Celebrating Three Decades of Worldwide Stock Market Manipulation
CCelebrating Three Decades of Worldwide Stock Market Manipulation
Bruce Knuteson
As the decade turns, we reflect on nearly thirty years of successful manipulation of the world’spublic equity markets. This reflection highlights a few of the key enabling ingredients and lessonslearned along the way. A quantitative understanding of market impact and its decay, which wecover briefly, lets you move long-term market prices to your advantage at acceptable cost. Hidingyour footprints turns out to be less important than moving prices in the direction most people wantthem to move. Widespread (if misplaced) trust of market prices – buttressed by overestimates ofthe cost of manipulation and underestimates of the benefits to certain market participants – makesprice manipulation a particularly valuable and profitable tool. Of the many recent stories heraldingthe dawn of the present golden age of misinformation, the manipulation leading to the remarkableincrease in the market capitalization of the world’s publicly traded companies over the past threedecades is among the best.
Markets are supposed to make sense. When you seeanomalies in the market, it is probably a place wherewe should look further.—
United States Securities and ExchangeCommission Chairman Jay Clayton [1]
Contents
I. Market manipulation II. Market impact III. Human manipulation IV. Human impact V. Be careful References I. MARKET MANIPULATION
Today we celebrate three decades of the Strategy [2, 3],a type of market manipulation employing a specific pat-tern of round-trip trading to create mark-to-market gainson a large existing portfolio. The Strategy, shown in car-toon form in Figure 1 [21] [22], exploits a general featureof worldwide public equity markets: early in the tradingday (near market open), spreads are wide and depths arethin; late in the trading day (near market close), spreadsare narrow and depths are thick [4]. An order placed nearmarket open thus moves the price more than an equallysized order placed near market close [5] [23] [24].The Strategy is to accumulate a large portfolio andthen systematically and repeatedly expand it a bit nearmarket open (when spreads are wide and depths are thin)and contract it a bit near market close (when spreads arenarrow and depths are thick) [25]. Think of your portfo-lio breathing, expanding its lungs near market open andcontracting its lungs near market close. On each breath,the number of shares in equals the number of shares out, but in units of dollars, your lungs expand just a bit morewhen you inhale than when you exhale, leaving the valueof your portfolio (in units of dollars) a bit bigger at theend of each breath.If you are a scientist or engineer, think of the marketas a simple mechanical system. Placing an order nearmarket open perturbs this system. The system then re-laxes (with part of this relaxation happening quickly andsome of it happening slowly, as discussed further in Sec-tion II). Placing a similarly-sized order near market closeperturbs the system less. The response of the system toyour perturbations (and the way it subsequently relaxes)can be modeled and understood the same way you wouldmodel and understand any other system: systematicallyperturb it and note what happens. There is no magichere.Each of your daily round trips, considered in isolation,is a money-losing effort. Properly and repeatedly done,however, your daily round trips push prices in your fa-vor, causing mark-to-market gains on your large exist-ing portfolio. The Strategy’s pattern of round-trip trad-ing – expanding your existing portfolio when your trad-ing moves prices more, contracting it when your tradingmoves prices less, losing money on your round-trip trad-ing, and posting mark-to-market gains on your large ex-isting portfolio due to the price impact of your trades –is market manipulation under any reasonable definition.Note that the cartoon form of the Strategy drawn inFigure 1 creates a suspicious return pattern. The in-traday return (from market open to market close) cor-responds to the negative return from peak to dip duringeach intraday period (marked with a green horizontal baralong the x-axis). The overnight return (from marketclose to the next day’s market open) corresponds to thepositive return from the bottom of each dip to the topof the peak the next day during each overnight period(marked with a blue horizontal bar along the x-axis).Stitching these together day after day, this cartoon viewof the Strategy produces a sequence of positive overnightreturns and negative intraday returns similar to thoseseen in the world’s major stock market indices over thepast three decades, as shown in Figures 2 and 3 [26].The first plot in Figure 2 appeared in Ref. [10] twelve a r X i v : . [ q -f i n . GN ] N ov t (days)505101520 Y o u r c u m u l a t i v e p r i c e i m p a c t ( × ) The Strategy overnightintraday * (1) Accumulate a large portfolio.(2) Move prices in your favor.Toy example: Sell at close. Buy at open. Repeat. Lose on your daily round-trip trading. Win on your mark-to-market gains. * Advanced version: Eliminate the suspicious positive overnight returns negative intraday returns by buying after (rather than at) market open. * not shown FIG. 1: A cartoon view of the Strategy [2, 3]. (Real-life implementation involves more complicated trading in greater volume.)The black curve shows the cumulative effect of your trading on price. (A long position in a single stock is shown for simplicity.)Prior to day 0, accumulate a large position in this stock. On day 0 at market close ( t ≈ .
66, at the end of the first greenhorizontal bar), sell a small fraction of the shares you hold. This pushes the price down by a bit less than 0.05% (leftmost dipin black curve). The system quickly starts to relax back to 0. On day 1 at market open ( t = 1 .
0, at the end of the first bluehorizontal bar), buy the same number of shares you just sold. This pushes the price up (leftmost peak in the black curve) by0.15%. The system again quickly starts to relax back to 0. Sell the same number of shares at market close ( t ≈ . t = 2 . years ago. Most of the remaining content of Figures 2and 3 appeared in Ref. [11] over four years ago.The obvious, mechanical explanation of the highly sus-picious return patterns shown in Figures 2 and 3 is some-one trading in a way that pushes prices up before orat market open, thus causing the blue curve, and thentrading in a way that pushes prices down between mar-ket open (not including market open) and market close(including market close), thus causing the green curve.The consistency with which this is done points to theactions of a few quantitative trading firms rather thanthe uncoordinated, manual trading of millions of people.Ref. [10] concluded as much twelve years ago, ending withthe paragraph:Hopefully, future extensions of our results willhelp explain further the sources of the dayand night effect. Potential explanations maycome from an examination of the effects ofthe growing and widespread practice of al-gorithmic trading by hedge funds and otherfinancial institutions; perhaps price pressureeffects from algorithm generated trading mayaccount for some of the observed price pat- terns we document.The existence of the Strategy explains how such a firm(with the qualities described in Ref. [3]) can benefit fromthis seemingly pointless and costly price pushing. Theliterature currently contains zero plausible alternative ex-planations for these highly suspicious return patterns inthe world’s major stock market indices [27] [28].The last plot in Figure 3 is a fun variation on the gen-eral theme in Figures 2 and 3. China is unique in havinga “T+1” trading rule that prohibits your buying a shareof a company and then selling it later the same day [8],making the “expand your long positions in the morningand contract them in the afternoon” half of the Strat-egy explicitly-and-easily-enforceably illegal [29]. China,which legalized short selling in 2010 [9], is totally finewith your shorting a share of a company and then buy-ing it back later the same day, making the “expand yourshort positions in the morning and contract them in theafternoon” half of the Strategy legal (or, equivalently,not-easily-enforceably illegal). As expected, the patternof overnight and intraday returns in China in the last plotin Figure 3 [30] is consistent with firms executing the not-explicitly-and-easily-enforceably illegal half of the Strat- United StatesS&P 500 SPDR ETF overnightintraday 0 +3098%-43%
United StatesNASDAQ Composite
CanadaiShares TSX 60 ETF
United KingdomiShares Core FTSE 100 ETF
FranceCAC 40
GermanyDAX
NetherlandsAEX
NorwayDNB OBX ETF
Overnight and Intraday Returns to Major Stock Market Indices
FIG. 2: Cumulative overnight (blue curve) and intraday (green curve) returns to eight major stock market indices over threedecades. The overnight (blue) curve cumulates returns from market close to the next day’s market open. The intraday (green)curve cumulates returns from market open to market close. The horizontal axis of each plot extends from January 1, 1990 toOctober 31, 2019. The (linear) vertical scale in each plot extends from a return of -100% (bottom of plot) through 0 (explicitlymarked, at left) to the largest cumulative overnight return achieved (top of plot). On each plot, the cumulative overnight andintraday returns on October 31, 2019 (or the last date available) are explicitly marked, at right. Several curves start on thefirst day for which data are available: S&P 500 (1993-01-29), TSX 60 (1999-10-04), FTSE 100 (2001-01-02 to 2018-06-20), CAC40 (1990-03-01), DAX (1993-12-14), AEX (1992-10-12), and DNB OBX (2009-05-08). The code used to make this figure isavailable at Ref. [6]. Data are publicly available from Yahoo! Finance.
IsraelTA-125 overnightintraday 0 +3314%-93%
AustraliaSPDR S&P/ASX 200 Fund
IndiaNIFTY 50
IndiaS&P BSE SENSEX
Hong KongHang Seng Index
SingaporeSPDR Straits Times Index ETF
JapanNikkei 225
ChinaSSE Composite Index
Overnight and Intraday Returns to Major Stock Market Indices (continued)
FIG. 3: Cumulative overnight and intraday returns to eight more major stock market indices, prepared in the same manneras Figure 2. Several curves start on the first day for which data are available: TA-125 (2007-01-08), ASX 200 (2001-08-27),NIFTY 50 (2007-09-17), SENSEX (1997-07-01), Straits Times (2008-01-10), and SSE (1997-07-02 to 2017-08-25). SENSEXprices from the Bombay Stock Exchange (available from 2009 onwards) [7] match those from Yahoo! Finance used for this plot.China’s return pattern can be understood in terms of China’s “T+1” trading rule (which makes the “expand your longs in themorning and contract them in the afternoon” half of the Strategy explicitly-and-easily-enforceably illegal [8]) and China’s banon short selling before 2010 (making the “expand your shorts in the morning and contract them in the afternoon” half of theStrategy impossible before 2010) [9]. egy from 2010 onward.An earlier caution [3] bears repeating: implementingthe Strategy in a manner creating the highly suspiciousreturn patterns shown in Figures 2 and 3 is unnecessary.In Figure 1, move your morning buy from just beforeor at market open to just after market open so you donot leave the glaringly obviously problematic return pat-terns shown in Figures 2 and 3 [31]. Trading in a waythat leaves Figures 2 and 3 is roughly equivalent to com-mitting crimes and leaving smoking guns right in frontof police stations in fourteen separate jurisdictions, withyour fingerprints all over the guns. In India, where youcan take the Strategy to a completely different level [32],you have left a smoking bazooka. Don’t do this. Thereare plenty of ways to implement the Strategy withoutleaving ridiculous price patterns for anyone to see in datapublicly available from Yahoo! Finance.Fortunately, human nature being what it is, most arehappy to ignore smoking guns if doing so increases thebalance in their retirement accounts. The disconnect be-tween this willful blindness and the quote starting thisarticle is stark.
II. MARKET IMPACT
Market manipulation is similar in nature to other mis-information campaigns in that effective implementationis aided by an accurate understanding of the costs andbenefits of actions available to you. With the Strategy,accurately predicting both costs (losses on your dailyround-trip trading) and benefits (mark-to-market gainsresulting from you pushing prices in your favor) requiresunderstanding how much the market changes when youperturb it by submitting an order. The impact your orderhas on the market is called “market impact” [33].When celebrating a technical achievement, it is permis-sible to indulge in a few details as long as they are keptshort. We permit ourselves a one-paragraph summary ofthe existing literature on market impact.Considering the limit order book of an active market,define a “fair market price” that probably lies somewherebetween the best bid and best offer. Imagine placing asingle order. Given your knowledge of the current state ofthe limit order book, let δ denote the fractional changein fair market price you expect placing your order to haveimmediately upon placement [34]. Letting t denote thetime elapsed after the placing of your order and introduc-ing the parameter s ( t ) = √ t for the sole purpose of usingfewer square root signs, Refs. [12–14] imply your priceimpact asymptotically decays as s − . That is, your bestestimate of your order’s initial price impact is (tautolog-ically) δ , and the existing literature claims this impactdecays as s − for sufficiently large s . Graphing this on alog-log scale [35], as in Figure 4, the only other point ofinterest is the knee, which occurs at s ≈ λ [36].For the manipulator, the takeaways from the precedingone-paragraph literature summary and Figure 4 (in light Square root of time ( s ) I m p a c t ( s ) / s Log-log scale
Market impact and decay
FIG. 4: The blue curve shows the price impact δ ( s ) of asingle order (vertical axis, logarithmic scale) as a function of s , the square root of the time elapsed after order placement(horizontal axis, logarithmic scale). The intercept δ (0) is,tautologically, your best estimate of your order’s initial priceimpact, δ . Refs. [12–14] imply this initial market impactasymptotically decays as s − . These two boundary conditionsconstrain δ ( s ) to something like the blue curve shown. Thecurve must have a “knee” (power law cutoff), the abscissa ofwhich we denote by λ . These facts together imply δ ( s ) s (cid:29) λ −−−→ δ λ/s , up to a multiplicative constant of order unity. of the predictable variation in spreads and depths overthe course of the trading day noted in Section I) are that(i) you can submit two similarly-sized orders with signifi-cantly and predictably different price impact by carefullychoosing when you place them (through δ , and perhapsalso through λ ), and (ii) some of your price impact sticksaround a long time, allowing persistent price pushes toaccumulate materially in your favor. These are the fea-tures of market impact that make the Strategy possiblein practice. III. HUMAN MANIPULATION
All subjects of celebration have a supporting cast. Wewish to acknowledge two in particular.First, the academic peer review process has supportedcontinued use of the Strategy with both false positivesand false negatives. Accepting studies of questionableaccuracy, the peer review system has provided tradingstrategies quants have used to construct portfolios. Sincewhat matters most is accumulating a large portfolio andthen trading at the margin to move prices in your favor(as shown in Figure 1), the (in)accuracy of the majorityof these studies is not a problem. Constructing signalsfrom the same pool of academic work has helped alignthe portfolios of multiple firms, facilitating constructiveinterference in their use of the Strategy. On the flip side,failure to accept Refs. [10, 11] into the peer-reviewed liter-ature has contributed to a surprising but very fortunategeneral ignorance regarding the existence of the highlysuspicious return patterns shown in Figures 2 and 3. Theacademic peer review process appears to have failed pre-cisely where it was needed most, and exactly as we mighthave hoped and expected given its design.Second, and more importantly, the successful use ofthe Strategy over three decades has required the carefuland thoughtful manipulation of individual people. Thismust continue.The incentives described in Ref. [3] keep those in yoursurrounding ecosystem in check and provide near immu-nity to prosecution as long as your use of the Strategy issuccessful, but nothing good lasts forever. Avoiding sub-sequent prosecution requires your not being among thetopmost decision makers in your firm who are on recordas being aware of your use of the Strategy [37].If you are already on record as being aware of yourprevious or ongoing use of the Strategy, you must com-municate this fact as a potential problem (clearly and onthe record, multiple times) to the person directly aboveyou, transferring as much of your personal liability aspossible to him [38].If you are among the top of the chain of command,you must ensure none of the people below you put youclearly on record as being aware of your material use ofthe Strategy. Maintaining plausible deniability requiresattentive and dexterous manipulation of the people im-mediately around you.Any potential problem is apt to develop along a pre-dictable path. An underling with overly rigid ethicalsensibilities understanding the materiality of your use ofthe Strategy will begin agitating to his peers and man-ager. Earlier restriction and careful culling of your di-rect reports will ensure this quant’s manager is not you.Observing such agitation, you must eliminate this em-ployee as quickly as possible. Your action, which must beframed to appear to others as well-reasoned and thought-fully justified, and in which you will need to involve oth-ers (including human resources and perhaps other col-leagues) to diffuse your liability, must be brutal in itsspeed and effectiveness. The time elapsed should be days,not weeks, and certainly not months. Every passing daypotentially adds to the document trail you must ensureyou are not on.If the problematic employee agitates to colleagues inother departments (including legal and compliance) orregulators, they will come to you, and you will be ableto satisfy their concern with some appropriate versionof there being nothing to see here. Penetrating follow-upquestions, the formulation of which requires at least a ba-sic level of knowledge, will not be forthcoming. Such in-teractions are problematic in the long term to the extentthey further entangle you in an incriminating documenttrail.
IV. HUMAN IMPACT
Three decades of worldwide stock market manipulationis quite an accomplishment, but it is the wider humanimpact explained in Ref. [3] that makes this achievementparticularly special.The tens of trillions of dollars your use of the Strat-egy has created out of thin air have mostly gone to thealready-wealthy: company executives and existing share-holders benefitting directly from rising stock prices; own-ers of private companies and other assets, including realestate, whose values tend to rise and fall with the stockmarket; and those in the financial industry and elsewherewith opportunities to “privatize the gains and socializethe losses,” as those in the business of doing so like tosay. These gains to capital over the last three decadeshave contributed directly and significantly to the currentlevel of wealth inequality in the United States and else-where [15, 16]. As a general matter, widespread mispric-ing leads to misallocation of capital and human effort,and widespread inequality negatively affects our socialstructure and the perceived social contract.The fact that three decades of small price nudges by sofew can have such far-reaching consequences in so manyareas of human life is truly marvelous.
V. BE CAREFUL
There is little more we can do here. You are now fin-ishing the last of a trilogy of articles [2, 3], written overas many years, of gradually increasing scope. We haverepeatedly brought this matter to the attention of rele-vant regulators, both domestic and foreign, and to hun-dreds of journalists, academics, and other professionals.None have offered an alternative plausible explanationfor the highly suspicious return patterns shown in Fig-ures 2 and 3. None have offered evidence disfavoring theexplanation provided in this article. These efforts, spreadover half a decade, have led to a grand total of one-thirdof an article written by somebody else [17].Our younger selves would have been confident thatsomebody somewhere was working to fix this [39], andthat there will always be at least a few willing to workagainst their immediate self-interest to protect othersfrom easily avoidable harm. Older and wiser, we are nolonger so sure.Perhaps somebody else will step up. Perhaps not.Today we celebrate nearly three decades of worldwidestock market manipulation. Tomorrow circumstancesmay change. At some point hundreds of millions of peo-ple will realize they have been had. Their anger will befully justified. You will not want it directed at you. [1] The Securities and Exchange Commission: Priorities Go-ing Forward. September 5, 2017, New York, NY. . youtube . com/watch?v=ln0hlt2IywY&t=2951 .[2] B. Knuteson, Information, Impact, Ignorance, Illegal-ity, Investing, and Inequality (2016), URL https://arxiv . org/abs/1612 . .[3] B. Knuteson, How to Increase Global Wealth Inequal-ity for Fun and Profit (2018), URL https://ssrn . com/abstract=3282845 .[4] D. Strumpf and C. Driebusch, Why Morning Is the WorstTime to Trade Stocks , The Wall Street Journal , Septem-ber 14, 2015, URL . wsj . com/articles/early-birds-suffer-in-market-1442273794 .[5] R. Cont, A. Kukanov, and S. Stoikov, Journal of Fi-nancial Econometrics , 47 (2014), URL https://arxiv . org/abs/1011 . .[6] https://bruceknuteson . github . io/spy-day-and-night .[7] . bseindia . com/indices/IndexArchiveData . html .[8] K. Qiao and L. Dam, The Overnight Return Puzzleand the “T+1” Trading Rule in Chinese Stock Markets (2019), URL https://ssrn . com/abstract=3418356 .[9] E. C. Chang, Y. Luo, and J. Ren, Journal of Bank-ing & Finance , 411 (2014), URL https://ssrn . com/abstract=2135067 .[10] M. J. Cooper, M. T. Cliff, and H. Gulen, ReturnDifferences Between Trading and Non-trading Hours:Like Night and Day (2008), URL http://ssrn . com/abstract=1004081 .[11] M.-E. Lachance, Night Trading: Lower Risk But HigherReturns? (2015), URL https://ssrn . com/abstract=2633476 .[12] J. Gatheral, Quantitative Finance , 749 (2010), URL https://ssrn . com/abstract=1292353 .[13] J. Donier, J. Bonart, I. Mastromatteo, and J.-P.Bouchaud, Quantitative Finance , 1109 (2015), URL http://arxiv . org/abs/1412 . .[14] M. Benzaquen and J.-P. Bouchaud, Quantitative Fi-nance , 1781 (2018), URL https://arxiv . org/abs/1710 . .[15] M. Kuhn, M. Schularick, and U. Steins (2017),CEPR Discussion Paper No. DP12218., URL https://ssrn . com/abstract=3018472 .[16] T. Piketty, Capital in the Twenty-First Century (Har-vard University Press, 2014), ISBN 9780674369559, URL https://books . google . com/books?id=J222AgAAQBAJ .[17] M. Levin, Nobody Knows What Palantir Is Worth , Bloomberg Opinion , November 15, 2018, URL . bloomberg . com/opinion/articles/2018-11-15/nobody-knows-what-palantir-is-worth .[18] J. Sommer, The Stock Market Works by Day, But ItLoves the Night , The New York Times , February 2, 2018,URL . nytimes . com/2018/02/02/your-money/stock-market-after-hours-trading . html .[19] D. McCrum, Someone is Wrong on the Internet, DayVersus Night Edition , Financial Times , February 6,2018, URL https://ftalphaville . ft . com/2018/02/06/2198427/someone-is-wrong-on-the-internet-day-versus-night-edition/ .[20] J. Clayton, Statement on Status of the Consol- idated Audit Trail , September 9, 2019, URL . sec . gov/news/public-statement/statement-status-consolidated-audit-trail .[21] Practical implementation of the Strategy is significantlymore complicated than the cartoon shown in Figure 1.A sensible risk profile is achieved with a suitably lever-aged, market-neutral portfolio covering many stocks.Your trading, perhaps totaling in the ballpark of one per-cent of total market volume, will be spread throughoutthe day, rather than concentrated solely just before andat market open and market close as shown in Figure 1.However complex the details of your trading, the impor-tant component (as far as your profits are concerned) isthe expansion of your existing portfolio when the impactof your trading on the market is large and the contrac-tion of your existing portfolio when the impact of yourtrading on the market is small. (If you have lots of newmoney coming in from outside investors, just expand; noneed to contract.)[22] The black curve in Figure 1 shows the price change youexpect in this stock due to your trading, averaging over(ignoring) stochastic variation.[23] This intraday predictability is a reasonable, perfectly un-derstandable, and not-intrinsically-problematic feature ofpublic stock markets. Market makers, not wishing tobe on the wrong side of overnight news they may havemissed, make wider markets early in the trading day. Themarket, viewed as an information aggregator, respectsthe information content of orders placed near the startof the trading day more than the information content oforders placed later in the trading day.[24] Considering a snapshot in time near market open (andspeaking loosely to help the non-expert build intuition),“spreads are wide and depths are thin” means there arefew other orders near the fair market price. Considering asnapshot in time near market close, “spreads are narrowand depths are thick” means there are many other ordersnear the fair market price. You can expect your impactto be greater when you are one of a few than when youare one of many.[25] That such market manipulation is possible is not in ques-tion. The cost of each round-trip trade depends on howmuch you trade, but does not depend on the size of yourexisting portfolio. Your mark-to-market gains are propor-tional to the size of your existing portfolio. Your mark-to-market gains will therefore exceed the cost of yourround-trip trading as long as your existing portfolio issufficiently large. The practical threshold for “sufficientlylarge” is in question, but for the world’s stock markets,roughly one billion dollars of capital (suitably leveraged,and used to form a market-neutral equity portfolio cov-ering many stocks) appears sufficient [2].[26] Most of the plots in Figures 2 and 3 actually underes-timate the true divergence between overnight and intra-day returns over the past three decades. Many of theplots of indices (such as the NASDAQ Composite index)do not include dividends with reinvestment, the inclu-sion of which leaves the green (intraday) curve unchangedand further increases the height of the blue (overnight)curve. (The most recent mainstream news article weare aware of covering the first plot in Figure 2 [18] ex- cludes dividends, thereby understating overnight returnsby nearly a factor of two.) As a separate matter, correct-ing stale opening prices in index constituents (see Table3 of Ref. [11]) further increases the divergence betweenthe overnight and intraday curves in the plots of indicesshown in Figures 2 and 3, which do not include this cor-rection.[27] The attempted explanation we hear most frequently isthat “company news” (particularly quarterly earnings)is often announced overnight and over the past threedecades this news has generally been good [19]. Sec-tion 4.1 of Ref. [10] dispensed with this attempted expla-nation twelve years ago: removing the days correspond-ing to company earnings announcements does not changethe overnight/intraday split shown in the first plot inFigure 2 in the slightest. Separately, no analysis whatso-ever is required to see that the release of company newsovernight does not explain the consistently negative in-traday returns shown in Figures 2 and 3.[28] We would obviously be very happy to find that the causeof the highly suspicious return patterns in Figures 2 and 3is innocuous. We consider this unlikely, in part because itwould be the first time in the history of financial marketsthat highly suspicious return patterns turned out to befine.[29] The Strategy typically involves systematically expandingand contracting a market-neutral equity portfolio con-sisting of both long and short positions. In some cases,including when regulations differ in their treatment ofyour long and short positions, it may be convenient totalk about the expansion and contraction of your longpositions as one “half” of the Strategy, and the expan-sion and contraction of your short positions as the other“half” of the Strategy.[30] We thank Kenan Qiao and Lammertjan Dam, the au-thors of Ref. [8], for helpful comments and analysis pro-vided in private correspondence related to the last plotin Figure 3.[31] Doing less of your morning expansion before and at mar-ket open in the United States after Ref. [10] pointed outthe first plot in Figure 2 has reduced the highly suspi-cious divergence between overnight and intraday returnsin the United States from 2008 onward (a fact more ob-vious when the top row of Figure 2 is plotted starting in 2008). This shift in your morning trading, combinedwith your regulator’s inability to see how you are tradingwithout your explicit assistance [20], has facilitated yourcontinued, unhampered use of the Strategy in the UnitedStates.[32] The Bombay Stock Exchange provides SENSEX pricesfrom 2009-01-01 onward [7]. These prices agree (with afew immaterial exceptions) with the prices provided byYahoo! Finance during this time.[33] The phrase “initial impact” (or “instantaneous impact”)refers to the immediate impact of your order (at t = 0,the time your order is placed, before any subsequent re-laxation). The phrase “impact decay” refers to the mar-ket subsequently relaxing (during times t > t ≥ t = 0)and “impact decay” (during subsequent times t > δ = 0 . . − . / . λλ