مدلسازی فشار قیمت در بازارهای مالی

We present experimental evidence that, unlike traditional assumptions in economic theory, security prices do not respond to pressure from their own excess demand. Instead, prices respond to excess demand of all securities, despite the absence of a direct link between markets. We propose a model of price pressure that explains these findings. In our model, agents set order prices that reflect the marginal valuation of desired future holdings, called “aspiration levels.” In the short run, as agents encounter difficulties executing their orders, they scale back their aspiration levels. Marginal valuations, order prices, and hence, transaction prices change correspondingly. The resulting price adjustment process coincides with the Global Newton Method. The assumptions of the model as well as its empirical implications are fully borne out by the data. Our model thus provides an economic foundation for why markets appear to search for equilibrium according to Newton’s procedure.

Economists have generally focused on the equilibrium implications of their models, leaving little time to consider how markets attain equilibrium. This focus is motivated by the claim that prices “move in accord with the excess demand (demand minus supply) in each market” (Negishi, 1962, 638). If excess demand is positive (there is more demand than supply), prices tend to increase. Conversely, if excess demand is negative (supply outstrips demand), then prices tend to decrease. As a result, price adjustment only stops at the point where excess demand equals zero, the equilibrium. The above process is what Walras first developed in his Élements d’Économie Politique Pure ( 1874) and what has subsequently remained one of the most studied price-adjustment processes. 2 As Gode and Sunder (1993, 120) proclaim, “Standard economic theory is built on two specific assumptions: utility-maximizing behavior and the institution of Walrasian tatonnement.” The Walrasian tatonnement theory builds on the intuitive premise that prices react to the demand in their own market only. Since the demand for a given asset already incorporates the substitution and complementarity effects between this and the other traded assets, there is no compelling reason why prices should react to anything but own excess demand. Unfortunately, if it is true that prices adjust only in the direction of own excess demand, the adjustment process may not converge. It is easy to construct counterexamples (see, e.g., Scarf, 1960). The counterexamples exploit the fact that no general shape restrictions exist for excess demand as a function of prices (this fundamental result is known as the Debreu–Mantel–Sonnenschein Theorem). Thus, for price adjustment to be generically converging and, hence, for equilibrium to be the natural state towards which markets tend, it better be that prices adjust to something else besides own excess demand. Evidence is presented here that markets do adjust differently. We study the outcomes in financial markets experiments where up to 70 (human) subjects traded four (three risky and one risk-free) securities for real money. Prices of none of the risky securities correlate significantly to their own excess demand, contrary to the tatonnement theory. The lack of correlation is caused by the presence of excess demand in other securities. Evidently, prices in one market react to excess demand of all markets, not only their own market, even if there is no direct link between markets. 3, 4 Such cross-security effects can arise within the well known Global Newton Method for the numerical computation of general equilibrium (see Arrow and Hahn, 1971, and Smale, 1976a). However, unlike the Walrasian tatonnement that is constructed to mimic the “invisible hand,” the Newton procedure lacks any economic intuition, and there is no reason to believe that the outcome of real trading will coincide with such an adjustment process.5 The main goal of this paper is to develop and further test a behavioral theory that explains the cross-security effects and as such provides an economic foundation for the price adjustment based on Newton’s method. We model excess demand assuming that individuals want to trade off expected return against variance. That is, excess demands are computed as in the capital asset pricing model (CAPM). This choice is justified because the CAPM explains eventual pricing in the experiments as well as end-of-period portfolio holdings. 6 Our model postulates that, in the short run, agents attempt to trade towards aspiration levels. Although not necessary, we take the aspiration levels to be the optimal positions at last transaction prices: aspiration levels equal current positions plus excess demands. These are also the aspiration levels in the classical Walrasian tatonnement, but the subsequent price adjustment in the Walrasian model is mechanical and fictitious: prices are assumed to change in proportion to excess demand. In contrast, our model spells out how agents would react when their orders (which are based on their aspiration levels) fail to be executed. Specifically, we conjecture that agents scale back their aspiration levels towards their current holdings. Marginal valuations are updated correspondingly (i.e., prices at which agents are willing to trade are revised). 7 Mean order prices and as a result prices at which subsequent transactions are likely to occur, change. Mathematically, the set of difference equations resulting from the aggregation of the individual reactions to unfilled excess demands coincides with the set of equations of the Newton’s numerical procedure. Thus, our theory provides an explanation for why it appears that real markets use the Newton procedure in their search for equilibrium.8 More specifically, price pressure in our model is driven by local changes in marginal valuations, which in turn are dictated by the Hessian of agents’ utility functions. In the case of mean-variance preferences, the Hessian is proportional to the covariance matrix of the final payoffs. When covariances are nonzero, not only does our model therefore predict the presence of cross-security effects, but also that these cross-security effects are related to the sign and even the magnitude of these covariances. In our subsequent empirical analysis we first test whether the main assumptions of our model about individual behavior hold in the data.9 We find significantly positive correlations (equal to 40 percent on average) between individual cancelation rates in different markets, thus obtaining support for the assumption that agents proportionally scale back their unfilled orders. The model further assumes that the more risk averse individuals scale back less than the agents with higher risk tolerance. We find that there is indeed (albeit weak) positive correlation between individual risk tolerance and the rate of order cancelations in the experimental markets. The empirical predictions of the model are fully borne out by the experimental data: there is a systematic relationship between, on the one hand, the cross-security effects and, on the other hand, the covariances of the final payoffs of the securities. In particular, if two securities have negatively correlated payoffs, then their prices tend to be negatively correlated with each other’s excess demands (vice versa if the correlation is positive); moreover, the magnitude of the cross-security effects is related to that of the payoff covariances. We recognize that the scope of the model is limited. It deals only with the mechanics of the direction in which prices change given unattainable aspiration levels. That is, ours is not a model of equilibration, but it could be embedded in a model of equilibration. One possibility is the following. As aspiration levels are scaled back and marginal valuations change, the average order prices change as well, to the point that agents may decide to cancel their orders altogether and re-submit new orders that reflect their excess demands at these revised average order prices. We will not explore the implications, but in the conclusion, we speculate to what extent this extension of our model would guarantee stability. Likewise, our model takes aspiration levels to be globally optimal demands given last transaction prices. One could define aspiration levels differently. For instance, in the models of Bossaerts (2008), Ledyard (1974), Smale (1976b), aspiration levels are current allocations plus changes that are locally optimal given previous transaction prices. In the context of mean-variance preferences, however, the empirical implications (in particular, the link between cross-security effects and payoff covariances) can be shown to be the same qualitatively. Our experimental findings are related to the recent field findings of cross-security effects. For the Tel Aviv stock exchange, Kalay and Wohl (2009) show that the relative slopes of the demand and supply sides of the book in one security could be used to predict subsequent price changes in other securities. Although our cross-security effects are about the relation of price changes to Walrasian excess demand, earlier work (Asparouhova et al., 2003) has revealed that there is correlation between relative slopes of the demand and supply sides of the book and Walrasian excess demand. The findings in our experiments as well as our theory are different from the studies of multi-security asset pricing under heterogeneous information such as Admati (1985) (theory) and Biais et al. (forthcoming) (theory and empirics). These studies concern equilibrium. While they do predict that signals and noise in one market affect pricing in other markets, these are equilibrium effects. The cross-security correlation between excess demand and price changes in our experiments is an off-equilibrium phenomenon: it clearly occurs before markets reach their equilibrium. The remainder of this paper is organized as follows. The next section describes the experiments. In Section 3, the excess demands are derived within the CAPM. Subsequently, we present empirical evidence of the extent to which prices in our experiments fail to change in the direction of excess demand because of cross-security effects. In Section 5 we develop a theory of price pressure that explains the observed cross-security effects. Key model assumptions along with further implications about the signs and magnitudes of the cross-security effects are verified in Section 6. Section 7 concludes.

Data from large-scale market experiments with four securities reject the simple price adjustment story in the Walrasian model because of significant cross-security effects: price changes correlate not only with own excess demand but with excess demands of other securities as well. This extends the findings of Asparouhova et al. (2003) and Bossaerts (2008). In this paper, we study a model of price pressure that enriches the basic Walrasian model, replacing its mechanical price adjustment rule with a model of price changes that better reflects the realities of competitive, decentralized markets. The agents in our model in the short run scale back their aspiration points in response to delays in execution and change order prices accordingly to reflect corresponding changes in their marginal valuations. Our model of price pressure implies the very cross-security effects present in the data. The resulting price adjustment process coincides with that of the Global Newton Method. Consequently, the model predicts the sign and relative magnitude of the cross-effects. Basically, as agents scale back their aspiration points, their marginal valuations change. The Hessian of the utility functions dictates how marginal valuations change. In the context of the mean-variance preferences we use here, the Hessian is proportional to the covariance matrix of final payoffs. This means that covariances provide the natural linkage between marginal valuation changes in one security and adjustments of desired quantities in another. Since changes in marginal valuations are revealed in changes in order prices, the pattern of covariances in payoffs show up in the way prices drift as a response to excess demands. The experimental data confirm the hypothesized link between cross-security effects and the structure of the covariance matrix. Although our model fits the data well, we leave many questions unanswered. Foremost, ours is a model of local price pressure, not of equilibration. It is meant simply as a more compelling and empirically relevant story of changes in prices given excess demands than the mechanistic adjustment in the original Walrasian model. Still, it could be embedded in the standard Walrasian model, replacing the Walrasian auctioneer, thus creating a model of equilibration. Its stability properties may be very different from those of the standard Walrasian model because the link between excess demands and price changes is provided by the Hessian of the utility function. The latter conveys crucial information about derivatives of the excess demand function. As a consequence, price adjustment in our model reflects the very information that Saari and Simon (1978) prove to be needed for generic stability of equilibration mechanisms and crucial for the link between our adjustment process and the Global Newton Method. In other words, replacing the standard, mechanistic price adjustment rule with our model of price pressure in the Walrasian equilibration model may generate the very stability that is needed to claim persuasively that general equilibrium is the natural state to which competitive markets tend. We leave this conjecture for future investigation. While we do stress the link between the Newton Method and our price adjustment process, we refrain from generalizing our model to non-additively separable utility functions because we would lose the interpretation of our assumptions in terms of risk attitudes. We do consider the issue of whether there is a general economic foundation for Newton’s method that holds for a large class of utility functions to be an excellent topic for future research. In our model, we take aspiration points (desired portfolio holdings) to be globally optimal positions given past transaction prices. Alternatives can be imagined, such as aspiration points based on locally optimal movements, see, for example, Bossaerts (2008), Bossaerts et al. (2003), Ledyard (1974), Smale (1976b). In these papers, orders are proportional to locally optimal excess demands, but as in the Walrasian model, price changes are mechanical: prices change in the direction of the net order flow. If we were to embed our model of price pressure into a model with aspiration points based on locally optimal movements, we would generate a complete model of price adjustment. Preliminary investigation of the implications of such an approach demonstrates, however, that the empirical implications of a model based on locally optimal aspiration points makes qualitatively similar predictions as one based on globally optimal aspiration points. This is because locally optimal movements are proportional to globally optimal movements, at least in the context of mean-variance preferences. More general preferences need to be contemplated in order to generate discriminatory power. We are working on such extensions at present.