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- 1. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR Richard H. Day* ABSTRACT Most exchanges in a decentralized economy are mediated by agents who make markets. This paper applies the elementary theory of such mechanisms to equity markets. Based on stylized institutional and behavioral facts and exploiting the methods of nonlinear dynamics, it explains salient properties of stock market dynamics. 1. INTRODUCTION involved in the exchange of goods and those in the exchange of ﬁnancial assets, constitute the visible In the real world, the exchange of goods rarely occurs hands of the market economy. between producers and consumers, and only in spe- It is natural that, as a ﬁrst step, this visible hand cial markets are prices arrived at through a bidding should be modeled in terms of general equilibrium process. Instead, transactions are most often medi- theory, a step taken by Clower and Friedman (1984) ated by merchants, brokers, or trading specialists. and by Friedman (1986). But such an approach tells Sometimes these ‘‘go-betweens’’ supply demanders us little about how the market economy ﬁnds an equi- out of inventory at announced prices and then re- librium and nothing at all about how it functions plenish inventories by purchasing from suppliers, when out of equilibrium. In my recent book, Complex again at an announced price. Such market mediators Economic Dynamics (Day 1994), I introduced an al- adjust prices in response to changes in inventory or ternative approach based on an explicit mediation order backlogs that reﬂect excess demand or supply. mechanism that is designed to work out of equilib- This is the case with retail stores for most con- rium. In it, buyers and sellers are not perfectly coor- sumer goods, for fuels such as petroleum and coal, dinated by prices. Rather, their actions are mediated and for various kinds of tools and machinery used in by a stylized ‘‘mediator,’’ who sets the price for a brief farming and industry. In such markets inventories are period of time, accumulates inventory from sellers, goods on display in stores or available on order and decumulates it in response to demand. The model through catalogs from wholesale distributors. Sup- is abstract and simpliﬁed—as a theory should be—but pliers and demanders do not bargain with each other. it reﬂects—at least in an approximate way—certain They simply carry out their desired transactions with structural characteristics of real markets and appears the mediator, given the current price (and whatever to explain, in a qualitative sense, salient features of expectations they may have about the future). They market behavior. then modify their actions in response to whatever Whatever its shortcomings as a theory of markets, prices are announced as time passes. The merchant it provides a useful pedagogical introduction to some must, of course, invest in this inventory, earn a wage fundamental concepts of complex economic dynam- on the management of it and a reasonable rate of ics. By the latter term, I mean the study of economic return on the capital invested. models whose trajectories exhibit one or both of two Likewise, markets for equities and other ﬁnancial properties: (1) irregular nonperiodic ﬂuctuations and assets also operate—at least in substantial measure— (2) endogenous phase-switching. These properties through systems of brokers and traders who accu- would, on the face of it, seem to be particularly rele- mulate orders to buy and sell, determine prices, and vant for describing behavior in ﬁnancial markets. mediate the ﬂow of assets among the various de- In this paper I ﬁrst summarize the basic idea of manders and suppliers. These mediators, both those market mediation and the requirements for a market to exist out of equilibrium. Next, the model of a styl- *Richard H. Day, Ph.D., is Professor of Economics at the University ized equity market is described and its properties of Southern California, Los Angeles, California 90089-0253. 6
- 2. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 7 summarized. Then I review some initial work in- demand and supply functions (that is, those that are tended to determine whether this approach is indeed downward and upward sloping, respectively): consistent with the facts. After the conclusion, an ap- pendix brieﬂy summarizes the central concepts of The Global Dynamics of Price Adjustments (A) complex dynamics used in the theory. (i) For a ‘‘robust’’ range of parameter values, the process converges to a unique, positive, station- 2. MARKET MEDIATION OUT OF ˜ ary state, p, which gives a market-clearing equi- librium. EQUILIBRIUM (ii) For a robust range of parameter values, the pro- cess exhibits cyclic or chaotic, out-of-equilib- 2.1 The Global Dynamics of Mediator rium price sequences. Price Adjustments (iii) When the price adjustment strategy satisﬁes cer- Assume that the mediator purchases from suppliers tain technical properties, then the price adjust- at an announced price and marks this price up by a ment process is strongly ergodic; that is, relative fraction for sales to demanders. Suppliers, therefore, price frequencies converge to an integrable den- receive p per unit and supply S(p). Demanders pay sity function. (1 )p and purchase D (p): D[(1 )p]. Demanders and suppliers belong to separate groups, and both de- In short, anything can happen in market mediation mand and supply must be non-negative. The param- processes. Prices may converge to a market-clearing eter is called the markup and (1 )p the markup equilibrium, or they may not. Periodic cycles may price. It is assumed that is a constant. The excess emerge, or there may be nonperiodic ﬂuctuations. It demand ( excess supply) is all depends on the proﬁles of demand and supply, on the mediator’s price markup, and the strength or e(pt) D (pt) S(pt). (1) speed of the mediator’s response to inventory The mediator’s inventory at the beginning of period changes. Moreover, as parameters of the underlying t is st. Sales to demanders reduce inventory, and pur- structural relationships change, those that appear in chases from suppliers increase inventories. Invento- the supply and demand functions or that govern spe- ries change, therefore, by the amount of excess cialist behavior—the qualitative behavior—will also supply, change, perhaps in a complex way. These results may explain—at least in part—why st 1 st e(pt). (2) real world prices and inventories ﬂuctuate and why The mediator does not know the demand or supply markets can work, even though supply and demand functions nor does he know the price at which they are not perfectly coordinated at prevailing prices. are equated. He adjusts his price in response to the Three questions must be considered, however. change in inventories, which he does observe accord- First, is mediation proﬁtable for the mediator and un- ing to the strategy, der what conditions of stability and instability? Sec- pt (pt) pt e(pt). (3) ond, can inventory outages occur; that is, could 1 excess demand exceed supply plus inventories so that I call this process mediator tatonnement. (If is zero, not all desired sales and purchases could be consum- the process reduces to the standard Walrasian form.) mated? Third, could the mediator experience a kind The parameter is a coefﬁcient indicating the of gambler’s ruin; that is, could the mediator go into ‘‘speed’’ or magnitude of price adjustment. bankruptcy and could this happen in ﬁnite time even An alternative relative mediator tatonnement is if mediation is proﬁtable on average? These questions based on percentage changes in price and excess de- add up to a fundamental issue: Can a market based mand, that is, on mediator tatonnement function indeﬁnitely? e(pt)pt pt 1 pt . (4) max{D (pt), S(pt)} 2.2 Existence out of Equilibrium In this case, prices are changed in proportion to the 2.2.1 Inventories excess demand relative to the long side of the market. For market mediation to work indeﬁnitely, the me- Elsewhere I have shown that these two strategies diator must have at any one time a sufﬁciently large possess the following properties for normal aggregate
- 3. 8 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 initial inventory to cover the largest excess demand of carrying inventory, and the cost of the mediation that can occur. Over time, the accumulation of inven- process itself. Suppose as a crude approximation that tories when excess supply occurs must replenish the these latter costs do not vary with sales but constitute decumulation that occurs during periods of excess de- a ﬁxed overhead cost per period in amount H. Then mand. If excess demand exceeds supply plus inven- the total cost of mediation is tory, then some desired transactions cannot be C(p) : pS(p) H. executed. An outage is said to occur. It can be shown that the price adjustment process Gross proﬁt is then is bounded and that prices eventually lie in a trapping (p) : R (p) C(p) pD (p) pe(p) H. set given by an interval [q, Q]. For each positive ini- tial p0, there is a minimum number of periods, T(p0), ˜ Let there be a stationary state, p, at which supply before price enters this set. The ﬁrm must be able to ˜ and demand are equal. Then e(p) 0, so cover the largest excess demand that could occur as ˜ (p) ˜ ˜ pD (p) H. well as the total accumulated excess demand for this interval. For the two-tatonnement processes under Consequently, the mediator makes a proﬁt at a sta- consideration, this amount is tionary state if, and only if, ˜ ˜ pD (p) H. Σ e[ T(p) sm max i (p)] . p i 1 The cash reserve of the ﬁrm at time t is given by Then it can be shown that if initial inventories are Σ t large enough; that is, if wt 1 wt (pf) w0 (pn) n 0 1 s0 (Q q) sm, If the mean proﬁt is positive, that is, if Σ t 1 no outage will occur. This implies that the slower the E[ ] lim (pi) 0, t→ t i 1 speed of adjustment, the wider the range of long-run price variation; and the more prices are perturbed be- then for each p0, there exists a smallest time, say low the range of long-run variations, the greater initial T(p0), such that wt 0 for all t≥T. The worst accu- inventories must be to prevent an outage. This is the mulated capital loss the ﬁrm can experience until case for mediator tatonnement. time T is reached, for any initial p0 and w0, is For relative mediator tatonnement, the initial in- Σ s ventory sufﬁcient to prevent outage is wm min [ i(p0)] 0 s T i 1 Y (Q q) s0 ≥ sm , Consequently, if w0 wm, then wt 0 for all t. These q assumptions imply that the cash reserve grows with- where Y is the upper bound on demand and supply out bound. However, if dividends are incorporated and where , q, Q, and sm have the meanings given into overhead and adjusted upward to absorb the ac- above. In a manner analogous to absolute mediator cumulated proﬁt, then the cash reserve will be tatonnement, the slower the speed of adjustment, the bounded. wider the relative range of long-run price variation, To sum up these ﬁndings, we can state the following: the greater the bounds on demand and supply, and the closer the initial price to zero, then the greater The Viability Conditions (B) the initial inventory must be to prevent an outage. If a given price adjustment process is ergodic (which includes convergence to a stationary state or cycle) 2.2.2 Proﬁts and Cash Reserves and if the statistical expectation of proﬁt is positive, The mediator’s revenues are a function, then there exists a large enough inventory and a large enough initial wealth such that the market mediator R (p) : (1 )pD (p). can avoid outages, sustain costs, and withdraw a pos- Costs include the expenditure on inventory accumu- itive dividend indeﬁnitely. lation, which is given by µpS(p) each period, the cost
- 4. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 9 3. A STYLIZED EQUITY MARKET or order backlog is therefore exactly the same as (1) except no markup is assumed; that is, v 0. Turn now to an application of the above theory to To keep inventory in balance is essential because ﬁnancial markets, in particular to the markets for eq- excess buying will exhaust the market maker’s ﬁnan- uities. These markets have two striking characteris- cial resources and excess selling will exhaust the in- tics that suggest the presence of complex dynamics. ventory. Consequently, the price is adjusted from First, stock prices ﬂuctuate in a highly irregular man- period to period so as to balance holdings over time, ner. Second, they generate alternating ‘‘bull’’ and while at the same time moderating price changes so ‘‘bear’’ market regimes that switch from one to the as not to destabilize the market excessively in the other at irregular intervals. In the bull markets, prices process. The pricing strategy is therefore analogous to move upward for some time accompanied by rela- Equation (3). tively steep ‘‘pullbacks,’’ with higher transaction vol- The mediator’s service to the public is to ‘‘maintain umes near the highs. an orderly market’’ and to make a proﬁt by doing so. The current fundamental value, v, of a share is Suppose the mediator charges buyers and sellers a based on the most recent ‘‘fundamental’’ data such as commission, , per unit transaction. This commission earnings, dividends, debt-equity ratio, and so on. A generates a positive ﬂow of income at any time of the short-cut formula would be just the present value of amount [| (pt)| | (pt)|], whether the market is ris- current earnings capitalized at the current rate of in- ing or falling! The value of stock purchases minus the terest or multiplied by a suitable price-earnings ratio value of stock sales presents a capital gain or loss in (Black 1986, p. 533). The investment value, u, is an amount pte(pt). Supposing, for simplicity, that the estimate of the expected future value of v based on cost of mediation is a ﬁxed amount per period, say H, comprehensive information and a sophisticated anal- then the proﬁt consists of ysis of ‘‘long-run’’ considerations. Stocks almost never trade at v or u but rather at a (pt) [| (pt)| | (pt)|] pte(pt) H 0. (6) market price, p, determined by a mediator, which re- Just as in the ‘‘general’’ case, if the statistical ex- ﬂects investors’ expectations and the aggregate forces pectation of of market supply and demand. The mediator is the abstract counterpart of a trading specialist. Epe(p) H E[| (p)| | (p)|], (7) In addition to the mediator, I include two investor then E[ (p)] 0 and mediation is proﬁtable on average. types called - and -investors, each of whom uses a Although I have speciﬁed the mediator’s strategy as distinct buy/sell strategy. Since each market partici- a simple price-inventory adjustment rule, it is possi- pant can be a demander or a supplier of shares, then ble to show how it is related to optimal behavior in a excess demand ( excess supply) is manner analogous to the argument of Section 2. More e(p) (p) (p), (5) of that is discussed below. At this point, let us con- sider the demand and supply for shares. where (p) and (p) are positive for demand and neg- ative for supply. In reality, a great many strategies are used by market participants. Two, however, are suf- 3.2 -Investors ﬁcient for the initial purpose of giving a theoretical The -investor uses a strategy based on an indepen- explanation of the stylized facts entirely in terms of dent, sophisticated estimate of ‘‘long-run’’ investment intrinsic market behavior. More general models can value, u, in relation to current price and on an esti- be discussed later. Consider each of the market par- mate of the chance for capital gains and losses. Such ticipants in turn. investors try to buy when prices are well below in- vestment value, entering the market when the chance 3.1 Mediators of a capital gain appears to be high; they try to sell when prices are above their estimated investment To simplify the analysis, we suppose that the media- value, exiting the market when the chance of a capital tor announces a price at ﬁxed intervals of time, say loss appears to be great. They have to consider the daily or hourly, and executes orders at that an- chance of lost opportunity either to fail to buy when nounced price. An excess of orders-to-buy over or- the market is low or to fail to sell when the market is ders-to-sell appears as a reduction of inventory or high. The chance of gain or loss may be supposed to order backlog of inventory. The change in inventory be estimated on the basis of anticipated topping price
- 5. 10 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 M u and an anticipated bottoming price m u with succeed in selling when prices are going up, nor do the following interpretation. When p is close to the they typically buy when prices are falling. Instead, in topping price M, the chance of losing a capital gain a characteristic manner they enter the market when and of experiencing a capital loss is great; when p is prices are rising, under the belief that the market will close to the bottoming price m, the chance of missing continue to go up, and exit when it is falling, under a capital gain by failing to buy is great; when p is close the belief that it will continue to go down. This means to the investment value u, the perceived chance of that -investors ‘‘chase’’ prices up and down, making capital gain or loss is small or zero. bull and bear markets. Their behavior would seem to Of course, each -investor differs in the estimates be irrational, but, as we shall see, they are actually of u, m, and M. As an approximation, we can repre- correct in their forecasts most of the time. sent their aggregate behavior by a demand function that depends on an ‘‘average’’ or representative val- 3.4 The Market Mechanism ues. Therefore, -investor demand, (p), is monoton- ically decreasing, falling rapidly near m, ﬂattening out For the time being, two additional background as- and approaching 0 near u, then falling rapidly again sumptions are made. First, we assume that a ﬂow of near M. Thus, -investors exits the market. Their assets are distributed to new -investors, who enter in a steady '(p) 0 and (u) 0. ﬂow. The latter replace those -investors who learn When market price equals investment value, -inves- how to become -investors and those who go bank- tors hold their current stocks so their demand is zero. rupt or get discouraged and abandon the stock market permanently. We assume these ﬂows are just enough to keep the aggregate market excess demand function 3.3 -Investors at a ﬁxed proﬁle. Second, we examine the dynamics The behavior of -investors is expensive: it takes of the market when the parameters u and v are con- time, uses costly information, and requires substantial stant and equal, and when the parameters m and M investment in intellectual and computational capital. that enter the -investor’s strategy are constant. Most market participants cannot afford to pursue be- Given these background assumptions and the sev- havior of this kind and in reality they do not. The eral behavioral assumptions about our market partic- great majority, whom we call -investors, instead use ipants described above, stock prices are generated by relatively simple rules and relatively low-cost advice. the process (3) which, given (5), becomes In particular, many investors and many investment pt 1 pt [ (p) (p)] (8) advisors (broker representatives and ﬁnancial con- sultants) base their expectations on an extrapolation of a current price and fundamental value. For this 3.5 The Implied Price Dynamics purpose, the spread p v can be taken as a signal of To keep the analysis simple, assume u v. Assuming the future price. When it is positive, the ‘‘market’’ is that the ( ) and ( ) functions are smooth, the graph projecting a rising trend; when it is negative, the of the process can take on two distinctly different ‘‘market’’ is projecting a falling trend. The rule is to shapes, depending on whether 0 '(v) 1, so prices buy into a rising (bull) market and sell into a falling would converge to the fundamental, investment (bear) market, which is captured by the -strategy, value, or '(v) 1, so the fundamental value is unsta- denoted by (p), where '(p) 0 and (v) 0. This ble and ﬂuctuations must ensue. In the former case, form implies that changes in -investor demand are irregular price ﬂuctuations could occur only if a ran- positively correlated with changes in price. dom ‘‘news’’ term were added to the estimate of u (or Given this simple extrapolative rule, our unsophis- to some other parameter). In the latter case, how- ticated investors respond, also like their sophisticated ever, irregular price ﬂuctuations can be generated -counterparts, to the spread between current price endogenously. and the investment value, as they estimate the latter A numerical simulation of Equation (8), using spe- to be. Unlike the -investors, the -investors do not ciﬁc functional forms that satisfy our assumptions and take into account any well-formed chance of capital illustrating the unstable case, is shown in Figure 1. In gain or loss nor specify any well-formed strategies for this experiment the initial price, p0, is just above v, buying or selling at carefully thought-out topping or so -investors enter the market. They create an ex- bottoming prices. As a result, they do not usually cess aggregate demand, because -investors have little
- 6. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 11 fear of a capital loss at this point and hold their po- FIGURE 2 sition. The market maker must sell from inventory A HISTOGRAM OF STOCK PRICES GENERATED BY THE MODEL and adjust the price upward. A sequence of such price increases is driven by the bullish behavior of -inves- tors whose expectations of a rising price are thereby fulﬁlled until -investors sell in sufﬁcient amounts to create an excess supply. Price is then pulled back. The pullback is relatively minor, however, on this ﬁrst runup. Cycles of varying period and amplitude occur as excess demand alternates in sign. In this way a ‘‘bull market’’ is generated, which is characterized by price runups followed by more or less sharp pullbacks. Eventually, however, the price reaches close to the - investors’ anticipated topping price. They ‘‘sell off’’ in so large an amount that the specialist is induced to drop the price drastically, which falls below v, thus precipitating a bear market, a series of price declines of various durations, followed by more or less sharp jumpups. FIGURE 1 A NUMERICAL SIMULATION This numerical simulation has a precise theoretical counterpart, which can be summarized informally as follows: Complex Stock Market Dynamics (C) If -investor demand is strong enough and if -inves- tor demand decreases sharply enough near the topping and bottoming prices, then for almost all ini- tial conditions, the following properties pertain gener- ically: (i) Chaos. Erratic, speculative ﬂuctuations occur. (ii) Switching regimes. Stock prices switch between bull and bear markets at more or less random intervals. (iii) Observability and ergodicity. The relative fre- quencies of prices converge to stable density functions. (iv) Robustness. The above properties are robust with respect to changes in the parameters. (v) Law of large numbers. Price averages obey the central limit theorem. (vi) Deceptive order. Price trajectories pass close to By extending the simulation, it is possible to esti- cycles of varying periodicities, giving the tem- mate the long-run behavior of the model in statistical porary appearance of order, but prices move terms. Figure 2 shows the histogram of stock prices away from any such order and will do so in an generated by the model, while Figure 3 shows four unpredictable way. distributions of stock price averages.
- 7. 12 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 FIGURE 3 DISTRIBUTIONS OFSTOCK PRICE AVERAGES a. 5 ‘‘DAY’’ AVERAGES b. 20 ‘‘DAY’’ AVERAGES c. 60 ‘‘DAY’’ AVERAGES d. 240 ‘‘DAY’’ AVERAGES
- 8. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 13 These results can be derived more easily for a FIGURE 5 piecewise version of the model, as shown in my paper THEORETICAL DENSITY FUNCTIONS FOR THE STOCK PRICE DYNAMICS: with Huang (1993). Then the price adjustment pro- THE PIECEWISE LINEAR CASE cess takes the form shown in Figure 4, and for a spe- cial class of parameters, the exact form of the density function can be derived. Indeed, we have shown that it is a step function whose step widths and heights depend on the parameters of our three agent types. See Figure 5. FIGURE 4 THE MEDIATOR PROCESS FOR THEPIECEWISE LINEAR VERSION (a) (b) 3.6 Mediator Proﬁts and Viability of the Market An equilibrium is not of much value either for spe- cialists or for -investors, because the opportunity for (c) speculative capital gains would not exist and trans- action fees would be nil. But -investors’ strategies mitigate against such an outcome, if their buying and Of course, the -investors always lose when v and selling strategies are highly concentrated in the neigh- u are constant. But notice that even in this extreme borhoods of the bottoming and topping prices. Such case, their expectations of rising and falling prices are behavior leads to a self-fulﬁlling expectation of capital right most of the time. That is, they do in fact cor- gains. Similarly, the mediator can beneﬁt by choosing rectly buy into a rising market and sell into a falling a sufﬁciently great price adjustment parameter, be- one. Their failure is one of acting too late, precisely cause that will lead to ‘‘market-churning’’ and a con- tinuing volume of buy/sell transactions.
- 9. 14 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 either because they do not form or adhere religiously functions. Short-term investors, for example, do not to a strict buy threshold in falling markets and a strict take into account long-run trends but only very near- sell threshold in rising markets. So, if one buys near term conditions. Their guesses for u might be close to the peak or sells near the trough, one can easily be v but be strongly inﬂuenced by current shocks. Their convinced that the mistake was merely one of timing, anticipated spread between m and M might be rather of buying or selling too late. In this way, the market narrow. They try to make money from short-term reinforces the behavior of -investors. Moreover, in a price changes that take place over a few hours, days, large economy there is likely to be a continual supply or weeks. They buy and sell frequently, jumping in of new -investors to replace those who drop out of and out of the market. On the other hand, long-term the market or who learn the -strategy. These are the -investors take into account longer trends. They an- newly established net savers who are entering the ticipate investment values that could be much market for the ﬁrst time. They are easy prey to their higher—or lower—than v and think in terms of a own self-generated interpretation of the lessons the much wider spread between m and M. They are pa- market is teaching. tient for anticipated price movements to materialize To show that the mediator mechanism can work and are content to make money over the long term, indeﬁnitely, it must be shown that conditions analo- usually entering and exiting at infrequent intervals of gous to those of Proposition (B) hold. Although - months or even years. investors’ demand functions are not normal, this can In between these two types is a range of medium- be done given the conditions that pertain in (C): term investors. The sum of all such investors is what we have in mind for the function (p). Similarly, - Market Viability (D) investors might vary in the shapes of their demand Given the conditions in (C), then generically responses to market signals, entering the market or (i) Mediation is proﬁtable on average exiting the market only when price signals are strong (ii) If the mediator has enough initial wealth in terms enough. both of stock inventory and liquidity—or ready Suppose then that there are a -investor types, access to them—the process is viable. each of whose aggregate demand function is repre- sented by a strategy i(p, ui), i 1, . . . , a. Suppose also there are b -investor types, each of whose ag- 3.7 Heterogeneous Investors gregate demand function is represented by a strategy, In the numerical experiments and theoretical analy- i(p, v), i 1, . . . , b. Excess demand is ses so far, it has been assumed for simplicity that Σ Σ a b u v. When u v(v u), we could say that -investors e(p) i(p, u ) i (p, v), i i 1 i 1 are bullish (bearish). The respective bull and bear markets are essentially like those already discussed: and the map that governs price-setting, ﬂuctuations with an occasional long runup followed by a sharp pullback in the case of bulls, or ﬂuctuations (p) : p e(p), at a lower level with an occasional long descent fol- would have a wiggly shape made up of pieces that look lowed by a sharp rebound in the case of bears. The like tilted z’s. It is possible to obtain results analogous conditions for these cases are essentially the same as to Propositions (A) and (B) for this case. the more general situation with switching bull and My students, Mu Gu and Rajesh Srinivasan, studied bear markets. Their study involves no new principles. a piecewise linear version of this more general model However, if u and v were changed with u falling ran- for three -investor types and one -investor type. domly (at random intervals) above and below v, we Figure 6 shows the price adjustment process for one would then get a switching between bull and bear set of parameter values. They analyzed the stochastic markets caused by external forces in addition to those properties of its generated price trajectories to deter- caused by the internal forces of market demand. mine to what extent it corresponded with real world Also, -investors may have very different time ho- data. The ﬁndings are, I think, quite startling. rizons that could lead to quite different demand
- 10. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 15 FIGURE 6 trends are ‘‘bull’’ or ‘‘bear’’ markets that may last for PRICE DYNAMICS WITH SHORT-, MEDIUM-, AND LONG-TERM more than a year and sometimes for a number of -INVESTORS: THE PIECEWISE LINEAR MODEL years. Secondary reactions involve sharp declines or advances that interrupt bull market or bear market trends or advances, respectively. These may last sev- eral weeks or months, during which prices may re- trace about half their primary movement prior to the beginning of the reaction. Superimposed on these long- and medium-term movements are daily ﬂuctu- ations. Technical analysts have identiﬁed at least eight more detailed patterns that also seem to appear in stock data. A ‘‘head and shoulder top’’ can appear when a bullish trend is reversed. A ‘‘head and shoul- der bottom’’ can appear when a bear trend is reversed. ‘‘Ascending triangles’’ are thought to signal a rising market, while conversely the ‘‘descending triangle’’ suggests a falling market. ‘‘Wedges’’ are thought to in- dicate a situation in which buyers are relatively more aggressive than sellers and that an overall bullish trend can develop. A ‘‘breakaway gap’’ is a sudden jump in price after a phase of ‘‘consolidation.’’ A ‘‘run- away gap’’ appears when a very strong move in the direction of trend continues for some time. An ‘‘ex- haustion gap’’ occurs at the termination of a primary 4. MIMICKING MARKET PRICE trend and signals a reaction or trend reversal. PATTERNS To explore the possibility that our theory can explain the existence of such patterns, we generated a very long Stock price patterns have long been studied intensi- time series of prices using the multiple - and -investor vely by technical market analysts. Many investment types discussed in the preceding section and illustrated theories have been proposed on this basis such as the in Figure 6. We then scanned the data to determine ‘‘Dow’’ or the ‘‘Gann.’’ The model presented here does whether patterns like those described by Rhea (1932) not explicitly include an investor type whose buy/sell and by Shaw (1975) could be identiﬁed. strategy is based directly on technical analysis in the Figures 7–12 present a comparison of these tech- sense of pattern recognition. However, it might pro- nical patterns with slices of model-generated price vide a possible explanation for the popularity of such trajectories. We have not yet attempted to test strategies. Indeed, if there does exist a deterministic whether a formal pattern recognition algorithm could component in stock price formation of the kind im- be ‘‘taught’’ to recognize these patterns in the real or plied by our theory, then model-generated data model data, or if such procedures would be able to should produce patterns similar to those found in ac- distinguish between the two. A visual comparison, tual stock price series. however, suggests that our simple theory has a re- According to the ‘‘Dow Theory,’’ stock price move- markable ability to mimic the kind of patterns viewed ments can be classiﬁed into ‘‘primary trends,’’ ‘‘sec- as important by the experts. ondary reactions,’’ and ‘‘daily ﬂuctuations.’’ Primary
- 11. 16 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 FIGURE 7 BULL MARKETS a. EMPIRICAL b. MODEL BASED FIGURE 8 BEAR MARKETS a. EMPIRICAL b. MODEL BASED
- 12. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 17 FIGURE 9 ASCENDING TRIANGLES b. MODEL BASED a. EMPIRICAL FIGURE 10 DESCENDING TRIANGLES b. MODEL BASED a. EMPIRICAL FIGURE 11 WEDGES a. EMPIRICAL b. MODEL BASED
- 13. 18 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 FIGURE 12 BREAKAWAY, RUNAWAY, ANDEXHAUSTION PATTERNS: EMPIRICAL a. BREAKAWAY b. RUNAWAY c. EXHAUSTION d. MODEL GENERATED It is well-known that stock market data exhibit (i) 5. CONCLUSION positive correlations that decay as the lag increases, (ii) excess volatility with respect to the efﬁcient mar- Technical analysts attempt to use patterns to predict ket hypothesis, (iii) strong evidence that they are gen- future price movements. However, in view of the ‘‘de- erated by a nonlinear process, and (iv) lepto-kurtic ceptive order’’ property of the present theory, one distribution (that is, extreme price variations occur must be skeptical about the long-run success of such more often than in normal distributions). All these a practice. Certainly, our model does suggest that ﬁndings have been conﬁrmed by Gu (1992) and Sri- characteristic patterns will indeed appear and will be nivasan (1993), and in a detailed analysis of model- approximated repeatedly—given enough time—but generated stock price series they ﬁnd exactly the never in exactly the same way. The explanation given same properties. by analysts for various patterns also differs from our Moreover, Srinivasan’s exploratory attempt to test own. For example, technical analysts often attribute the model’s validity using the self-excited threshold primary trend reversals and secondary reactions to and regression (SETAR) model suggests that the type external news. The current theory attributes them to of regime-switching due to the existence of short-, the actions of -investors, whose expectations of pos- medium-, and long-term investors predicted by the sible capital gains and losses based on excessively theory is indeed present in the data. under- or overvalued shares, drive strong buying or selling activity. In reality, it seems to us, both expla- nations play a role. If we were to add random shock
- 14. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 19 terms to our model, we could explore the extent to 1. Dynamical Systems which each factor is involved. Let ( , X) be a dynamical system deﬁned by a differ- This hardly proves, however, that the theory in the ence equation in a state variable xt, mathematical form used here is adequate to explain what is going on. Its shortcomings are too substantial. xt 1 (xt) (1) In particular, it does not account for the total amount with domain X. of securities or their ﬂow among the market partici- A trajectory is a sequence (x0) {xt}0 that satisﬁes pants. It does not incorporate the random effects of (1). We note that (x) {x, (x), n(x), . . . ,} is a tra- news, nor does it incorporate investor types whose jectory for x x0. A periodic ﬂuctuation or cycle is a strategies are based on technical trading criteria. Fi- trajectory in which for some integer p 1, xt p xt for nally, it models expectations in a trivial way. all t but xt q / xt for any q with 0 q p. A stationary However, the fact that it does incorporate salient trajectory is a trajectory for which xt x for all t for ˜ characteristics of real market participants and of the some x∈X. A nonperiodic trajectory is one that is nei- ˜ mediator mechanism gives it an a priori plausibility. ther stationary nor cyclic. The fact that it mimics well-known patterns in time series of stock market prices would suggest that it is on the right track. I conclude that the theory of out- 2. Essentially Nonlinear Systems of-equilibrium market mediation and the methods of A nonlinear dynamical system is a dynamical system complex, nonlinear dynamics have much to offer for ( , X) for which the map is not afﬁne. Using terms explaining economic behavior. suggested by Blatt, an essentially linear dynamical system is one for which there exists a transformation f:x→f(x) and an afﬁne map : f(X)→f(X), such that REFERENCES BLACK, F. 1986. ‘‘Noise,’’ The Journal of Finance 41:519–43. (f )(x) ( f(x)) all x ∈ X. (2) CLOVER, R., AND FRIEDMAN, D. 1985. ‘‘Trade Specialists and An essentially nonlinear system is one that is not es- Money in an Ongoing Exchange Economy.’’ Chapter 5 in sentially linear. The Dynamics of Market Economies, edited by R. Day and For example, consider a simpliﬁed version of Solow’s G. Elverson. Amsterdam: North-Holland. model in discrete time, DAY, R.H. 1994. Complex Economic Dynamics. Cambridge, Mass.: The MIT Press. xt 1 Bxt , (3) DAY, R., AND HUANG, W. 1993. ‘‘Chaotically Switching Bear and Bull Markets, The Derivation of Stock Price Distributions where x is the capital/labor ratio; B and are con- from Behavioral Rules.’’ Chapter 12 in Nonlinear Dynam- stants reﬂecting production; and is the savings ratio. ics and Evolutionary Economics, edited by R. Day and P. Let f(x) log x. Then Chen. New York: Oxford University Press. f(xt 1) B f(xt) [f(xt)]. FRIEDMAN, D. 1986. ‘‘Two Microdynamic Models of Exchange,’’ Journal of Economic Behavior 7:129–47. Therefore, (3) is essentially linear. Suppose, however, GU, MU. 1992. ‘‘A Theory of Stock Price Behavior.’’ Ph.D. thesis, there are absolute diseconomies that change (3) to University of Southern California. HUANG, W. 1989. ‘‘Distributional Dynamics for Chaotic Eco- xt 1 Bxt (k xt) , (4) nomic Systems.’’ Ph.D. thesis, University of Southern Cal- where k 0 is an upper bound on possible capital/ ifornia. labor ratios and a positive coefﬁcient. Then (4) is SRINIVASAN, R. 1993. ‘‘An Econometric Study of Stock Market Prices.’’ Ph.D. thesis, University of Southern California. essentially nonlinear. 3. Chaos in the Sense of Li and Yorke APPENDIX Sensitivity to initial conditions or perturbations is COMPLEX ECONOMIC DYNAMICS sometimes deﬁned to mean that two trajectories de- part exponentially, but such departure is local in sys- This appendix summarizes the concepts utilized in tems that are globally stable. Another criterion of this paper. For a complete discussion and references sensitivity is that for given initial conditions x, y∈X, to the related literature, see Day (1994). then
- 15. 20 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 lim sup | t(x) t (y)| 0. (5) I call the dynamical system ( , X) strongly ergodic t→ on the set S if there exists a unique absolutely con- This means in effect that no matter how close y is to tinuous invariant measure µ represented by the den- x, and no matter how close points t(x) and t(y) sity f, such that µ(A) A fdm for all A S. might come at time t, they will move apart a ﬁnite That f( ) is absolutely continuous means that it is distance. If y is a cyclic point and x is not, then (5) positive on a set of intervals and that is nonperiodic means that the trajectory (x) must move away from for almost all initial conditions in these intervals. Let any cycle no matter how close it may come to one. g( ) be an integrable function, then Condition (5) is therefore a strong instability property. Σ g( T 1 1 If nonperiodic trajectories are bounded in X, they lim (x)) t S g(x) f(x)dx, (8) T→ T t 0 may ﬂuctuate ‘‘around one another,’’ perhaps coming close from time to time. If they almost touch inﬁnitely and in particular, often, we have Σ T 1 1 lim s ( t(x)) fdx, (9) lim inf | t(x) t (y)| 0. (6) T→ T t 0 S t→ where XS(x) 1 if x∈S and zero otherwise. These Li and Yorke and later Li, Misiurewicz, Pianigiani equations mean that one can deal with the expected and Yorke presented constructive conditions for the value of a function of the state variable in the same existence of an uncountable scrambled set S X 1 , manner that one would for a random variable even such that for all x, y∈S, (5) and (6) are satisﬁed, and though it is generated by a deterministic process. for all x∈S and for any periodic y, (5) is satisﬁed. Evidently, strongly ergodic systems have some Trajectories that satisfy these conditions are called characteristics very much like stochastic processes, ‘‘chaotic in the sense of Li and Yorke’’ and systems not only in being representable by density functions, that generate them are said to possess topological but also in additional properties that follow, such as chaos. various laws of large numbers and the central limit If there exist periodic trajectories in X with station- theorem. ary cyclic initial conditions, then nonperiodic trajec- tories could wander near some or even all of them. In particular, suppose y is a cyclic point and x is non- 5. Deceptive Order periodic. If If a system ( , X) is strongly ergodic, then it can usu- lim inf | (x) t y| 0, (7) ally be shown that for some integer m the system ( m, t→ X) is topologically chaotic, in which case cycles of all then the trajectory beginning at x will come close to orders mn, n 1, 2, 3, . . . exist as well. For such sys- the periodic point y inﬁnitely many times (in princi- tems, all nonperiodic trajectories that belong to the ple), so that even if (5) holds, (x) may for a time support of µ are sensitive to initial conditions in the approximate the cycle that emanates from y, espe- sense of (5) and all nonperiodic trajectories in support cially if the period of y is fairly small. of µ possess deceptive order in the sense of (7). They pass close to periodic trajectories inﬁnitely often but in a nonperiodic way! An example is the tent map on 4. Ergodic Behavior the interval [0, 1], whose limiting distribution is the A type of nonperiodic behavior also called chaotic and rectangular or uniform distribution. discussed in the mid-19th century arises when the values in a nonperiodic trajectory can be character- 6. Statistical Equilibrium ized in the long run by a stable probability distribu- tion that gives the relative frequencies (or fraction of A strongly ergodic economic system possesses a kind time) a system may enter a given subset of states. of statistical equilibrium, in contrast to a stationary Modern existence proofs originate with Birkhoff and or steady-state equilibrium. In the statistical equilib- von Neuman in the 1930s. More recently Lasota, Li, rium a stationary density function of states replaces Pianigiani, Misiurewicz and Yorke have provided con- the stationary or steady state. Just as it is helpful for structive conditions for the existence of and the tech- some purposes to derive an equilibrium theory and to niques for deriving the density functions themselves. compare equilibria, so it is also helpful for other pur- poses to derive density functions and to compare how
- 16. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 21 they might change as parameters of the system Discussion change. IRWIN T. VANDERHOOF* 7. Multiple-Phase Systems I have for some time been interested in the concept Let Xi, i∈ {0, 1, . . . , n} be a partition of a state of complexity and its use as a paradigm of ﬁnancial space X n and let i, i∈ be a collection of maps, i: and actuarial contexts. The poetic deﬁnition of com- Xi→X. By a (discrete time) multiple-phase dynami- plexity is ‘‘order emergent at the edge of chaos.’’ How- cal system, I mean the following difference equation: ever, when we talk of complexity and chaos in the same sentence, we are using terms of art, not words xt 1 (xt) : (xt) i if xt ∈ Xi. (10) in normal usage. The sets Xi∈ are called phase zones; the maps, i, In this usage, chaos refers, not to random numbers, phase structures; and the pairs ( i, Xi) regimes. The but rather to deterministic chaos. In this situation the phase indicated by i 0 is called the null phase and future is a determined function of the past, but there has the property that 0(x)∈X0 for all x∈X0. In appli- is a nonlinear behavior that can cause the results to cations it represents a state in which the system un- seem to be totally random; this is deterministic chaos. der investigation cannot work or breaks down. Further, if we are dealing with a system described by Multiple-phase systems are essentially nonlinear, nonlinear equations, we may ﬁnd that for some values even though individual regimes may be governed by of controlling parameters the development of the sys- linear maps. tem over time seems to be single-valued, strictly de- terministic, quasi-periodic, or chaotic. Professor Day brieﬂy covers this in his discussion of ergodic behavior. 8. Frequency in Phase The crucial thing, from my point of view, is that If ( , X) is a strongly ergodic multiple-phase dynami- these systems can change from one mode of behavior cal system, and if the ergodic set S intersects more to another as a result of changes in parameters that than one phase zone, then one can give the ‘‘proba- we do not normally focus on. These changes are now bility’’ of visiting various regimes and the conditional characterized as changes in regime—changes in the probabilities of switching from one regime to another. nature of the game, the nature of the reality we are In this case we are permitted to speak quite rigorously dealing with. about frequency in phase, for each phase zone is ‘‘vis- The paper by Professor Day is an illustration of the ited’’ with a frequency in proportion to its measure. groping towards a new kind of understanding of the problems of the actuary working within a ﬁnancial in- termediary. The current level of understanding of the 9. Escape and Evolution problems of the insurance and pension funds is a For the broader purposes of studying economic de- great achievement. Our achievement is so great that velopment, it would seem to be important to account we can now start to understand how great a step is for structural change and irreversible developments before us in reaching towards the now evident new that are not encompassed by the idea of a statistical level of understanding. equilibrium. To capture this idea within the frame- Let me be explicit. We have used statistical meth- work of multiple-phase dynamics, we need not as- ods and equations to develop the necessary safety sume ergodic behavior but only that escape from one margins for ﬁnancial intermediaries on account of sta- regime to another is possible and that the reversion tistical ﬂuctuations in mortality, morbidity, interest to previous regimes is eventually blocked for some ini- rates, and other operational and ﬁnancial variables. tial conditions chosen with positive measure; that is, However, when have companies failed because of sta- for every regime there is a regime that can be reached tistical ﬂuctuations? Was that the case of Mutual Ben- with positive measure from which a reversion to the eﬁt, or Executive Life, or Charter? While I admit that given regime cannot occur. the occasional P&C company can be destroyed by one Using the techniques of ergodic theory, one can tropical storm too many, the usual cause for a signif- sometimes derive probability statements about vari- icantly large company becoming insolvent is some ous kinds of phase-switching, the possibility of con- tinuing evolution or for getting stuck with positive *Irwin T. Vanderhoof, F.S.A., Ph.D., is Clinical Professor at the Stern probability or of doing one or the other ‘‘almost School of Business at New York University. His mailing address is surely.’’ 18 Two Bridges Road, Towaco, NJ 07082.
- 17. 22 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 3 kind of a change in regime, and traditional statistical Beneﬁt and most of the Japanese ﬁnancial system. methodology provides little guidance in providing Again, there seems to have been a change in the way margin for, or even recognizing, these changes in re- the asset was valued, and this change in regime led to gime—the nature of the current reality. the result. I am not so familiar with the details here. If this argument seems a little metaphysical, con- In all these cases more sophisticated statistical sider a speciﬁc example. The recent problems of the analysis of the past would have accomplished little. Monarch Life group were related to failure of sales of What was needed was an understanding of the possi- variable life. The public lost conﬁdence in the safety of ble sudden change in the way the stochastic game such a product after October 19, 1987 and October 13, would be played. Professor Day opens the ﬁrst under- 1989. According to Jackwerth and Rubinstein (1996), standings of how this change can take place. on the ﬁrst date the two-month S&P futures fell 29%. The paper itself builds a model of ﬁnancial markets ‘‘Under the lognormal hypothesis, this is a 27 stan- that assumes agents with different methodologies. dard deviation event with probability 10 160 . . .’’ On This is, of course, in stark contrast with modern ﬁ- October 13, 1989 the S&P fell 6%, a 5 standard de- nancial theory wherein all agents are assumed to have viation event. Aaron Tenenbein, of New York Univer- the same utility functions, information, and analytic sity, wrote an article for The Actuary in early 1988 technology. This may not bother you, but I ﬁnd it pointing out that the October 19 behavior of the mar- offensive. It may be that you are as intelligent as I am, ket was inconsistent with the assumption of lognor- but I resent the argument that everyone is. Perhaps mality of returns for the reasons given above. you are less sensitive. Most of modern ﬁnancial theory uses a normal or The paper presents a clear answer to the challenge lognormal distribution to describe returns. This be- presented by the highly respected Jose Sheinkman havior cannot be explained within this context, be- (1990). Sheinkman argues that nonlinear dynamics cause it constitutes a change in regime. had not then been shown capable of producing results We may be able to get some further ideas about that would mimic the actual movements of the mar- what is really happening from Longin’s paper (1996). ket. Professor Day has done exactly that in his illus- Extreme value theory is attractive because the ex- trations of ‘‘head and shoulders’’ conﬁgurations and treme returns over a period must fall into one of only so on. I have an anecdotal experience with head and three distributions: the Gumbel, the Frechet, or the shoulders formation. Many years ago I was trading Weibull. The conclusion here is that the extremes fol- stocks using technical analysis. At one point I found low the Frechet distribution and that the market can- a perfect head and shoulders in American Airlines. not be either a Gaussian or stable Paretian market, Several months later I was able to determine that dur- but something in between. It would seem to me that ing that period a large mutual fund had divested itself the changes in regime consistent with a complex sys- of its American Airlines’ shares. In this case the story tem would be a possible explanation of this result. We that technicians tell justifying the formation was true. could have changes in regime between a well-behaved Unfortunately, these same formations can be created Gaussian distribution and a jump process. Merton has by the interplay of market participants, as Professor developed mathematics for such a market. We could Day has shown. also have changes in regime created by a catastrophic For criticisms of the paper, I can cite only one se- change in parameters, such as the demand function. rious one—one I cannot suggest how to remedy. The This latter situation was the apparent cause of the author cites it also, that is, the failure to include any failure of the Executive Life group. In 1990 the de- effect of the random news events that also have an faults on junk bonds were far higher than they had impact on the market. The author has made a step been during the past 20 years. In addition, the Wash- forward. However, until we are also able to model the ington regulators were forcing the S&Ls to divest impact of chance events and recognize the possibility themselves of holdings of these same junk bonds. I of such an event precipitating a change in regime, we believe that, had it not been for this change in regime, still have much further to go. I hope we continue. the California department would eventually have been For fun, ask yourself, what traditional statistical able to straighten out affairs for Executive Life with- technique would have helped avoid any of the great out the insolvency. insolvencies of the last decade more than the simple A ﬁnal example was the disastrous change in valu- argument that more surplus is always better? ations of real estate that caused the failure of Mutual
- 18. COMPLEX DYNAMICS, MARKET MEDIATION AND STOCK PRICE BEHAVIOR 23 REFERENCES TENENBEIN, A. 1988. ‘‘After the Crash: Statistical Implications,’’ The Actuary 22, no. 2 (February): 1–3. JACKWERTH and RUBINSTEIN. 1996. ‘‘Recovering Probability Dis- tributions from Option Prices,’’ Journal of Finance (De- Additional discussions on this paper will be ac- cember):1611–31. LONGIN, F. M. 1996. ‘‘The Asymptotic Distribution of Extreme cepted until January 1, 1998. The author reserves the Stock Market Returns,’’ Journal of Business (July):383– right to reply to any discussion. See the Table of Con- 408. tents page for detailed instructions on the prepara- SHEINKMAN, J. 1990. ‘‘Nonlinearities in Economic Dynamics,’’ tion of discussions. Economic Journal (Conference?).

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