4A.4 Gaps

Motivation

India is closed for about 17.75 hours between sessions while the US, Europe, crude oil and GIFT Nifty trade. A large part of Nifty's daily variance arrives overnight and shows up at the open as a gap. Gaps decide whether stops fill near their trigger, and "gap fill" is one of the oldest trading claims.

4A.4.1Gap fills under a random walk#

Definition 4A.6gap and fill

The gap on day $t$ is $g_t=\ln(O_t/C_{t-1})$ . It is a full gap up if $O_t>H_{t-1}$ . An up-gap ( $g_t>0$ ) is filled on day $t$ if $L_t\le C_{t-1}$ ; a down-gap is filled if $H_t\ge C_{t-1}$ .

Proposition 4A.4fill probability under a random walk

If, after the open, the intraday log price is a driftless Brownian motion with session volatility $\sigma_{in}$ , then the probability that a gap of size $g$ is filled within the session is

P(\text{fill})=2\Phi\!\left(-\frac{|g|}{\sigma_{in}}\right), \tag{4A.4}

and the probability that an up-gap day closes below the previous close is $\Phi(-g/\sigma_{in})$ .

Proof

Filling an up-gap means the intraday minimum of $\sigma_{in}W_s$ , $s\in[0,1]$ , falls to $-g$ ; by the reflection principle $P(\min W\le -g/\sigma_{in})=2\Phi(-g/\sigma_{in})$ . Closing below the previous close means $\sigma_{in}W_1<-g$ .∎

So small gaps are filled most of the time by noise alone, and a gap of one intraday standard deviation is filled 32% of the time. "Gaps get filled" is true and uninformative unless the fill rate beats (4A.4).

4A.4.2Nifty gap statistics#

Worked example 4A.3Nifty gap statistics, 2011–2026

From Yahoo ^NSEI daily bars, 1 Feb 2011 to 4 Nov 2026 (3,866 days; computed for this module, script in the course repository's scratch work; Yahoo's index opens were checked for the stale-open artefact where the open equals the previous close, which affects under 1% of days, almost all before 2011):

Gap standard deviation 0.625%, mean +0.102%; close-to-close standard deviation 1.038%. Overnight returns account for 36% of daily variance. The correlation between the gap and the subsequent open-to-close return is −0.04.
$|g|>0.5\%$ on 25.2% of days; $|g|>1\%$ on 6.9%; $|g|>2\%$ on 1.6%.
Fill rates against the model (4A.4), with $\sigma_{in}$ the trailing 20-day Parkinson estimate (no look-ahead):

$\lvert g\rvert/\sigma_{in}$	days	actual fill rate	random-walk prediction
0–0.25	1,285	87.3%	90.2%
0.25–0.5	1,024	72.4%	71.2%
0.5–1.0	1,058	49.1%	48.5%
1.0–1.5	324	28.7%	23.9%
1.5–2.5	126	19.8%	7.0%
above 2.5	47	8.5%	0.3%

Up to one intraday standard deviation, the random walk predicts fill rates almost exactly. Large gaps fill far more often than the model says. The main reason is not mean reversion but volatility: on days with $|g|>1.5\sigma_{in}$ , that day's own Parkinson volatility averaged 1.31 times the trailing estimate. Big gaps arrive on news days, and news days are volatile in both directions. Conditional on a large gap, use a higher intraday volatility, not a reversal story.

Practitioner notes

Stops and gaps. A stop is a promise about the trigger, not the fill. A stop-loss market order triggered by a gap fills at the open or worse. Over 2011–2026, 1.6% of Nifty days gapped by more than 2%; for single stocks around results (M06A) the gap distribution is much wider. Size positions on the loss at a plausible gap, not at the stop (4A.12, Case 2).
Gaps and stop placement. A stop placed inside the typical overnight range (about $\pm0.6\%$ for Nifty) is hit by gaps alone on a large fraction of days.
GIFT Nifty (4A.11) trades through most of the Indian night, so its level at 09:00 is the market's estimate of the gap before the pre-open auction sets it.

The gap is $g_t = \ln(O_t/C_{t-1})$ . An up-gap is filled on day $t$ if $L_t \le C_{t-1}$ , a down-gap if $H_t \ge C_{t-1}$ (Definition 4A.6). Each day's gap is measured against $\sigma_{in}$ , the trailing 20-day Parkinson estimate of intraday volatility computed only from earlier days, and the fill rate is compared with the random-walk prediction $2\Phi(-|g|/\sigma_{in})$ of Proposition 4A.4. Days whose open equals the previous close (stale index opens) are excluded. Worked example 4A.3 reports the same table for 1 Feb 2011 to 4 Nov 2026.

What to look at

The bar chart: for gaps up to one intraday standard deviation the random walk predicts the fill rate almost exactly. Large gaps fill more often than predicted.
The share of daily variance that arrives overnight, and the correlation between the gap and the rest of the session.

4A.4.1Gap fills under a random walk#

4A.4.2Nifty gap statistics#

Nifty opening gaps and gap fills

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