not establish it. A direct test would need dealer inventory or order-imbalance data we do not have.

4.2 Necessary, not sufficient

Live observation (June 2026) added a qualitative refinement that is not yet backtested: the count passes on muted two-day drifts with small ranges and mid-bar closes that do not plausibly load dealers, and those nights did not drift. The conviction filter (range expansion, volume, close near the low) is recorded as an open hypothesis with a stated test, not as a result.

5. Two set-ups on the underlying

The threshold is the gate. The set-ups add location: a pullback to the lower Keltner channel on 4-hour bars (EMA(20) minus 2.0 times EMA(20) of true range), split by whether the 4-hour EMA is rising or falling over the prior thirty bars.

Floor Play runs at about three times the unconditional overnight mean. Deep Dip's mean is near zero with a positive median: the typical night pays and a few nights cost a great deal. Strike-touch rates for Floor Play fall quickly with distance, 87% at +0.2%, 62% at +0.5%, 28% at +1.0%. The two set-ups fire about 3.4 times a month combined, with five zero-signal months in eighteen years.

Two refinements were measured against a contemporaneous rather than a frozen reference. Price reverts to the moving 240-minute EMA on 47% of armed nights, and the rate is flat across depth (39%, 36%, 36% at more than 1.0, 1.5 and 2.0 ATR below). Against the same level frozen at entry, which is the question a fixed strike asks, the rate is 32%. Both are correct answers to different questions and the earlier version of this work conflated them.

6. Option expression

6.1 Why a call collects a left-skewed drift

Let X be the underlying's move from entry to exit. Any random variable splits into positive and negative parts,

The future earns (3) and pays the whole left tail. A call struck k above entry for premium C earns

and the left tail is replaced by the fixed cost C. The call beats the future whenever E[(X − k)⁺] − C > E[X⁺] − E[X⁻], which is the case when E[X⁻] is large relative to C. Deep Dip is that case: a positive median with a near-zero mean says the left tail cancels many small gains. Truncating it at C is what makes the structure positive.

6.2 The take-profit inversion

On a one-day out-of-the-money call bought at the 18:00 CT entry, premium about 4 ES points, 2021 onward, n = 106. The strike rule is fixed and mechanical and is the one element of the method this paper does not disclose; the result below does not depend on it, only on the strike being out of the money at entry:

Write the exit value as a multiple of premium, V = v·C, with measured E[v] = 1.58 and win rate 0.17. If losers recover about a tenth of the premium on average, an assumption we state rather than measured, the winners must average

A take-profit at m·C sells every path that touches it, so

With winners near 8.8·C and m = 2, each capped winner gives back about 6.8·C. The win rate rises to the measured 23–28% because paths that touch 2× and fade now count as wins worth only 2·C. Putting a 25% win rate into (6) gives 0.25·2 + 0.75·0.1 ≈ 0.58, a mean of about −42%, against the measured −43%. The arithmetic is a check on the measurement, not an independent result; its point is structural. When expectancy sits in a few large outcomes, any cap that binds before them removes the only positive term.

The practical consequence is that this exposure cannot be "managed" in the usual sense. A rule that improves the win rate destroys the mean. The set-ups therefore carry no resting take-profit and are exited by hand or at a fixed time boundary.

7. Null results and boundaries

• Scheduled news does not stack on the drift. On nights with zero or one down day in a rising 4-hour trend, a next-morning CPI or NFP release lifts the overnight win rate from 50.7% (n = 1,337) to 60.2% (n = 201), and the gain accrues before the 08:30 ET print (63.7% win, t = 1.5 exiting at 07:00 CT; holding through the print doubles the standard deviation). This is the macro-announcement premium of Savor and Wilson (2013) and is a separate edge. On two-or-more-down nights it adds nothing: 51.2% pre-print on news nights (n = 82) against 57.9% with t = 2.82 on no-news nights. Pre-FOMC drift is dead in the modern sample, 64% in 2007–2015 and 36% in 2016–2024, consistent with the documented decay of the Lucca and Moench (2015) effect.
• No pit-bar close filter helps. Three attempts to condition on how the regular session's bar closes were null each time. Depth below the band at entry is what matters; the filter is retired.
• The edge is an index-futures edge. SPY shows the overnight premium (59% win, +5.2 bps per night, about 70% of its 2024–2026 return). A single stock, AAPL, shows none, in line with Berkman et al. (2012). On the 2026 cross-instrument study none of the six set-ups validated on ETF tapes; the arming read stays on the ES tape by design.
• Sequencing. A completing dip-buy raises the odds of the following day's continuation set-up by 1.57× (n = 122). It is used as a look, not a size-up.

8. Out-of-sample test

The set-ups were designed on data ending 6 November 2024. We re-ran Tables 2 and 3, through the same code path, on every later night we hold: a vendor 5-minute series resampled to the same hourly bars (14 November 2024 to 4 November 2025, 197 nights) and our own /MES tape (21 May to 5 October 2026, 78 nights). No file covers 6 November 2025 to 13 May 2026. Before reading any out-of-sample number, the in-sample window was re-run through the new file and reproduced Tables 2 and 3 exactly.

Reading. The sign survives: two-or-more-down nights are positive against a flat baseline, and the zero-down cell is at or below zero, as in-sample. The magnitude is about half of Table 2 and none of it is statistically significant. The set-up cells, at n = 9 and n = 6, are below any floor we use and are not evidence either way.

Power. The contrast with the design sample is the point: Table 2 rests on 738 two-plus-down nights and Table 5 on 66. The in-sample two-down cell has a per-night standard deviation of about 89 bps (from 9.3 / 2.12 × √413). Confirming a +9.3 bps mean at t = 2 therefore needs roughly

Sixteen months cannot confirm or refute the effect. We record the result as consistent and underpowered, and we will keep appending nights. A result that moves the mean of the pooled two-plus cell below zero over a multi-year window would falsify the claim; nothing in this window does.

8.1 Stability across the eighteen years

The true out-of-sample window is short, but the design sample is eighteen years long, and the second filter of our validation doctrine asks whether an effect holds in both halves of it. It does. The threshold gap also has the right sign in 14 of 18 individual years, which a sign test puts at p ≈ 0.015 one-sided.

The threshold effect is the stable object: near +9 bps in both halves, with the dead zone flat in both. Floor Play is positive in both halves but with half the sample it is not individually significant in either. Deep Dip's mean is zero in both halves, which is the shape Section 6.1 relies on: its value as an option comes from truncating the left tail, not from drift in the underlying. The four years in which the gap reversed were 2009, 2014, 2018 and 2023. The year-by-year Floor Play means are noisy at n ≈ 10, as expected, and were negative in 2007, 2008, 2010, 2022 and 2023, four of them bear years, consistent with Section 3's insurance reading.

9. Limitations

1. Hourly bars approximate the session boundaries and understate intrabar excursions.
2. One data vendor, one continuous series, vendor roll treatment. No transaction costs on the futures legs.
3. The set-up encodings were chosen by the author who trades them, and multiple conditioning variables were examined in-sample; the step in Table 2 should be read with that in mind. The out-of-sample window in Section 8 is sixteen months and underpowered.
4. The option results rest on 106 events from one strike rule, and the cells in Table 4 share those events. Stated assumptions in (5) are the author's.
5. The dealer-inventory mechanism is borrowed from BLW and the microstructure literature. We observe a pattern consistent with it and do not test it.

10. Conclusion

The overnight premium documented by BLW survives on an independent retail dataset, concentrates where they said it would, and switches on after two consecutive down days rather than rising smoothly with the prior day's loss. Sixteen months of later data keep the sign at half the size, without the power to settle it; across the eighteen design years the threshold holds in both halves and in 14 of 18 single years. It reverses in crisis years. Carried through to a one-day option, its expectancy lives in the right tail, and every take-profit rule we tested converted a positive mean to a negative one. The trading consequences follow directly: arm only on two or more conviction down days, hold through the European open, own convexity without capping it, and keep the risk defined because the same premium is clawed back in the years that hurt.

References

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Boyarchenko, N., Larsen, L. C., and Whelan, P. (2023). Overnight drift. Review of Financial Studies, 36(9), 3502–3547. Earlier version: Federal Reserve Bank of New York Staff Report No. 917, 2020, revised 2022.

Cooper, M. J., Cliff, M. T., and Gulen, H. (2008). Return differences between trading and non-trading hours: like night and day. Working paper, SSRN 1004081.

Grossman, S. J., and Miller, M. H. (1988). Liquidity and market structure. Journal of Finance, 43(3), 617–633.

Ho, T., and Stoll, H. R. (1981). Optimal dealer pricing under transactions and return uncertainty. Journal of Financial Economics, 9(1), 47–73.

Lucca, D. O., and Moench, E. (2015). The pre-FOMC announcement drift. Journal of Finance, 70(1), 329–371.

Savor, P., and Wilson, M. (2013). How much do investors care about macroeconomic risk? Evidence from scheduled economic announcements. Journal of Financial and Quantitative Analysis, 48(2), 343–375.