Asset Pricing Using Finite State Markov Chain Stochastic Discount Functions

This article fuses two pieces of theory to make a tractable model for asset pricing. The first is the theory of asset pricing using a stochastic discounting function (SDF). This will be reviewed. The second is to model uncertainty in an economy using a Markov chain. Using the semi-martingale dynamics for the chain these models can be calibrated and asset valuations derived. Interest rate models, stock price models, futures pricing, exchange rates can all be introduced endogenously in this framework.     

Probabilistic stit logic and its decomposition

We define an extension of stit logic that encompasses subjective probabilities representing beliefs about simultaneous choice exertion of other agents. This semantics enables us to express that an agent sees to it that a condition obtains under a minimal chance of success. We first define the fragment of XSTIT where choice exertion is not collective. Then we add lower bounds for the probability of effects to the stit syntax, and define the semantics of the newly formed stit operator in terms of subjective probabilities concerning choice exertion of other agents. We show how the resulting probabilistic stit logic faithfully generalizes the non-probabilistic XSTIT fragment. In a second step we analyze the defined probabilistic stit logic by decomposing it into an XSTIT fragment and a purely epistemic fragment. The resulting epistemic logic for grades of believes is a weak modal logic with a neighborhood semantics combining probabilistic and modal logic theory.     

Time inconsistency and learning in bargaining games

This paper studies an alternating-offers bargaining game between possibly time-inconsistent players. The time inconsistency is modeled by quasi-hyperbolic discounting and the “naive backwards induction” solution concept is used in order to obtain the results. Both naive agents who remain naive and those who learn about their own preferences are considered. Offers of the players who are naive or partially naive are never accepted by any type of player in either no learning or gradual learning cases. The game between a naive or partially naive player who never learns and a time-consistent agent ends in an immediate agreement if the time-consistent agent is the proposer. A one period delay occurs if the time-consistent agent is the responder. The more naive the player is, the higher the share received. In addition, two naive agents who never learn disagree perpetually. When naive and partially naive agents play against exponential or sophisticated agents and they are able to learn their types over time, there exists a critical date before which there is no agreement. Therefore, the existence of time-inconsistent players who can learn their types as they play the game can be a new explanation for delays in bargaining. The relationship among the degree of naivete, impatience level and bargaining delay is also characterized. Specifically, for sufficiently high discount factors, agreement is always delayed. On the other hand, if the naive agent has sufficiently firm initial beliefs (slow learning or high degree of naivete), agents agree immediately.     

The Dynamic Interaction of Speculation and Diversification

A discrete time model of a financial market is developed, in which heterogeneous interacting groups of agents allocate their wealth between two risky assets and a riskless asset. In each period each group formulates its demand for the risky assets and the risk‐free asset according to myopic mean‐variance maximizazion. The market consists of two types of agents: fundamentalists, who hold an estimate of the fundamental values of the risky assets and whose demand for each asset is a function of the deviation of the current price from the fundamental, and chartists, a group basing their trading decisions on an analysis of past returns. The time evolution of the prices is modelled by assuming the existence of a market maker, who sets excess demand of each asset to zero at the end of each trading period by taking an offsetting long or short position, and who announces the next period prices as functions of the excess demand for each asset and with a view to long‐run market stability. The model is reduced to a seven‐dimensional nonlinear discrete‐time dynamical system, that describes the time evolution of prices and agents' beliefs about expected returns, variances and correlation. The unique steady state of the model is determined and the local asymptotic stability of the equilibrium is analysed, as a function of the key parameters that characterize agents' behaviour. In particular it is shown that when chartists update their expectations sufficiently fast, then the stability of the equilibrium is lost through a supercritical Neimark–Hopf bifurcation, and self‐sustained price fluctuations along an attracting limit cycle appear in one or both markets. Global analysis is also performed, by using numerical techniques, in order to understand the role played by the chartists' behaviour in the transition to a regime characterized by irregular oscillatory motion and coexistence of attractors. It is also shown how changes occurring in one market may affect the price dynamics of the alternative risky asset, as a consequence of the dynamic updating of agents' portfolios.     

Bargaining in endogenous trading networks

We study a model in which heterogeneous agents first form a trading network where linking costs are positive but infinitesimally small. Then, a seller and a buyer are randomly selected among the agents to bargain through a chain of intermediaries. We determine both the trading path and the allocation of the surplus among the seller, the buyer and the intermediaries at equilibrium. We show that, under the initiator bargaining protocol, a trading network is pairwise stable if it is a core–periphery network where the core consists of all impatient agents who are linked to each other and the periphery consists of all patient agents who have a single link towards an impatient agent. Once agents do not know the impatience of other agents, each bilateral bargaining session may involve delay. Then, core–periphery networks may not be pairwise stable because agents may prefer to add links for reducing the length of trading paths and so avoiding costly delays in reaching a global agreement.     

二阶耗散动力系统的降维对解长期行为的误差估计

基于非线性动力学理论,对一类高维二阶耗散自治动力系统的降维及其对解的长期行为的影响进行了理论分析．该分析将方程的解投影到控制方程的线性算子的特征向量所张成的完备空间中,并在相空间中引入一距离的概念,方便地解决了缩减后系统与原始系统解之间的误差或距离的描述．基于此距离定义,首先,分析了由于高阶模态的截取对解的长期行为的影响,并推导出了相应的误差估计,该估计表明由于降维对系统长期行为的影响不仅与系统的高阶子空间中的固有频率和阻尼比乘积的最小值有关,并且与高阶子空间中的某一最大固有频率有关．然后,将一般的模态截取视为对原系统的解的一个扰动,对一些文献中由于降维程度的不同而造成解的拓扑性质发生变化的现象进行了定性的解释．     

NEGOTIATING OVER EXTERNALITIES

We present a simple model in which two perfectly informed, risk neutral agents will not negotiate an efficient agreement to lessen the effects of an externality and for which the outcome of negotiation depends on the legal assignment of property rights. The model permits agents to pre-commit themselves to refuse to negotiate particular agenda issues. The result is obtained because we prove that one player is always made strictly worse off from the addition of side-payments to a bargaining game. Along the way, we devise a supporting hyperplane for the n-person Nash bargaining game solution. We also display a simple game which establishes that our main result holds true—for at least some games—for an array of alternative bargaining game solutions such as that of Raiffa, Kalai, and Smorodinski.     

Complex dynamics in equilibrium asset pricing models with boundedly rational,heterogeneous agents

We study a simple model based upon the Lucas framework where heterogeneous agents behave rationally in a fully intertemporal setting but do not know other investors' personal preferences, wealth or investment portfolios. As a consequence, agents initially do not know the equilibrium asset pricing function and must make guesses, which they update via adaptive learning with constant gain. We demonstrate that even in this simple environment the economy can, depending on parameters, exhibit either stable convergence to equilibrium, or chaotic dynamical behavior of asset prices and trading volume without converging to the rational expectations equilibrium of the Lucas model. This contradicts the assertion that the Lucas model is stable in the face of modest deviations from the strong assumptions required to compute the equilibrium. © 2013 Wiley Periodicals, Inc. Complexity 19: 38–55, 2014     

Global exponential stability of almost periodic solution for a large class of delayed dynamical systems

Research on delayed neural networks with variable self-inhibitions, interconnection weights, and inputs is an important issue. In this paper, we discuss a large class of delayed dynamical systems with almost periodic self-inhibitions, inter-connection weights, and inputs. This model is universal and includes delayed systems with time-varying delays, distributed delays as well as combination of both. We prove that under some mild conditions, the system has a unique almost periodic solution, which is globally exponentially stable. We propose a new approach, which is independent of existing theory concerning with existence of almost periodic solution for dynamical systems.     

Delay in the alternating-offers model of bargaining

Alternating offers bargaining has been extensively used to model two-sided negotiations. The celebrated model of Rubinstein [Econometrica 50(1):97–109, 1982] has provided a formal justification for equitable payoff division. A typical assumption of these models under risk is that the termination event means a complete and irrevocable breakdown in negotiations. In this paper, the meaning of termination is reinterpreted as the imposition to finish negotiations immediately. Specifically, bargaining terminates when the last offer becomes definitive. While Rubinstein’s model predicts an immediate agreement with stationary strategies, we show that the same payoff allocation is attainable under non-stationary strategies. Moreover, the payoffs in delayed equilibria are potentially better for the proposer than those in which agreement is immediately reached.     

Productive public expenditures, expectation formations and nonlinear dynamics

In this paper, we investigate the dynamic properties of an overlapping generations’ model with capital accumulation and publicly funded inventions under three different expectations: perfect foresight, myopic expectations and adaptive expectations. We show that considering productive public expenditures in the model will increase the dimension of the dynamical system. To study the dynamic behavior of a high-dimensional dynamical system, we focus on the case when the elasticity of publicly funded invention to output is small and approximate the system by using a one-dimensional dynamical system. This approximation method provides an efficient way to rigorously prove the existence of chaos in high-dimensional dynamical systems. We show that when agents are perfectly foresighted, there exists a unique, nontrivial steady state which is a global attractor. Cycles or even chaos may occur under myopic and adaptive expectations when the inter-temporal elasticity of substitution of consumption is large enough. Furthermore, we find that the impact of fiscal policy is sensible to the expectation formation.     

Optimal investment under partial information

We consider the problem of maximizing terminal utility in a model where asset prices are driven by Wiener processes, but where the various rates of returns are allowed to be arbitrary semimartingales. The only information available to the investor is the one generated by the asset prices and, in particular, the return processes cannot be observed directly. This leads to an optimal control problem under partial information and for the cases of power, log, and exponential utility we manage to provide a surprisingly explicit representation of the optimal terminal wealth as well as of the optimal portfolio strategy. This is done without any assumptions about the dynamical structure of the return processes. We also show how various explicit results in the existing literature are derived as special cases of the general theory.     

Agreeing to disagree with lexicographic prior beliefs

The robustness of Aumann’s seminal agreement theorem with respect to the common prior assumption is considered. More precisely, we show by means of an example that two Bayesian agents with almost identical prior beliefs can agree to completely disagree on their posterior beliefs. Besides, a more detailed agent model is introduced where posterior beliefs are formed on the basis of lexicographic prior beliefs. We then generalize Aumann’s agreement theorem to lexicographic prior beliefs and show that only a slight perturbation of the common lexicographic prior assumption at some–even arbitrarily deep–level is already compatible with common knowledge of completely opposed posterior beliefs. Hence, agents can actually agree to disagree even if there is only a slight deviation from the common prior assumption.     

Controlled random fields,von Neumann–Gale dynamics and multimarket hedging with risk

We develop a model of asset pricing and hedging for interconnected financial markets with frictions – transaction costs and portfolio constraints. The model is based on a control theory for random fields on a directed graph. Market dynamics are described by using von Neumann–Gale dynamical systems first considered in connection with the modelling of economic growth [13,24]. The main results are hedging criteria stated in terms of risk-acceptable portfolios and consistent price systems, extending the classical superreplication criteria formulated in terms of equivalent martingale measures.     

Formulation,Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems

In this paper, we present a unified treatment and analysis of a dynamic traffic network model with elastic demands formulated and studied as a projected dynamical system. We propose a travel route choice adjustment process that satisfies the projected dynamical system. Under certain conditions, stability and asymptotical stability of the equilibrium patterns are then derived. Finally, two discrete-time algorithms, the Euler method and the Heun method, are proposed for the computation of the solutions, and convergence results established. The convergence results depend crucially on stability analysis. The performance of the algorithms is then illustrated on several transportation networks.     

Bargaining and negative externalities

Two important issues in distributive bargaining theory are, first, the conditions under which a negotiation breakdown occurs, and second, what and how source of parties’ bargaining powers influences the properties of a possible agreement. Research based on classic Nash’s demand game has explored both questions by sophisticating the original game a lot. As an attempt to deal with both issues under a simpler framework, we propose a modification of the Nash demand game in which bargainers suffer negative externalities proportional to the share of the surplus captured by their rival. It is shown that the negotiator experiencing a relatively high externality level has greater bargaining power and thus, appropriates a larger proportion of the surplus at stake. However, if externality levels are sufficiently high, bargaining powers become incompatible and a negotiation breakdown emerges from the bargaining process. We compare our results with the previous literature, and argue that they can be especially relevant in negotiations held under highly polarized environments.     

Sentiment lost: the effect of projecting the pricing kernel onto a smaller filtration set

Abstract

This paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward-looking estimate, extracting the real world measure from historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. As a consequence of this disalignment, the two measures no longer share the same nullset, thus distorting the investors’ risk premium and the validity of the pricing measure. From a probabilistic viewpoint, the missing beliefs are totally unaccessible stopping times on the coarser filtration set, so that an absolutely continuous strict local martingale, once projected on it, becomes continuous with jumps. Some empirical examples complete the paper.     

Endogenous reference points in bargaining

We allow the reference point in (cooperative) bargaining problems with a reference point to be endogenously determined. Two loss averse agents simultaneously and strategically choose their reference points, taking into consideration that with a certain probability they will not be able to reach an agreement and will receive their disagreement point outcomes, whereas with the remaining probability an arbitrator will distribute the resource by using (an extended) Gupta–Livne bargaining solution (Gupta and Livne in Manag Sci 34:1303–1314, 1988). The model delivers intuitive equilibrium comparative statics on the breakdown probability, the loss aversion coefficients, and the disagreement point outcomes.     

The Dynamics of a Prey-dependent Consumption Model Concerning Integrated Pest Management

A mathematical model for the dynamics of a prey-dependent consumption model concerning integrated pest management is proposed and analyzed. We show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than some critical values. Furthermore, the conditions for the permanence of the system are given. By using bifurcation theory, we show the existence of a nontrival periodic solution if the pest-eradication periodic solution loses its stability. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamics, which implies that dynamical behaviors of prey-dependent consumption concerning integrated pest management are very complex, including period-doubling cascades, chaotic bands with periodic windows, crises, symmetry-breaking bifurcations and supertransients.