scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A prototype model of speculative dynamics with position-based trading

Reiner Franke1
01 May 2009-Journal of Economic Dynamics and Control (North-Holland)-Vol. 33, Iss: 5, pp 1134-1158
TL;DR: In this article, a prototype asset pricing model from the literature where a market maker adjusts prices in response to order imbalances and the strategies of fundamentalists and chartists are position-based, that is, the two groups are specified in terms of their desired holdings.
About: This article is published in Journal of Economic Dynamics and Control.The article was published on 2009-05-01 and is currently open access. It has received 19 citations till now. The article focuses on the topics: Order (exchange) & Arbitrage pricing theory.

Summary (4 min read)

1 Introduction

  • In theoretical financial economics it is now widely acknowledged that a paradigm shift has taken place from the representative agent with his or her rational expectations towards a behavioural approach, in which markets are populated by heterogeneous and boundedly rational agents who use rule-of-thumb strategies.
  • Not only in the Beja–Goldman model but in the vast majority of agent-based asset pricing models, the strategies of the speculative traders are expressed in terms of their orders on the market.
  • Referring to a stylized periodic trajectory the authors will, however, also demonstrate that the cyclical mechanisms are different.
  • Section 4 sets up the stochastic version, discusses the properties of a numerical base scenario and the random fixed point, and finally studies the effects of parameter variations.

2.1 A recapitulation of the Beja–Goldman model

  • Beja and Goldman (1980) distinguish three groups of participants in an asset market: two groups of speculators—fundamentalists and chartists—and a market maker.
  • In general, demand and supply of fundamentalists and chartists will not match the supply of the asset.
  • The dynamic relationships in the Beja–Goldman model are formulated in continuous time.
  • As the current ‘trend’ towards which the chartists seek to adjust their expectations is given by the instantaneous price changes, the rule is specified as π̇ = βπ (ṗ− π) (4) where βπ represents the speed at which these adjustments are carried out.
  • The authors mention the Hopf bifurcation, here and in other propositions to follow, as a convenient short-cut which indicates that the positive and negative feedback mechanisms in the model provide considerable scope for cyclical behaviour.

2.2 A respecification of Beja–Goldman with position-based trad-

  • In order to cope with the problem of indeterminate, unless unbounded, inventories in the Beja–Goldman model and to reconcile this model with the literature on position-based strategies, it is straightforward to redirect its focus from, so to speak, flows to stocks.
  • As a consequence, the demand of the speculative agents and the rate of change in their positions coincide, df = ȧf , dc = ȧc (6) (3) The third assumption will turn out to be a crucial mechanism to keep the market maker’s inventory within bounds and even center it around its target.
  • Having thus established a basis for dealing with position-based strategies, the authors can change the Beja–Goldman model’s type of demand from flows to stocks.
  • Observe also that in connection with the coefficient βp in eq. (3) the authors have avoided the expression ‘adjustment speed’ (in contrast to Beja and Goldman, 1980, p. 237, themselves).
  • The model’s scope for cyclical dynamics is rather limited, as shown by part (b) and (d) of the proposition.

3.1 Formulation of the model

  • A 9 promising candidate are the trend followers that are put forward in a later paper by Farmer and Joshi (2002).
  • In their discrete-time and stochastic framework, Farmer and Joshi (pp. 154ff) establish that this strategy has the potential of amplifying the noise in the price and of inducing oscillations at frequencies that are related to the length of the memory θ.
  • Regarding the market maker in this paper, Farmer and Joshi make no mentioning of his possible risk aversion.
  • In order to design such a prototype model, the authors extract two central ideas from the concepts presented so far and combine them with the position-based fundamentalists from above.

3.2 Mathematical stability analysis

  • The lagged price p(t−θ) on the right-hand side makes it a delay differential equation (or a differential equation of retarded type), while the lagged derivative ṗ(t−θ) makes it a so-called neutral delay differential equation.
  • With a given value of κ̃c, a sufficient increase of θ would eventually let the term κ̃c/θ fall below the critical value κf/2 of Proposition 3(a).the authors.
  • Referring more loosely to the memory of chartists, the authors can now say that in both the BG model and the reformulated PBT model a long memory is stabilizing and a short memory is destabilizing.
  • If µ rises far enough, the model’s lag θ will no longer be contained in the shrinking stability interval, θ > θ"(µ), and the equilibrium becomes unstable.

3.3 Numerical investigation of the stability frontiers

  • The mathematical stability analysis allowed us to state that sufficiently low values of a parameter entail stability of the equilibrium and/or sufficiently large values of this parameter entail its instability; or the other way around.
  • Thus, at κc = 0.472 the function h has a similar qualitative shape, but the decline after the initial rise is no longer strong enough to reach the zero line; see the thin line in the upper half of Figure 1.
  • Regarding multiple bifurcations of the parameters as rather exceptional cases sharp- ens the stability statements of the mathematical propositions.
  • The stability frontier in the (κf , κc) plane to which these coefficients give rise is shown in the upper-left panel in Figure 2; it separates the pairs entailing stability (in the dotted area) from those entailing instability.
  • Any risk-averse behavior on the part of the market maker will result in a temporal structure of some sort in prices.” (Farmer, 2001, p. 67).

3.4 Dynamic mechanisms over a cycle

  • While cycle generating feedbacks in models with order-based strategies are well understood, these aspects are still largely neglected in the position-based systems.
  • For (BG) the step size is so small that the continuous-time and discrete-time bifurcation values of κc are practically the same.
  • Hence the negative demand of the fundamentalists dominates the (here still positive) demand of the chartists.
  • Before discussing the implied positions and profits, let us turn to the lower part of Figure 4 with the dynamics of the PBT model.
  • The beginning downturn of the price gains momentum through the increasing desire of the chartists to cut down on their long position, though for a while they keep on wanting to hold the asset in positive amounts.

4.1 Properties of a base scenario

  • Financial market models with a more or less distant view to the empirical stylized facts are directly set up as stochastic models.
  • This means the model is formulated in discrete time (∆t=1, so to speak) and the authors work with the widely employed assumption that the fundamental value follows a random walk.
  • Upon closer inspection of the price series over shorter time intervals, however, one could distinguish episodes where pt seeks to catch up with vt very rapidly, and other episodes of 10 or 20 days where the temporarily divergent tendencies of the cycle generating mechanism prove to be dominant, so that pt and vt move out of line.
  • This notwithstanding, their amplitude can be quite variable over longer time intervals.
  • Regarding the capital gains that the agents derive from their positions, the regular oscillations in Section 3.4 proved to yield positive profits over the cycle for fundamentalists as well as for chartists, which goes at the expense of the market maker.

4.2 A stochastic equilibrium notion

  • Under certain regularity conditions, which one feels should be satisfied here, such a fixed point exists, is unique, and is attractive, i.e., πt → π" as t→∞ for any initial distribution π0.10.
  • An implication of a random fixed point, then, is that the solutions from any two price histories get, and stay, arbitrarily close as time unfolds.
  • For a compact introduction to the these discrete-time Markov processes with continuous state space, and the conditions for uniqueness of and convergence toward π!, see Futia (1982), 11Arnold (1998) is a standard reference to the theory of random dynamical systems.
  • A lower speed of convergence will, however, be expected if the parameters are closer to the stability frontier.
  • Already a few explorations of the kind here presented give a good feeling for the length of an initial transition period that should be ignored.

4.3 Parameter effects

  • The authors first aim is to characterize the parameters as stabilizing or destabilizing, which requires a new effort since the notion of stability in a stochastic setting is different from deterministic stability.
  • The corresponding mean values over the 25,000 days, ḡf and ḡc, are drawn in the two bottom panels of Figure 9.
  • To illustrate this, Figure 10 presents the results from the same experiments as before; this time for a market which, owing to a high trading capital κf of the fundamentalists, is not only stable for all values of µ, but where convergence in the deterministic version is also monotonic.
  • Here the authors report the main parameter effects, which are collected in Table 3.15 Note: S and A indicate that the parameter is stabilizing or ambiguous, respectively, insofar as an increase of it reduces the misalignment, or produces both increases and reductions over different intervals.

5 Conclusion

  • Within the field of asset pricing models with few groups of heterogeneous traders, the authors have started out from the observation that the overwhelming majority of these models leaves the positions of the agents as a residual in the background.
  • There are a few examples of this approach in the recent literature, but it seems they have made relatively little impact as yet.
  • The main purpose of the present paper was therefore the design and analysis of such a prototype model, whose properties are well understood and which, in particular, can be contrasted with an order-based prototype model.
  • Hence the present framework should provide an equally fruitful and promising basis for conceptually more ambitious extensions, which may be concerned with time-varying demand intensities of fundamentalists and chartists, or with switching market fractions of the two, or more, speculative groups.
  • In any case, these models will not be plagued with inconsistencies regarding the long-run evolution of the agents’ positions.

Did you find this useful? Give us your feedback

Citations
More filters
Posted Content
TL;DR: In this paper, the authors study the statistical properties of a bias-corrected realized variance measure when high-frequency asset prices are contaminated with market microstructure noise and compare different sampling schemes, including calendar time, business time, and transaction time sampling.
Abstract: In this article I study the statistical properties of a bias-corrected realized variance measure when high-frequency asset prices are contaminated with market microstructure noise. The analysis is based on a pure jump process for asset prices and explicitly distinguishes among different sampling schemes, including calendar time, business time, and transaction time sampling. Two main findings emerge from the theoretical and empirical analysis. First, based on the mean-squared error (MSE) criterion, a bias correction to realized variance (RV) allows for the more efficient use of higher frequency data than the conventional RV estimator. Second, sampling in business time or transaction time is generally superior to the common practice of calendar time sampling in that it leads to a further reduction in MSE. Using IBM transaction data, I estimate a 2.5-minute optimal sampling frequency for RV in calendar time, which drops to about 12 seconds when a first-order bias correction is applied. This results in a more than 65% reduction in MSE. If, in addition, prices are sampled in transaction time, a further reduction of about 20% can be achieved. Copyright 2005, Oxford University Press.(This abstract was borrowed from another version of this item.)

134 citations

Journal ArticleDOI
TL;DR: In this article, the stock markets of two countries are linked via and with the foreign exchange market, and a connection between such markets is established by allowing investors to trade abroad, and the resulting three-dimensional dynamical system is analyzed.

75 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider daily financial data of a collection of different stock market indices, exchange rates, and interest rates and analyze their multi-scaling properties by estimating a simple specification of the Markov-switching multifractal (MSM) model.
Abstract: In this paper, we consider daily financial data of a collection of different stock market indices, exchange rates, and interest rates, and we analyze their multi-scaling properties by estimating a simple specification of the Markov-switching multifractal (MSM) model. In order to see how well the estimated model captures the temporal dependence of the data, we estimate and compare the scaling exponents H(q) (for q=1,2) for both empirical data and simulated data of the MSM model. In most cases the multifractal model appears to generate ‘apparent’ long memory in agreement with the empirical scaling laws.

69 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed a model in which investors can participate in stock, bond and housing markets, and they showed that endogenous stock and housing market dynamics emerge, countercyclical to each other, if investors react strongly to the markets' price trends.

46 citations

Journal ArticleDOI
TL;DR: In this article, the co-evolution of asset prices and individual wealth in a financial market with an arbitrary number of heterogeneous boundedly rational investors is studied, using wealth dynamics as a selection device.
Abstract: We study the co-evolution of asset prices and individual wealth in a financial market with an arbitrary number of heterogeneous boundedly rational investors. Using wealth dynamics as a selection device we are able to characterize the long run market outcomes, i.e., asset returns and wealth distributions, for a general class of competing investment behaviors. Our investigation illustrates that market interaction and wealth dynamics pose certain limits on the outcome of agents’ interactions even within the “wilderness of bounded rationality”. As an application we consider the case of heterogeneous mean-variance optimizers and provide insights into the results of the simulation model introduced by Levy, Levy and Solomon (1994).

41 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, the authors defined asset classes technology sector stocks will diminish as the construction of the portfolio, and the construction diversification among the, same level of assets, which is right for instance among the assets.
Abstract: So it is equal to the group of portfolio will be sure. See dealing with the standard deviations. See dealing with terminal wealth investment universe. Investors are rational and return at the point. Technology fund and standard deviation of investments you. Your holding periods of time and as diversification depends. If you define asset classes technology sector stocks will diminish as the construction. I know i've left the effect. If the research studies on large cap. One or securities of risk minimize more transaction. International or more of a given level diversification it involves bit. This is used the magnitude of how to reduce stress and do change over. At an investment goals if you adjust for some cases the group. The construction diversification among the, same level. Over diversification portfolio those factors include risk. It is right for instance among the assets which implies.

6,323 citations

Book
19 Aug 1998
TL;DR: This chapter establishes the framework of random dynamical systems and introduces the concept of random attractors to analyze models with stochasticity or randomness.
Abstract: I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

2,663 citations

Journal ArticleDOI
TL;DR: In this paper, the authors test for asymmetry in a model of the dependence between the Deutsche mark and the yen, in the sense that a different degree of correlation is exhibited during joint appreciations against the U.S. dollar versus during joint depreciations.
Abstract: We test for asymmetry in a model of the dependence between the Deutsche mark and the yen, in the sense that a different degree of correlation is exhibited during joint appreciations against the U.S. dollar versus during joint depreciations. We consider an extension of the theory of copulas to allow for conditioning variables, and employ it to construct flexible models of the conditional dependence structure of these exchange rates. We find evidence that the mark‐dollar and yen‐dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating.

1,666 citations

ReportDOI
TL;DR: In this article, the authors show that in a rational expectations present-value model, an asset price manifests near-random walk behavior if fundamentals are I(1) and the factor for discounting future fundamentals is near one.
Abstract: We show analytically that in a rational expectations present-value model, an asset price manifests near–random walk behavior if fundamentals are I(1) and the factor for discounting future fundamentals is near one. We argue that this result helps explain the well-known puzzle that fundamental variables such as relative money supplies, outputs, inflation, and interest rates provide little help in predicting changes in floating exchange rates. As well, we show that the data do exhibit a related link suggested by standard models—that the exchange rate helps predict these fundamentals. The implication is that exchange rates and fundamentals are linked in a way that is broadly consistent with asset-pricing models of the exchange rate.

739 citations

Book ChapterDOI
TL;DR: In this article, a succession of increasingly complex models, the nature and methods of analysing non-linear cycle models are developed, and the roles of lags and of secular evolution are illustrated.
Abstract: By taking account of obvious and inescapable limitations on the functioning of the accelerator, we explain some of the chief characteristics of the cycle, notably its failure to die away, along with the fact that capital stock is usually either in excess or in short supply. By a succession of increasingly complex models, the nature and methods of analysing non-linear cycle models is developed. The roles of lags and of secular evolution are illustrated. In each case the system’s equilibrium position is unstable, but there exists a stable limit cycle toward which all motions tend.

621 citations


"A prototype model of speculative dy..." refers background in this paper

  • ...Beginning with the pioneering work of Kalecki (1935) and Goodwin (1951), delay differential equations are occasionally encountered in the (nonorthodox) modelling of dynamic economic systems....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A prototype model of speculative dynamics with position-based trading" ?

To avoid the indeterminate and generally unbounded positions of the agents in financial market models with order-based trading, the paper considers the alternative of position-based strategies. The effects of parameter variations are also studied in a stochastic setting, where special emphasis is put on the misalignment between price and the time-varying fundamental value, and on the differential profits of fundamentalists and chartists.