A brief summary of my doctoral thesis

The title of the thesis is System Dynamics Modeling of Stylized Features of Stock Markets. The thesis summary follows:

Stock market is an essential element of the financial sector of an economy. The growth of the stock market is closely related to the economic development. It also provides liquidity for the investors. It is of extreme importance for financial policy makers and individuals to understand the behavior of the market. The dynamics of a stock market is complex. Many models have been proposed over the years. These models may be classified as: “time series models” that study the past finance data and build a model to represent the empirical properties, “market models” that average the effects of individuals and concentrate only on the overall behavior, which are used in pricing, portfolio optimization, etc., and “agent-based models” that consider traders as agents and incorporate their interactions in explaining the market behavior.

“Behavioral finance” tries to understand the market behavior based on bounded rationality of traders, their singular decision strategies, etc. It also considers the individual characteristics while making decisions with partial information. Agent-based market models allow us to consider different types of individuals and investigate the emerging behavior of the market. A number of simulated artificial markets have been built using these models. Santa Fe Artificial Stock Market is one of them. It is based on complex decision rules, and trading mechanism used is computationally very intensive. Another agent-based market model, recently proposed, is Minority Game (MG) with simple rules for the agents. MG is based on El-Farol Bar problem where modeling a competition is involved.

In this thesis we attempt to model some stylized features of stock markets in the framework of a dynamical system. The features that are addressed in this thesis include “herding”, “limits to arbitrage” and “confidence bias”. A system like stock market besides having complex interactions among its agents also exhibits dynamic behavior. System dynamics approach has been successively used to model various complex phenomena in engineering and social sciences. This thesis attempts to model the stock market in the framework of system dynamics that allows us to address the stylized features of the stock market incorporating MG rules.

In the proposed system dynamics framework, the herding behavior in stock markets is interpreted as a manifestation of reduced order dynamics. It is also shown how this interpretation helps us to understand the effects of herding. There is a risk when the market price variation due to herding is thought of as entirely due to the portfolio fundamentals. A new risk measure is considered and is related to the dynamics associated with the decision-making by individual traders, and their impact on the market. Simulations have been carried out in this framework to study the features of herding and its associated risk. Though the transition from no herding to herding regimes can be controlled in the simulation, it is nearly impossible to notice such transition in reality. Simulation studies show that phase transitions in econophysics models that occur before herding is captured as reduced order dynamics in the system framework. Herding event should cease after some period. It is also shown that the risk associated with the herding has an impact on the sustenance of it. Simulation has been able to capture this important element, which is intuitively clear that the herding phenomenon cannot go on forever.

A generic dynamical system model that captures some aspects of the limits to arbitrage has been proposed. Since the objective was to model those limits through a dynamical system, we focus on generic aspects rather than the specific details. First, we make an unrealistic assumption that our system is a deterministic one. This makes our interpretations more clear. But, the proposed approach can be extended to include the stochastic nature of the real market using more sophisticated analysis. The sources of the risk associated in using the arbitrage opportunity are (i) fundamental risk, (ii) noise trader risk, (iii) implementation risk, and (iv) model risk. Fundamental risk was accounted by a change in the value input of the traders. An expression for ‘zero arbitrage’ condition, which depends on the traders’ profiles, is derived. This is one of the distinct features of the proposed framework. Noise trader risk was incorporated by different feedback gains in various time periods. It was found that these gains had to satisfy a condition; else infinite arbitrage would be available. The synchronization risk was considered by introducing a delay element. The delay could exist in the following: (i) in deciding a strategy based on new information, (ii) observing the market condition, and (iii) time taken to implement the decisions. In the proposed model, these delays were taken into account either by a delay element in the feedback loop or by a delay in the forward path. The impact of information on arbitrage was studied using the proposed model. The ways in which different delays can be implemented in the new framework have been described. It is shown through simulation that this model allows us to study the impact of various risks on the limits to arbitrage in a single framework. It is observed that noise trader risk and implementation risk limit the use of arbitrage much more than the other two risks. Empirical studies have also shown that implementation risk is the key factor in limits to arbitrage.

We have also developed a stochastic model for individual’s confidence bias incorporating concepts of Prospect Theory. The effect of confidence is studied by minimizing a criterion. The idea of excessive trading arising due to overconfidence and optimism has been derived analytically for a simple probability distribution. The strategy obtained is intuitive in the sense that if the trader is confident he would invest the maximum possible capital and he will hold on to his position if he is not confident. This non-optimal behavior can be attributed to their varying evaluation of status of the market, which has an impact on their confidence. The proposed model can be extended to include more complex distributions and expected utility functions.

In the dynamical system model of herding and the limits to arbitrage, we use the switching mechanism to include MG rules. The distinctive feature in the above framework is the dependence of the dynamics and observation equations on regimes. The regime transitions are assumed to be random and are described by its indicator function. The indicator is also piecewise constant and we assume a simple binomial model for its probability distribution. Switching is applied based on the slope of the response and parameters. We consider the following values during simulation: Number of traders is 10001 and the length of history is 15 values. In another simulation to study the effect of different types of traders, we have considered 2400 traders with 600 value traders, 600 momentum traders, 600 rule-based traders, and 600 noise traders. It is observed that noise traders cause more volatility as reported in many empirical studies. Simulation shows that rule-based traders and the value traders are the reason for forcing the stability of the price. Momentum traders, in simulation, push the price away from equilibrium.

Simulation studies made in this thesis do establish that dynamical system framework can give rise to rich behavior that could be used as the basic templates to model stock market features.

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