Motivated by the behavior of asset prices, trading volume and price volatility during historical episodes of asset price bubbles, we present a continuous time equilibrium model where overconfidence generates disagreements among agents regarding asset fundamentals. With short-sale constraints, an asset owner has an option to sell the asset to other overconfident agents when they have more optimistic beliefs. As in Harrison and Kreps (1978), this re-sale option has a recursive structure, that is, a buyer of the asset gets the option to resell it. Agents pay prices that exceed their own valuation of future dividends because they believe that in the future they will find a buyer willing to pay even more. This causes a significant bubble component in asset prices even when small differences of beliefs are sufficient to generate a trade. In equilibrium, large bubbles are accompanied by large trading volume and high price volatility. Our model has an explicit solution, which allows for several comparative statics exercises. Our analysis shows that while Tobin’s tax can substantially reduce speculative trading when transaction costs are small, it has only a limited impact on the size of the bubble or on price volatility. We also give an example where the price of a subsidiary is larger than its parent firm. This paper was previously circulated under the title “Overconfidence, Short-Sale Constraints and Bubbles.”

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modeling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated ‘third parties’. The mathematical contributions are a functional central limit theorem for stationary semi-Markov processes, and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.

Motivated by the behavior of internet stock prices in 1998-2000, we present a continuous time equilibrium model of bubbles where overconfidence generates agreements to disagree among agents about asset fundamentals. With a short-sale constraint, an asset owner has an option to sell the asset to other agents when they have more optimistic beliefs. This re-sale option has a recursive structure, that is a buyer of the asset gets the option to resell it, causing a significant bubble component in asset prices even when small differences of beliefs are sufficient to generate a trade. The model generates prices that are above fundamentals, excessive trading, and excess volatility. We also give an example where the price of a subsidiary is larger than its parent firm. Our analysis shows that while Tobin's tax can substantially reduce speculative trading when transaction costs are small, it has only a limited impact on the size of the bubble or on price volatility.

One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agent-based viewpoint in finance goes back at least to the work of Garman (1976) and shares the philosophy of statistical mechanics in the physical sciences. We discuss recent developments in market microstructure models. They are capable, often through numerical simulations, to explain many stylized facts like the emergence of herding behavior, volatility clustering and fat tailed returns distributions. They are typically queueing-type models, that is, models of order flows, in contrast to classical economic equilibrium theories of utility-maximizing, rational, “representative ” investors. Mathematically, they are analyzed using tools of functional central limit theorems, strong approximations and weak convergence. Our main examples focus on investor inertia, a trait that is well-documented, among other behavioral qualities, and modelled using semi-Markov switching processes. In particular, we show how inertia may lead to the phenomenon of long-range dependence in stock

We consider an agent-based model of financial markets with asynchronous order arrival in continuous time. Buying and seeling orders arrive in accordance with a Poisson dynamics where the order rates depend both on past prices and the mood of the market. The agents form their demand for an asset on the basis of their forecasts of future prices and where their forecasting rules may change over time, as a result of the influence of other traders. Among the possible rules are “chartist ” or extrapolatory rules. We prove that when chartists are in the market, and with choice of scaling, the dynamics of asset prices can be approximated by an ordinary delay differential equation. The fluctuations around the first order approximation follow an Ornstein-Uhlenbeck dynamics with delay in a random environment of investor sentiment.

In this paper we study multiagent models with time-varying type change. Assume that there exist a closed system of N agents classified into r types according to their states of an internal system; each agent changes its type by an internal dynamics of the internal states or by the relative frequency of different internal states among the others, e.g., multinomial sampling. We investigate the asymptotic behavior of the empirical distributions of the agents ’ types as N goes to infinity, by the weak convergence criteria for time-inhomogeneous Markov processes and the theory of Volterra integral equations of the second kind. We also prove convergence theorems of these models evolving in random environment.