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ECONOMIC SCIENCES SPECIAL FEATURE efficiency are, in some sense, small, but they persist even in A the long term, and cause extended deviations from fundamen- tal values and excess volatility (which, in extreme cases, becomes market instability). Our study provides a simple example of how analyzing markets in these terms and tracking market ecosys- tems through time could give regulators and practitioners better insight into market behavior. Model Description The structure of the model is schematically summarized in Fig. 1. There are two assets, a stock and a bond. The bond trades at a fixed price and yields r = 1% annually in the form of coupon payments that are paid out continuously. The stock pays a dividend D(t) at each time step that is modeled as a discrete time-autocorrelated geometric Brownian motion, of the form

D(t) = D(t − 1) + gD(t − 1) + σD(t − 1)U (t), [1] U (t) = ωU (t − τ) + (1 − ω2)Z (t),

where g is the average rate of dividend payments, σ is the vari- ance, ω is the autocorrelation parameter of the process, and Z is Gaussian noise. The auxiliary process U (t) introduces persis- tence. We choose parameters so that one time step is roughly equal to a day. We use estimates from market data by LeBaron (13), taking g = 2% per year for the growth rate of the dividend Trend Follower with a volatility of σ = 10% (see ref. 14, for example, for a review of the empirical evidence on dividends). The dividend process is ≤ 0% 2% 4%%%6 8 ≤ 10 % positively autocorrelated through the auxiliary process U (t) with return rate (annual) lag τ = 1 d. Trend Follower wealth We use market clearing to set prices. The stock has a fixed 38% 33% 28% 23% 18% 13% 8% supply Q, but the excess demand E(t) for the stock by each B trading strategy varies in time. We allow the trading strategies 2.5% to take short positions and to use leverage (i.e., to borrow in 0% order to take a position in the stock that is larger than their -2.5% wealth). We impose a strategy-specific leverage limit λ¯. Because we use leverage and because the strategies can have demand 7.5% Trend Follower functions with unusual properties, market clearing is not always 5.0% Value Investor straightforward—see Materials and Methods. 2.5% The size of a trading strategy is given by its wealth W (t), 0% that is, the capital invested in it at any given time. In ecology, Return Volatility 19% 24% 29% 34% 39% 44% 49% this corresponds to the population of a species, which is also Value Investor wealth called its abundance. Unless otherwise stated, the wealth of each strategy varies proportional to its cumulative performance. Let- Fig. 2. The profitability of the dominant strategy in the wealth landscape. A is a ternary plot which displays the returns achieved by the strategy with ting πi (t) be the return of strategy i at time t, the wealth changes the largest return. The axes correspond to the relative wealth invested in according to each strategy. The top corner is pure trend followers, the left corner is pure noise traders, and the right corner is pure value investors. The color indicates Wi (t + 1) = (1 + f πi (t))Wi (t). [2] the strategy with the highest returns at a given relative wealth vector w. The regions colored in red correspond to the noise traders, blue regions The reinvestment rate f models investor flows of capital. The correspond to value investors, and green regions correspond to the trend default value f = 1 corresponds to passive reinvestment, and f > followers. The intensity of the color indicates the size of the average return. 1 means that profitable strategies attract additional capital and (B) Upper shows the average returns to value investors (blue) and trend unprofitable strategies lose additional capital. followers (green) while holding the noise trader wealth at its equilibrium A trading strategy is defined by its trading signal φ(t), which value of 42%. Lower shows the volatility in the returns of each strategy. The can depend on the price p(t) and other variables, such as div- horizontal axis is the relative wealth of the trend follower (top axis) and value investor (bottom axis). idends and past prices. We modify φ by a tanh function to

ensure that the excess demand is bounded and differentiable. A strategy’s excess demand for the stock is

W (t − 1)λ¯  1  E(t) = tanh (c · φ(t))+ − S(t − 1), [3] p(t) 2

where S(t − 1) is the number of shares of the stock held at ¯ Fig. 1. The three trading strategies correspond to noise traders, value the previous time step, and λ is the strategy-specific leverage investors and trend followers. They invest their capital in a stock and a bond. limit. The parameter c > 0 determines the aggressiveness of the The mixture for each strategy changes with time as strategies accumulate response to the signal φ, and is strategy specific. When the sig- wealth based on their historical performance. nal of the strategy is zero, the agent is indifferent between the

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ECONOMIC SCIENCES SPECIAL FEATURE Table 1. Estimated community matrix near the equilibrium at This process reverts to√ the long-term mean µ = 1 with rever- w = (NT = 0.43, VI = 0.34, TF = 0.23) sion rate ρ = 1 − 6×252 0.5, meaning the noise has a half life

Gij NT, % VI, % TF, % of 6 y, in accordance with the values estimated by Bouchaud et al. (22). The variable  is a standard normal random vari- NT −0.89 0.89 0.82 able, and γ = 20% is a volatility parameter, chosen so that noise VI 26.6 −10.6 22.4 traders generate volatility in excess of the volatility of dividends, TF 11.1 15.2 −19.3 matching the level observed in the US stock market. NT, noise traders; VI, value investors; TF, trend followers. The parameters of the model are summarized later (see Table 5). We have chosen them for an appropriate trade-off between realism and conceptual interest, for example, so that each strat- V VI(t) The valuation used by the value investors is made by egy has a region in the wealth landscape where it is profitable. We g estimating the mean growth rate from past data using the will see that there are a few ways in which the properties of these [D(t)] = D(0)(1 + g)t classical dividend discount model (16) E . strategies do not match those observed in real markets (the trend ω This ignores the autocorrelation coefficient , and so, in general, strategy is very short term, the value investor uses somewhat high V VI(t) 6 =V (t) . leverage, and, at equilibrium, the noise trader has a surprisingly We define the trading signal for the value investor as the dif- low return-to-risk ratio). Nonetheless, they are realistic enough ference in log prices between the estimated fundamental value to make our key points. V VI(t) and the market price. Results VI φVI(t) = log2 V (t) − log2 p(t). [5] Density Dependence. An ecosystem is said to have density depen- dence if its characteristics depend on the population sizes of the This strategy enters into a long position when the proposed price species, as is typically the case. Similarly, a market ecosystem is is lower than the estimated fundamental value and enters into a density dependent if its characteristics depend on the wealths of short position when the proposed price is higher than the esti- the strategies (both its own wealth and that of other strategies). mated fundamental value. The use of the base two logarithm The toy market ecosystem that we study here is strongly density means that the value investor employs all of its assets when the dependent. stock is trading at half the perceived value (17). When the core ideas in this paper were originally introduced in ref. 3, prices were formed using a market impact function, which Trend followers. Trend followers expect that historical trends translates the aggregate trade imbalance at any time into a shift in returns continue into the short-term future. Several vari- in prices. This can be viewed as a local linearization of market ants exist in the literature, including the archetypal trend fol- clearing. The use of a market impact function suppresses density lower that we use here (3, 10, 18–20). There is evidence to dependence and neglects nonlinearities that are important for suggest that trend-based investment strategies are profitable understanding market ecology. over long time horizons, and ref. 21 argues that investors earn With market clearing, there is strong density dependence. a premium for the liquidity risk associated with stocks with This is evident in Fig. 2, which shows which strategy makes high momentum (momentum trading is a synonym for trend the highest profits as a function of the relative size of each following). of the three strategies. To control the size of each strategy, The trend-following strategy extrapolates the trend in price we turn off reinvestment, and instead replenish the wealth between recently realized prices p(t − 1) and p(t − 2) time steps of each strategy at each step as needed to hold wealth con- in the past. It always buys when prices trend upward and sells stant. We then systematically vary the wealth vector W = when they trend downward. (WNT, WVI, WTF). We somewhat arbitrarily let the total wealth 8 WT = WNT + WVI + WTF = 3 × 10 . For convenience of inter- φTF(t) = log p(t − 1) − log p(t − 2). [6] 2 2 pretation, we plot the relative wealth wi (t) = Wi /WT . The results shown are averages over many long runs; to avoid tran- The trend followers’ demand is a decreasing function of price. sients, we exclude the first 252 time steps, corresponding to one Unlike the value investor, who cannot observe autocorrelations trading year. in dividends, the trend follower can exploit the autocorrelation Roughly speaking, the profitability of the dominant strategy that dividends impart to prices. Because the positive autocor- divides the wealth landscape into four distinct regions. Trend relation in market prices is, at most, ω = 0.1, we multiply the followers dominate at the bottom of the diagram, where their signal by 1/ω. wealth is small. Value investors dominate on the left side of the diagram, where their wealth is small, and noise traders domi- Noise traders. Noise traders represent nonprofessional investors nate on the right side of the diagram, where their wealth is small. who do not track the market closely. Their transactions are There is an intersection point near the center where the returns mostly for liquidity, but they are also somewhat aware of value, so of all three strategies are the same. In addition, there is a compli- they are slightly more likely to buy when the market is underval- cated region at the top of the diagram, where no single strategy ued and slightly more likely to sell when the market is overvalued dominates. The turbulent behavior in this region comes about (this is necessary so that their positions do not diverge over long because the wealth invested by trend followers is large, and the periods of time). price dynamics are unstable. We do not regard this region as The signal function of our noise traders contains the product realistic, except perhaps in rare extreme market conditions. of the value estimate V VI(t) (which is the same as for the value investors) and a stochastic component X (t), Table 2. Estimated community matrix near VI w = (NT = 0.26, VI = 0.55, TF = 0.19) φNT(t) = log2 X (t)V (t) − log2 p(t). [7] Gij NT, % VI, % TF, % The noise process X (t) is a discretized Ornstein–Uhlenbeck NT 0.46 0.40 0.36 process, which has the form − VI 8.94 −0.77 −1.89 TF 6.81 6.87 9.65 X (t) = X (t − 1) + ρ(µ − X (t − 1)) + γ. [8] −

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ECONOMIC SCIENCES SPECIAL FEATURE A

B

C

D

Fig. 5. How ecosystem dynamics cause market malfunctions. A gives a color map of the price volatility over the wealth landscape; the volatility is low and constant throughout the lower right part of the diagram, where the system spends most of its time, but there is a high-volatility region running across the upper left. A sample trajectory spanning 200 y, beginning at the efficient equilibrium, is shown in black. The noise caused by statistical fluctuations in performance causes large deviations from equilibrium and excursions into the high volatility region. B shows the volatility of this trajectory as a function of time, plotted against the predicted volatility (see Eq. 12). C shows the actual mispricing plotted against the predicted mispricing. D shows the wealth of the value investors and trend followers.

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(24, w i G i htis, that , (t ji ) r positive, are netdin invested −0.89 i sthe as G 3 2) (3, π ji [9] i for = is yterelation the by let never species are we webs If food simple. real this that means detritivores, and three. omnivores level trophic have lions than and two, higher one, level level level trophic trophic one has have grass zebras system, definition, idealized this by in so, is, eats, it species what a of of The level grass. population with trophic interactions the direct no on have depends lions though then grass even grass, lions, of grass, of density density eat the the by zebras similarly, affected and, strongly and is zebras, lions of interactions eat population the lions the If understanding species. for between framework conceptual Level. tant Trophic and financial Webs Food in solutions oscillating question. open of simple an existence remains that ecosystems for The strong least so system. at approximation, is this poor system a are this that equations indicate in Lotka–Volterra here dependence results density Our (3). the Lotka–Volterra ecology derived market Farmer for dependence, equations density no of tion ahmtclterms, mathematical when is, strategy strategy strategy of of ence returns given the quantity of a analogous the that define role We ecosystem. the about think they ecosystem. to the but way in integers, plays useful species not a typically provide are still levels trophic resulting The needn aibeCoefficient variable Independent independent as wealth funds’ variables mispricing the and and volatility variables, with dependent at regressions as Multivariate levels trophic 3. Table compute under- we better to dependence, are value order strategies density the In three the from the changes. stand of this lot wealths unequal, a just the substantially quite profits where but follower equilibrium, trader the trend noise in the discussion the the and investor—see from trader, prof- little noise investor a value the the follower, from either trend from its the with profit or proximity strategies cannot investor three trader value The only noise the 2.99). The are levels: there = 1.00, wealth because = follower similar is trader trend values integer (noise 2.00, to are = levels investor trophic value the equilibrium, cient rn follower Trend investor Value 1/256, value, possible smallest size. the grid to simulation the it by set determined is but which zero to entries wealth diagonal the for order and os rdr2.4 trader Noise Mispricing, follower Trend investor Value trader Noise Volatility, A ij h xsec faiaswt oecmlctddes uhas such diets, more-complicated with animals of existence The ecnas opt rpi eesfrtesrtge nan in strategies the for levels trophic compute also can We Eqs. = A P ij 10 a [0, max i 0 = i hntetohclevel trophic the then , W k R twealth at max[0, 2 and R j = 2 0 = hntednmntri eo oeas ht in that, also Note zero. is denominator the when = 9 037observations 50,397 0.79; 11 3 037observations 50,397 0.33; π j u l fteohrwatsrmi h ae In same. the remain wealths other the of all but i ed hsb ipycmaigtertrsof returns the comparing simply by this do We . 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ECONOMIC SCIENCES SPECIAL FEATURE Table 4. Details of the balance sheet items used by all funds value investor’s wealth and the trend follower’s wealth have large Assets Liabilities coefficients (in absolute value), and the fit is overwhelmingly sta- tistically significant. The noise trader is also highly statistically Equity significant, but the coefficients and the t statistics are more than Cash C Capital K an order of magnitude smaller. In Fig. 5 B and C, we compare Debt a time series of the predicted volatility and predicted mispricing Margin M Loans L against the actual values. The predictions are very good. trading securities S+ borrowed securities S−

νˆ = −68wvi + 107wtf + 2.4wnt All securities use the most recent market value in valuation. . [12] mˆ = −1.02wvi + 1.5wtf − 0.15wnt each point in the wealth landscape. For three strategies, there Note that, in both cases, the coefficient for trend followers is are 3! = 6 possible orderings of the trophic levels. We display positive, indicating that they drive instabilities, and the coeffi- the ordering of the trophic levels across the wealth landscape cient for value investors is negative, indicating their stabilizing in Fig. 4. influence. The computation of trophic levels is complicated by the fact Nonetheless, due to their effect on the population of value that, for some wealth vectors, there are cycles in the food web. investors, the net effect of the trend followers on market mal- For example, for w = (0.05, 0.15, 0.80), value investors exploit functions is not obvious. In Fig. 5D, we plot the wealth of value noise traders (who cause reversion to fundamental value), trend investors and trend followers. The strong mutualism predicted followers exploit value investors (who induce autocorrelations), by the community matrix is clearly evident from the fact that the and noise traders exploit trend followers (who generate excess wealth of trend followers and value investors rise and fall together. volatility), to complete a cycle. When this happens, Eq. 9 does However, their dynamics are quite different—there are several not converge, and the trophic levels become undefined. Cycles in precipitous drops in the value investors’ wealth, whereas the the trophic web are not unique to markets—they can also occur trend followers tend to take more-gradual losses. As predicted, in biology, for example, due to cannibalism or detritovores. the highest-volatility episodes happen when the value investors’ A comparison of Figs. 4 and 3C makes it clear that, at long wealth drops sharply while the trend followers’ wealth is high. times (after transients have died out), the system divides its time between the region in which the trophic levels are ordered as Discussion (noise trader, value investor, trend follower), as they are at the equilibrium point, and the region where there are cycles, where Our analysis here demonstrates how understanding fluctuations the trophic levels are undefined. of the wealth of the strategies in the ecosystem can help us pre- dict market malfunctions such as mispricings and endogenously How Ecosystem Dynamics Cause Market Malfunction. The wealth generated clustered volatility. The toy model that we study here dynamics of the market ecosystem help explain why the mar- is simple and highly stylized, but it illustrates how one can import ket malfunctions and illuminate the origins of excess volatility ideas from ecology to better understand financial markets. Our and mispricing, that is, deviations of prices from fundamental analysis of this model illustrates several properties of market values. Both in real markets (2) and in the agent-based models ecosystems that we hypothesize are likely to be true in more mentioned earlier (including our model here), volatility and mis- general settings. pricings change endogenously with time—there are eras where Concepts from ecology give important insights into how devi- they are large and eras where they are small. Volatility varies ations from market efficiency occur and how they affect prices. intermittently, with periods of low volatility punctuated by bursts While the market may be close to efficiency in the sense that the of high volatility, called clustered volatility. The standard expla- excess returns to any given strategy are small, nonetheless, there nations for clustered volatility are fluctuating agent populations can be substantial deviations in the wealth of different strategies, (9, 10, 27) and leverage (28). We focus here on the first explana- that can cause excess volatility and market instability. tion; while we also observe that clustered volatility increases with Market ecology is a complement rather than a substitute for increasing leverage, we have not investigated this, in detail, here. the theory of market efficiency. There are circumstances, such Fig. 5A presents the variation of the volatility across the as pricing options, where market efficiency is a useful hypothe- wealth landscape. The landscape can roughly be divided into two sis. Market ecology, in contrast, provides insight into how and regions. On the lower right, there is a flat “low-volatility plain” why markets deviate from efficiency, and what the consequences occupying most of the landscape. On the upper left, there is a of this are. It can be used to explain the time dependence high-volatility region, with a sharp boundary between the two. in the returns of trading strategies, and, in some cases, it can As we will now show, excursions into the high-volatility region be used to explain market malfunctions. One of our main cause clustered volatility. A similar story holds for mispricing. Fig. 5A shows a sample trajectory that begins at the efficient Table 5. List of the model parameters and their values equilibrium and spans 200 y. This sample is representative of trajectories that fluctuate around the equilibrium point indefi- Parameter Value Description nitely. We choose a 200-y time span to show the scale at which r 1% annual Risk-free rate spikes in volatility and mispricing occur. The statistical fluctu- g 1% annual Dividend growth rate ations in the performance of the three strategies act as noise, k 2% annual Cost of equity causing large excursions away from equilibrium. The trajectory σ 10% annual dividend growth Volatility mostly remains on the volatility plain, but there are several epochs ω 0.1 Dividend autocorrelation parameter where it ventures into the high-volatility region, causing bursts of τ 1 d Dividend autocorrelation lag high volatility. f 1 Reinvestment rate q The wealth dynamics have strong explanatory power for both ρ 1 − 6·252 1 Noise trader mean reversion rate mispricing and volatility. This is illustrated in Table 3, where we 2 σNT 20% annual Noise trader volatility perform regressions of the strategies’ wealth against volatility λ¯ , λ¯ , λ¯ 1, 8, 1 Leverage limit using daily values for the time series shown in Fig. 5A. For volatil- NT VI TF c , c , c 5, 10, 4 Signal scale ity, R2 = 0.79, and, for mispricing, R2 = 0.33. In both cases, the NT VI TF

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Palmer, G. R. LeBaron, D. B. Holland, H. J. markets. Arthur, efficient B. informationally W. of impossibility 9. the On Stiglitz, E. J. value-tracking Grossman, of Dynamics J. MacKay, S. S. R. 8. Battey, S. H. Bashe, L. K. Gunton, trading. R. computer Beale, of N. future the 7. on perspective ecological An Skouras, S. Farmer, D. evolutionary J. an 6. from efficiency Schenk-Hopp Market R. K. Hens, hypothesis: T. market 5. adaptive The Lo, evolution. A. and 4. ecology force, prices? Market Farmer, stock D. moves J. What 3. Summers, Shiller, J. H. R. L. 2. Poterba, M. J. Cutler, M. D. 1. hr r aypsil xesost hswr.A obvi- An work. this to extensions possible many are There to approach the that is results striking most our of One all of sheets balance the to access have potentially Regulators of studies empirical of examples few a only far, so are, There .Tsaso,K .Jd,Es Esve,e.1 06,vl ,p.1187–1233. pp. 2, vol. 2006), 1, ed. (Elsevier, Eds. Judd, L. K. Tesfatsion, L. 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J. 101–123 37, 895–953 11, .Portfolio J. 1059 65, 9 .Krlno .S ye .Smd,T uu,Tefls rs:Hg-rqec trading High-frequency clustered crash: and flash The tails Tuzun, T. fat Samadi, M. causes Kyle, S. Leverage A. Kirilenko, Geanakoplos, A. J. 29. Farmer, dynam- D. pricing J. asset Thurner, S. in 28. volatility stochastic Structural Westerhoff, F. Franke, R. 27. interactions. mutualistic in outcomes Conditional Bronstein, L. J. 26. stable? be system Novak complex large M. a game. Will 25. May, reality M. The R. Lloyd, 24. S. Farmer, D. J. Cherkashin, D. 23. Bouchaud P. J.-P. 22. trading. Asness momentum S. toward C. bias a 21. and dynamics Wealth LeBaron, B. 20. 0 .Msito .Mrta .Pio .N atga ogtr clg fivsosi a in investors of ecology Long-term Mantegna, N. R. Piilo, J. Marotta, L. Musciotto, F. 30. strategies. trading common of dynamics price simple The a Joshi, in S. chaos to Farmer, routes D. and J. common beliefs 19. Heterogeneous on Hommes, policy H. dividend C. and Brock, yield A. W. dividend of 18. effects profit. The of Scholes, rate M. required Black, The F. analysis: 17. equipment Capital Shapiro, E. Gordon, J. M. 16. 00rsac n noainpormudrGatAreet952215. Agreement Horizon (TAILOR) Grant Union under Reasoning program European innovation and by and research Optimisation funded 2020 Learning, project Trust- Economic by a Integrating the supported (https://tailor-network.eu/), - partially We by was Baillie-Gifford, AI research funded work. This Awards, worthy program, Council. this Faculty Research Macroeconomics Social of AI and Rebuilding Morgan development the J.P. the and by during Svosve funding Cephas discussion acknowledge and Kleinnijenhuis, enlightening Alissa for Cont, Rama Wooldridge, Michael ACKNOWLEDGMENTS. (https://github.com/INET-Complexity/market-ecology). GitHub in deposited Availability. Data Library. default Software. the lists 5 Table insolvent. losses. are to funds due more condition parameters. or happen solvency one changes can the when assets meet This ends funds lation risky capital. all equity of that of require proportion We amount the the when than constraint faster leverage its late odns on,admri stesm sters-rert rmholding from rate risk-free the as same the bond. is the margin and 10. loans, and holdings, 1 cash common between borrowing indicate managers ratios by fund leverage institutional assets, use US risky of can filings additional regulatory funds Public purchase The to sets cash. fund funds of the borrowed form time, the later can in a fund rowings, amount at short-selling lender margin the the a that to aside guarantee securities to borrowed order the In return position. short a ate equity with endowed S are Funds 4. Table securities in presented is capital that sheet ance Sheets. Balance and Accounting Methods and Materials ‡ † h lcrncDt ahrn,Aayi,adRtivlsse EGR,publicly (EDGAR), system Retrieval and Commission. Analysis, Exchange and Securities Gathering, https://www.sec.gov/edgar.shtml , at Data available Electronic The noe-orelbayfraetbsdmdln hc spbil cesbeat accessible publicly is which modeling agent-based for https://github.com/INET-Complexity/ESL. library open-source An < elhi acltdas calculated is Wealth na lcrncmarket. electronic an in volatility. contest. model and (2012). Estimation ics: 1994). (6 214–217 matrix? community the (2016). is What bations: (2009). 1091–1105 2018). June 1 (Accessed https://arxiv.org/abs/1711.04717 (2017). [Preprint] (2013). Lett. Organ. Behav. nnilmarket. financial model. pricing asset returns. and prices stock Sci. 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ECONOMIC SCIENCES SPECIAL FEATURE