Speculation with memory

This thesis considers asset price evolution in financial markets, deriving the price dynamics from micro-economic considerations. In contrast to most models of speculative price evolution in which the Efficient Market Hypothesis is assumed valid, and thus the price process is Markovian, we introduce dependence on past prices via the action of speculators endeavouring to profit by extracting information from the price history. Such speculators, called chartists, form a large portion of the agents in financial markets. However, hitherto no satisfactory continuous time model of their impact on market behaviour has been developed. This thesis attempts to close that gap. By modelling price evolution by a linear stochastic integro-differential equation and making exogenous allowances for the fluctuation of agents’ participation levels and the periodicity present in such markets, we show th a t several properties of financial markets can be qualitatively mimicked: the relationship between heterogeneity and fat-tailed returns distributions, the autocorrelation term structure of the returns, the relationship between volatility and volume and the relative success of chart speculators. Our model outlines mechanisms by which speculative bubbles or crashes arise, and demonstrates th a t certain types of derivative pricing are robust to the violation of the Efficient Market Hypothesis. These applied results are based on new findings in the theory of stochastic integro-differential equations which are developed in this thesis. We establish that such equations have unique, continuous solutions whose paths have the same local topology as those of Brownian Motion and which can be expressed in terms of the resolvent of a related deterministic integro-differential equation. By using the techniques of stochastic analysis (in particular the Ito Calculus) and the theory of deterministic integro-differential equations, we determine the pathwise asymptotics, a.s. growth of the extrema and asymptotic distributional character of the evolution