---
title: "Lead–Lag and Cross-Venue Trading"
book: "Market Making and High-Frequency Trading"
subject: quant
language: en
chapter: 8
exercises: 8
source: https://one-course.com/books/quant/11/en/chapter/8-leadlag-and-cross-venue-trading
---

# Chapter 8 — Lead–Lag and Cross-Venue Trading

When the index future moves, the fund follows within milliseconds and the constituent stocks a little later. Who leads depends on the hour, the venue and the size of the move, and it has changed as the fastest firms have caught up. Between 2005 and 2011 the median life of an arbitrage between the S&P 500 future and its fund fell from 97 milliseconds to 7. In this chapter’s simulated markets, where a fund follows its future half a second late, the lead is recovered from the two mids to within 25 milliseconds. The future’s information share is 97–98%. A taker who trades the fund when the future has moved and the fund has not earns 0.49 ticks a share if it acts within a millisecond. The edge is gone before the half-second is.

## 8.1 Price discovery across markets

Two prices of one thing (a future and the fund that tracks the same index, one stock on two venues, a coin on two exchanges) share a common efficient price and differ by transitory noise. Price discovery is the process by which new information enters that common price; the market where it enters first leads, and the other follows. Hasbrouck found, for US equity indices, that most price discovery happens in the E-mini futures for the S&P 500 and the Nasdaq-100, that it is shared between the regular future and the fund for the S&P 400, and that the S&P 500 fund leads the sector funds far more than they lead it. Mizrach and Neely found the Treasury futures carrying most of the price discovery of five- and ten-year notes from 1999 on.

At the highest frequencies the two prices do not even move together. Budish, Cramton and Shim measured, for the S&P 500 future and its fund in 2011, a median return correlation of 0.10 at 10 milliseconds and 0.008 at one millisecond, against nearly one over a minute: the Epps effect of One Quant Book 4, chapter 21, turned into an opportunity for whoever reads the leader first.

**Definition 8.1 (Cross-venue lead).**

A *cross-venue lead* is the time by which the prices of one market (a venue, or an instrument sharing the same efficient price) move ahead of those of another: the lag at which the cross-correlation of their price changes peaks, estimated on their own asynchronous ticks.

## 8.2 Information shares

One Quant Book 10, chapter 8 defines the two measures of price discovery. The *information share* of a market is the share of the variance of the common efficient price’s innovations that comes from that market’s innovations; the *component share* is that market’s weight in the common efficient price. Both come from a vector error-correction model (One Quant Book 4, chapter 20) of the two prices $p^1,p^2$ on a regular grid, with the cointegrating vector fixed at $(1,-1)$:

$$
\Delta p_t=\alpha\,(p^1_{t-1}-p^2_{t-1})+\sum_{i=1}^{k}\Gamma_i\,\Delta p_{t-i}+e_t .
$$

The market that corrects towards the other has the large adjustment coefficient; the leader’s is near zero.

**Proposition 8.2 (The two shares of a pair).**

With $\alpha=(\alpha_1,\alpha_2)$ and $\alpha_\perp=(\alpha_2,-\alpha_1)$, the component share of market 1 is $\gamma_1=\alpha_2/(\alpha_2-\alpha_1)$. With $\Omega=\mathrm{Cov}(e_t)$ and $F$ its Cholesky factor for a given ordering of the markets, the information share of market 1 is $[(\gamma^\top F)_1]^2/\gamma^\top\Omega\gamma$, and the two orderings give its lower and upper bounds.

**Proof.** *Admitted here.* ∎

The derivation of the common-trend representation is Hasbrouck’s and Gonzalo and Granger’s; the form above uses the fact, for a pair with cointegrating vector $(1,-1)$, that the common trend’s weights are proportional to $\alpha_\perp$.

[Figure 8.1](#fig-hf-lead-lag-and-cross-venue-trading-shares) shows the leader’s shares on the chapter’s markets. At a lead of 50 milliseconds, shorter than the 0.1-second grid, the innovations of the two markets are simultaneous on the grid and the information share is not identified: its bounds are 0.52 and 0.73. At half a second they are 0.97 and 0.98; at one second both are 0.998. The bounds narrow as the grid resolves the lead, which is why information shares on one-second data say little about a millisecond race.

![Price-discovery shares of the leading market A, from a vector error-correction model on a 0.1-second grid with ten lags, by the lead planted in the simulation (log scale); mean of two twenty-minute sessions. Data: hf_leadlag.table.](https://one-course.com/images/onecourse/chapters/quant-11/hf-lead-lag-and-cross-venue-trading/fig-185ad3a67674.svg)

***Figure 8.1.** Price-discovery shares of the leading market A, from a vector error-correction model on a 0.1-second grid with ten lags, by the lead planted in the simulation (log scale); mean of two twenty-minute sessions. Data: `hf_leadlag.table`.*

## 8.3 Estimating lag on asynchronous ticks

Two markets’ prices change at different moments; sampling both on a grid adds the Epps bias of One Quant Book 4, chapter 21. The lead–lag estimator of One Quant Book 7, chapter 10 works on the ticks themselves: the Hayashi–Yoshida covariance of the two series’ increments, with one series’ clock shifted by a candidate lag, normalised into a correlation; the lag with the largest correlation is the lead ([Figure 8.2](#fig-hf-lead-lag-and-cross-venue-trading-ccf)).

![Hayashi–Yoshida cross-correlation of the two mids’ changes, on their own ticks, with B’s clock moved back by each candidate lag, in steps of 25 milliseconds; one twenty-minute session with a planted lead of half a second. Data: hf_leadlag.estimate.](https://one-course.com/images/onecourse/chapters/quant-11/hf-lead-lag-and-cross-venue-trading/fig-604274850bc7.svg)

***Figure 8.2.** Hayashi–Yoshida cross-correlation of the two mids’ changes, on their own ticks, with B’s clock moved back by each candidate lag, in steps of 25 milliseconds; one twenty-minute session with a planted lead of half a second. Data: `hf_leadlag.estimate`.*

On two sessions for each planted lead, the estimates are 0.05 and 0.25 seconds for 0.05, 0.225 and 0.25 for 0.2, 0.475 and 0.575 for 0.5, and 1.075 and 1.1 for one second. A market can be too slow for this: in Book 7’s default configuration each simulated book follows its efficient price over several seconds, the peak of the cross-correlation is flat, and the same estimator returned leads of $-1.65$ and $+1.0$ seconds for a planted 50 milliseconds. The chapter’s markets make informed traders four times as active and stale quotes cancel faster, so that each mid follows its efficient price within a fraction of a second; only then is a sub-second lead identified.

## 8.4 Trading the lead: quoting and taking

A market maker in the follower uses the lead to protect its quotes (chapter 7’s futures-led skew). A taker uses it to cross the follower’s spread before the follower moves. The taker’s rule here: when the leader’s mid has moved by at least a tick over the last second more than the follower’s, cross the follower’s spread for one lot, and mark the trade at the follower’s mid five seconds later, after half the spread and a fee of 0.1 tick. Acting $d$ seconds after the leader’s move ([Figure 8.3](#fig-hf-lead-lag-and-cross-venue-trading-edge); planted lead half a second, 135 trades in two sessions):

![Edge per share of taking one lot of the follower after the leader has moved and the follower has not, after half the spread and a 0.1-tick fee, marked five seconds later, against the taker’s latency (log scale), ± one standard error; planted lead half a second, 135 trades in two sessions of twenty minutes. Data: hf_leadlag.edge_curve.](https://one-course.com/images/onecourse/chapters/quant-11/hf-lead-lag-and-cross-venue-trading/fig-eea1187488ce.svg)

***Figure 8.3.** Edge per share of taking one lot of the follower after the leader has moved and the follower has not, after half the spread and a 0.1-tick fee, marked five seconds later, against the taker’s latency (log scale), $\pm$ one standard error; planted lead half a second, 135 trades in two sessions of twenty minutes. Data: `hf_leadlag.edge_curve`.*

The edge is 0.485 ticks a share at one millisecond, 0.43 at 100, 0.24 at 300, and $-0.22$ at 500 milliseconds, the planted lead: the follower has moved, the gap is gone, and the taker pays half the spread and the fee for nothing ($-0.54$ at one second). Most of the edge survives the first 100 milliseconds only because this market’s lead is long. Budish, Cramton and Shim’s numbers put the real future–fund lead at a few milliseconds in 2011, so the same curve, drawn for a real pair, is compressed into the first milliseconds, and speed decides who earns it (chapter 9).

## 8.5 When the leader changes

A lead is a property of a pair at a time. The venue that leads can change with the hour (a futures market that opens before the stocks leads at the open), with the move (large moves start in the deepest market), with a venue’s fees or outages, and with competition: when enough traders trade the lead, the follower’s market makers withdraw their stale quotes faster (chapter 6), and the lead shortens. The practical consequences are to estimate the lead continuously on recent data, by time of day, and to monitor the information share for a change of leader, as for any predictor’s decay (One Quant Book 7, chapter 13).

## 8.6 Strategy files

**Strategy file 8.1 — Index future to fund.**

**Who pays you, and why.** The fund’s liquidity providers whose quotes are stale for the milliseconds after the future moves.

**Instruments and venues.** An index future and the exchange-traded fund on the same index (and its options).

**Signal.** The future’s move, in fund units through the fair-value basis (chapter 2), minus the fund’s move over a short window.

**Sizing and execution.** Take one clip of the fund when the gap exceeds half the spread plus fees; or, as the fund’s market maker, withdraw the stale side.

**Costs.** Half the spread, taker fees, the hedge in the future if the position is held.

**How it dies.** The arms race: the median opportunity lasted 97 milliseconds in 2005 and 7 in 2011, while its median profit per opportunity stayed near 0.08 index points (Budish, Cramton and Shim); the edge goes to the fastest.

**Horizon, capacity, infrastructure.** Milliseconds; a clip per opportunity; the fastest link between the future’s and the fund’s matching engines (One Quant Book 14).

**Backtest honestly.** Both venues’ timestamps on one clock, the real propagation delay, and execution at the price available when the order arrives.

**Sources.** Hasbrouck (2003); Budish, Cramton and Shim (2015); this chapter: 0.485 ticks a share at one millisecond, gone before the planted lead.

**Strategy file 8.2 — Fund to constituents.**

**Who pays you, and why.** Market makers in single stocks whose quotes lag a market-wide move shown first in the fund or the future.

**Instruments and venues.** A broad or sector fund and its largest constituents.

**Signal.** The fund’s move times each stock’s beta, minus the stock’s own move.

**Sizing and execution.** Skew or withdraw the stale side of many stocks at once; take only in the largest gaps.

**Costs.** Many small positions and their fees; a hedge in the fund.

**How it dies.** Stock-specific news: a stock that did not follow may have its own reason. Sector funds follow the broad fund more than they lead it (Hasbrouck), so the direction matters.

**Horizon, capacity, infrastructure.** Milliseconds to seconds; a book of hundreds of names.

**Backtest honestly.** Point-in-time betas; removing stocks with news in the window only if the live system could.

**Sources.** Hasbrouck (2003).

**Strategy file 8.3 — Cash Treasury to Treasury futures.**

**Who pays you, and why.** Liquidity providers in the slower of the cash note and its future, on whichever side leads at the time.

**Instruments and venues.** On-the-run notes on the interdealer platforms (One Quant Book 2, chapter 4) and the Treasury futures.

**Signal.** The leader’s yield move translated by the conversion factor and the basis (One Quant Book 2, chapter 6) into the follower’s price.

**Sizing and execution.** Take in the follower; or skew quotes there.

**Costs.** Platform fees, the basis’s own moves, financing if held.

**How it dies.** The leader changes: Mizrach and Neely found futures carrying most price discovery for five- and ten-year notes from 1999 on, with the spot market’s share varying with spreads, activity and volatility.

**Horizon, capacity, infrastructure.** Milliseconds around data releases, seconds otherwise; access to both markets’ matching engines.

**Backtest honestly.** Estimate the leader on past data only, by period.

**Sources.** Mizrach and Neely (2008).

**Strategy file 8.4 — Leading crypto venue to lagging venue.**

**Who pays you, and why.** Makers on slower or smaller exchanges whose books follow the price-discovering venues late.

**Instruments and venues.** One coin on several centralised exchanges (One Quant Book 3, chapter 16), spot and perpetual.

**Signal.** The leading venue’s move minus the follower’s, estimated lead by venue pair.

**Sizing and execution.** Take on the follower with inventory prepositioned there (One Quant Book 3, chapter 16); or make there with a skew.

**Costs.** Taker fees in basis points, transfers to rebalance inventory, rate limits.

**How it dies.** When capital moves freely the gaps close; Makarov and Schoar found deviations much larger across countries than within them, with capital controls keeping them open.

**Horizon, capacity, infrastructure.** Milliseconds to seconds; servers near each venue (One Quant Book 14).

**Backtest honestly.** Each venue’s timestamps and its data latency to the trader, not a merged tape.

**Sources.** Makarov and Schoar (2020); chapter 24.

**Strategy file 8.5 — Primary FX venue to secondary platforms.**

**Who pays you, and why.** Liquidity providers on secondary platforms whose streams lag the primary venue (One Quant Book 2, chapter 14).

**Instruments and venues.** Major currency pairs across the primary venue, other platforms and streams to aggregators.

**Signal.** The primary venue’s move minus the platform’s, and triangular consistency across pairs.

**Sizing and execution.** Take where the platform allows it; last look (chapter 9) lets the provider reject.

**Costs.** Platform fees and rejections.

**How it dies.** Algorithmic trading itself: Chaboud and co-authors found it reduced triangular arbitrage opportunities, mainly through computers taking liquidity.

**Horizon, capacity, infrastructure.** Milliseconds; the matching sites of the primary venue (One Quant Book 14, chapter 24).

**Backtest honestly.** Include rejections under last look and hold times.

**Sources.** Chaboud, Chiquoine, Hjalmarsson and Vega (2014).

## 8.7 Tutorial: who moves first

**Goal.** Recover a planted lead from two asynchronous price series, measure price-discovery shares, and price the lead as a function of latency. **End state:** Figures [8.1](#fig-hf-lead-lag-and-cross-venue-trading-shares), [8.2](#fig-hf-lead-lag-and-cross-venue-trading-ccf) and [8.3](#fig-hf-lead-lag-and-cross-venue-trading-edge).

1. **Markets.** `hf_leadlag.pair(lag, seed)` simulates the leader and the follower, made responsive.
2. **Lead.** `estimate` applies Book 7’s Hayashi–Yoshida estimator to the two mids’ change points on a grid of candidate lags.
3. **Shares.** `firm.xvenue.shares` fits the error-correction model on a 0.1-second grid and returns the component share and the information share’s bounds ([Listing 8.1](#lst-hf-lead-lag-and-cross-venue-trading-shares)). `def shares (p1, p2, lags: int = 5 ) -> dict : p = np.column_stack([np.asarray(p1, float ), np.asarray(p2, float )]) dp = np.diff(p, axis=0 ) z = (p[:-1 , 0 ] - p[:-1 , 1 ]) rows = range (lags, len (dp)) X = [np.r_[z[k], [dp[k - i, j] for i in range (1 , lags + 1 ) for j in range (2 )], 1.0 ] for k in rows] X = np.array(X) Y = dp[lags:] B, *_ = np.linalg.lstsq(X, Y, rcond=None ) alpha = B[0 ] E = Y - X @ B omega = np.cov(E.T) a_perp = np.array([alpha[1 ], -alpha[0 ]]) gamma = a_perp / a_perp.sum() out = {" alpha " : alpha, " component " : float (gamma[0 ])} iss = [] for order in ((0 , 1 ), (1 , 0 )): Om = omega[np.ix_(order, order)] F = np.linalg.cholesky(Om) g = gamma[list (order)] contrib = (g @ F) ** 2 tot = contrib.sum() iss.append(float (contrib[order.index(0 )] / tot)) out[" is_low " ], out[" is_high " ] = min (iss), max (iss) out[" is_mid " ] = 0.5 * (out[" is_low " ] + out[" is_high " ]) return out` **Listing 8.1.** The error-correction model and the two price-discovery shares. code/firm/xvenue/firm_xvenue.py
4. **Edge.** `lead_edge` replays the two books without impact and trades the gap after a latency; `edge_curve()` sweeps it. `for i in change: t = lead_t[i] if t - last < window: continue d_lead = lead_mid[i] - sample(lead_t, lead_mid, [t - window])[0 ] d_fol = sample(fol_t, fol_mid, [t])[0 ] - sample(fol_t, fol_mid, [t - window])[0 ] gap = d_lead - d_fol if abs (gap) < move: continue s = 1.0 if gap > 0 else -1.0 te = t + latency if te + H > fol_t[-1 ]: break px = sample(fol_t, fol_ask if s > 0 else fol_bid, [te])[0 ] later = sample(fol_t, fol_mid, [te + H])[0 ] edges.append(s * (later - px) - fee) last = t` **Listing 8.2.** Trade the gap after a latency, at the follower’s price then; mark five seconds later. code/firm/xvenue/firm_xvenue.py

**What to change next.** Make the lead random from move to move and see the cross-correlation’s peak widen; give the follower its own information (a share of its efficient price moves first) and watch the information shares meet.

## 8.8 Build: the cross-venue toolkit

**Purpose.** Who leads, by how much, and what the lead is worth at a given latency.

**Interface.** `sample(t, x, grid)`; `lead(t1, x1, t2, x2, lags)`; `shares(p1, p2, lags)` with the component share and the information-share bounds; `lead_edge(lead_t, lead_mid, fol_t, fol_bid, fol_ask, latency, move, window, H, fee)`.

**Rules.** Leads are estimated on the series’ own ticks; shares on a grid whose step is stated; edges after half the spread and fees, one lot, no impact.

**Acceptance tests.** `code/firm/xvenue/tests/`: on a random walk seen by one market three steps before the other, the leader’s shares exceed 0.8 and swap when the markets are swapped; a planted grid lead is recovered exactly; the edge of a hand-built pair at two latencies.

**Stretch.** Shares by hour of the day; a lead that depends on the size of the move; several followers at once.

Sources and further reading

- J. Hasbrouck, “One security, many markets: determining the contributions to price discovery”, *Journal of Finance* 50(4), 1995.
- J. Gonzalo and C. Granger, “Estimation of common long-memory components in cointegrated systems”, *Journal of Business and Economic Statistics* 13(1), 1995.
- J. Hasbrouck, “Intraday price formation in U.S. equity index markets”, *Journal of Finance* 58(6), 2003.
- B. Mizrach and C. J. Neely, “Information shares in the US Treasury market”, *Journal of Banking and Finance* 32(7), 2008.
- I. Makarov and A. Schoar, “Trading and arbitrage in cryptocurrency markets”, *Journal of Financial Economics* 135(2), 2020.
- A. Chaboud, B. Chiquoine, E. Hjalmarsson and C. Vega, “Rise of the machines: algorithmic trading in the foreign exchange market”, *Journal of Finance* 69(5), 2014.
- E. Budish, P. Cramton and J. Shim, *Quarterly Journal of Economics* 130(4), 2015.

## 8.9 Exercises

**Exercise 8.1 ★.**

An error-correction fit gives $\alpha_1=-0.02$ and $\alpha_2=0.18$. Which market leads, and what is its component share?

**Solution of Exercise 8.1.**

Market 2 corrects (large $\alpha_2$), so market 1 leads; $\gamma_1=0.18/(0.18+0.02)=0.9$.

**Exercise 8.2 ★.**

A taker makes 135 trades in two twenty-minute sessions at 0.363 ticks a share, one lot of 100 shares, with a one-cent tick. What does it earn a session?

**Solution of Exercise 8.2.**

$67.5$ trades a session $\times0.363\times100\times\$0.01=\$24.50$.

**Exercise 8.3 ★.**

By what factor did the median life of the future–fund arbitrage shrink between 2005 and 2011?

**Solution of Exercise 8.3.**

$97/7\approx14$.

**Exercise 8.4 ★★.**

Why are the information share’s bounds wide at a 50-millisecond lead and narrow at one second, on a 0.1-second grid?

**Solution of Exercise 8.4.**

When the lead is shorter than the grid step, both markets’ reactions to a piece of news fall in the same interval: their innovations are correlated and the Cholesky ordering decides who gets the shared part. When the lead spans several steps, the leader’s innovation comes first and the follower’s is mostly the error correction, so the innovations are nearly uncorrelated and the bounds meet.

**Exercise 8.5 ★★.**

Why does the edge become negative, rather than zero, once the latency exceeds the lead?

**Solution of Exercise 8.5.**

Once the follower has moved, the trade is taken at the new price with no gap left to close: the expected move is zero and the taker still pays half the spread and the fee, $-0.6$ ticks, close to the $-0.54$ measured at one second.

**Exercise 8.6 ★★.**

From [Figure 8.3](#fig-hf-lead-lag-and-cross-venue-trading-edge), between which latencies does the edge cross zero?

**Solution of Exercise 8.6.**

Between 300 milliseconds ($+0.24$) and 500 ($-0.22$).

**Exercise 8.7 ★★★.**

*Coding.* With a planted lead of 0.2 seconds, compute the edge at latencies of one and of 100 milliseconds (`edge(0.2, 0.001)`, `edge(0.2, 0.1)`).

**Solution of Exercise 8.7.**

0.444 ticks a share at one millisecond and 0.142 at 100: with a lead of 0.2 seconds, 100 milliseconds already costs two thirds of the edge.

**Exercise 8.8 ★★★.**

*Find the flaw.* “On one-second data the future’s information share is 99%, so the fund contributes nothing to price discovery and we can ignore its book.”

**Solution of Exercise 8.8.**

The share depends on the sampling step: on one-second data a lead of milliseconds is invisible and the shares mostly reflect which market’s noise is smaller. The fund may matter at the frequencies where the desk trades; measure on its own ticks and at its own horizon.

## 8.10 Problem: Who Moves First

**Problem 8.1.**

Weekend problem — who moves first

A desk trades a fund against its index future and must decide how fast it needs to be.

**Part I — The evidence.**

1. What did Hasbrouck find about price discovery in US equity index markets?
2. What did Mizrach and Neely find in Treasuries?
3. What happens to the future–fund correlation at 10 and at 1 millisecond?
4. Define a [cross-venue lead](#def-hf-lead-lag-and-cross-venue-trading-lead) .

**Part II — Measuring.**

5. Write the error-correction model and the two shares of [Proposition 8.2](#prop-hf-lead-lag-and-cross-venue-trading-shares) .
6. Give the leader’s shares at each planted lead and explain their pattern.
7. Why must the lead be estimated on the series’ own ticks?
8. Give the lead estimates for each planted lead.
9. Why did the default simulated books hide a 50-millisecond lead?

**Part III — Trading.**

10. Describe the taker’s rule and how its edge is measured.
11. Give the edge at 1, 100, 300 and 500 milliseconds and at one second.
12. Why is the edge still large at 100 milliseconds here, and why would it not be for a real future–fund pair?
13. How does a market maker in the follower use the same signal?

**Part IV — The verdict.**

14. State the *named result* : the planted and estimated lead, the leader’s information share, and the latency at which the edge disappears.
15. Why does the leader change over time?
16. Which strategy file depends most on capital controls, and why?
17. How would you monitor a change of leader in production?
18. Why does the edge not depend on the lead’s length as long as the taker is faster than it?
19. What is the cost of a latency advantage that is not used?
20. In one sentence: what decides who earns a lead?

**Solution of Problem 8.1.**

1. Most price discovery in the E-mini for the S&P 500 and Nasdaq-100; shared between the regular future and the fund for the S&P 400; the S&P 500 fund leads sector funds far more than the reverse.
2. Futures carried most price discovery for five- and ten-year notes from 1999 on.
3. Median correlation 0.10 at 10 ms and 0.008 at 1 ms in 2011.
4. See [Definition 8.1](#def-hf-lead-lag-and-cross-venue-trading-lead) .
5. $\Delta p_t=\alpha(p^1_{t-1}-p^2_{t-1})+\sum\Gamma_i\Delta p_{t-i}+e_t$ ; $\gamma_1=\alpha_2/(\alpha_2-\alpha_1)$ ; the information share from the Cholesky factor, in both orderings.
6. Information share 0.52–0.73, 0.81–0.90, 0.97–0.98 and 0.998 for leads of 0.05, 0.2, 0.5 and 1 second; component share 0.57, 0.73, 0.87, 0.98; the bounds narrow as the grid resolves the lead.
7. Sampling on a grid adds the Epps bias and blurs leads shorter than the step.
8. 0.05 and 0.25; 0.225 and 0.25; 0.475 and 0.575; 1.075 and 1.1 seconds.
9. The default books followed their efficient price over seconds, so the cross-correlation peak was flat; the estimates were $-1.65$ and $+1.0$ seconds.
10. Cross the follower’s spread for one lot when the leader has moved at least a tick more over the last second; marked at the follower’s mid five seconds later, after half the spread and a 0.1-tick fee.
11. 0.485, 0.43, 0.24, $-0.22$ and $-0.54$ ticks a share.
12. The planted lead is half a second; a real future–fund lead was a few milliseconds in 2011, so the curve would be compressed into them.
13. It withdraws or skews its stale side when the leader moves (chapter 7).
14. Planted 0.5 s, estimated 0.475 and 0.575 s; the leader’s information share 0.97–0.98; the edge disappears between 300 and 500 ms.
15. Liquidity, hours, fees, outages and competition move price discovery; trading the lead itself shortens it.
16. Leading crypto venue to lagging venue: Makarov and Schoar found gaps much larger across countries, where capital controls slow arbitrage.
17. Rolling estimates of the lead and the information shares by time of day, with alerts when they move beyond their usual range.
18. What the taker earns is the move it trades ahead of; once faster than the lead, being faster still adds little in this market (0.485 at one millisecond against 0.463 at fifty).
19. Nothing is earned by the unused speed while its fixed cost is paid (chapter 1’s operating leverage).
20. Speed relative to the others trading the same lead.

## 8.11 Interview questions

**Interview question 8.1 ★ trader.**

The future is up two ticks and the fund has not moved. List three reasons not to buy the fund.

**Solution of Interview question 8.1.**

The move may already be priced in the fund’s other venues; the fund’s market makers may have withdrawn and the spread widened; the future’s move may be a single print that reverts; the gap may not cover the spread and fees at the time the order arrives.

*What the interviewer is looking for: costs, stale data, and the chance of being late.*

**Interview question 8.2 ★★ researcher.**

Why does sampling two asynchronous series on a grid bias their correlation down, and how does the Hayashi–Yoshida estimator avoid it?

**Solution of Interview question 8.2.**

Grid sampling pairs increments over intervals in which one series may not have moved yet, so part of the common move is missed (the Epps effect). Hayashi–Yoshida sums products of every pair of overlapping increments on the series’ own ticks, so no move is dropped.

*What the interviewer is looking for: the source of the bias and the overlap idea.*

**Interview question 8.3 ★★ researcher.**

Explain why the information share has an upper and a lower bound, and what makes them close.

**Solution of Interview question 8.3.**

The innovations of the two markets are correlated, and the Cholesky decomposition assigns the correlated part to whichever market is ordered first; the two orderings bound the share. They are close when the innovations are nearly uncorrelated: a fine grid, or a lead longer than the step.

*What the interviewer is looking for: the ordering argument.*

**Interview question 8.4 ★★ developer.**

Two venues’ feeds arrive on servers in two data centres. How do you timestamp them so that a lead of a few milliseconds can be measured?

**Solution of Interview question 8.4.**

Timestamp at the network card with hardware stamps disciplined by a common time source (precision time protocol to a satellite reference), record the exchanges’ own timestamps too, and measure the path delay between the sites (One Quant Book 14, chapter 4).

*What the interviewer is looking for: hardware timestamps and a shared clock.*

**Interview question 8.5 ★★ trader.**

Your edge from trading a lead halves in a month while its frequency is unchanged. What happened, and what do you check?

**Solution of Interview question 8.5.**

Someone faster trades the same gaps: your fills come later in each opportunity. Check your latency rank (fill times against the leader’s move), the follower’s quote-withdrawal speed and the lead’s own length.

*What the interviewer is looking for: competition and how to measure it.*

**Interview question 8.6 ★★★ researcher.**

In a pair cointegrated with vector $(1,-1)$, show that if market 2 alone corrects ($\alpha_1=0$), market 1’s component share is one, and discuss its information share.

**Solution of Interview question 8.6.**

$\alpha_\perp=(\alpha_2,0)$, so $\gamma=(1,0)$. With market 1 ordered first its information share is one; ordered second it is $1-\rho^2$, with $\rho$ the correlation of the two innovations: the bounds are $[1-\rho^2,1]$.

*What the interviewer is looking for: the algebra and the role of innovation correlation.*