We analyse financial market models in which 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. Agents will switch from one rule to
another stochastically, and the price and profits process will reflect these switches. Among
the possible rules are “chartist” or extrapolatory rules. Prices can exhibit transient behaviour
when chartists predominate. However, if the probability that an agent will switch to being a
“chartist” is not too high then the process does not explode. There are occasional bubbles
but they inevitably burst. In fact, we prove that the limit distribution of the price process
exists and is unique. This limit distribution may be thought of as the appropriate equilibrium
notion for such markets. A number of characteristics of financial time series can be captured
by this sort of model. In particular, the presence of chartists fattens the tails of the stationary
distribution.
In the first part of the article, we characterize distribution-invariant risk measures with convex
acceptance and rejection sets on the level of distributions. It is shown that these risk measures
are closely related to utility-based shortfall risk.
In the second part of the paper, we provide an axiomatic characterization for distribution-invariant
dynamic risk measures of terminal payments. We prove a representation theorem and
investigate the relation to static risk measures. A key insight of the paper is that dynamic consistency
and the notion of "measure convex sets of probability measures" are intimately related.
This result implies that under weak conditions dynamically consistent dynamic risk measures can
be represented by static utility-based shortfall risk.
We derive a continuous time approximation of the evolutionary market selection model of Blume &
Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous
time approximation equals the solution of an integral equation
in a random environment. For constant asset returns, the integral equation reduces to an autonomous
ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to
Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.
We analyze an interactive model of credit ratings where external shocks, initially
affecting only a small number of firms, spread by a contagious chain reaction to the
entire economy. Counterparty relationships along with discrete adjustments of credit
ratings generate a transition mechanism that allows the financial distress of one firm
to spill over to its business partners. Such a contagious infectious of financial distress
constitutes a source of intrinsic risk for large portfolios of credit sensitive securities that
cannot be “diversified away.” We provide a complete characterization of the fluctuations
of credit ratings in large economies when adjustments follow a threshold rule. We also
analyze the effects of downgrading cascades on aggregate losses of credit portfolios. We
show that the loss distribution has a power-law tail if the interaction between different
companies is strong enough.
Stability of Linear Stochastic Difference Equations in Strategically Controlled Random Environments
(2004)
We consider the stochastic sequence fYtgt2N defined recursively by the linear relation
Yt+1 = AtYt+Bt in a random environment. The environment is described by the stochastic
process f(At;Bt)gt2N and is under the simultaneous control of several agents playing a
discounted stochastic game. We formulate sufficient conditions on the game which ensure
the existence of Nash equilibrium in Markov strategies which has the additional property
that, in equilibrium, the process fYtgt2N converges in distribution to a stationary regime.
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 modelling 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 market
participants. 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.
We consider a financial market model with a large number of interacting
agents. Investors are heterogeneous in their expectations
about the future evolution of an asset price process. Their current
expectation is based on the previous states of their “neighbors” and
on a random signal about the “mood of the market.” We analyze the
asymptotics of both aggregate behavior and asset prices. We give sufficient
conditions for the distribution of equilibrium prices to converge to
a unique equilibrium, and provide a microeconomic foundation for the
use of diffusion models in the analysis of financial price fluctuations.
We consider general economies in which rational agents interact locally. The local aspect
of the interactions is designed to represent in a simple abstract way social interactions, that
is, socioeconomic environments in which markets do not mediate all of agents' choices, and
each agent's choice might be in part determined, for instance, by family, peer group, or ethnic
group effects. We study static as well as dynamic infinite horizon economies; we allow for
economies with incomplete information, and we consider jointly global and local interactions,
to integrate e.g., global externalities and markets with peer and group effects. We provide
conditions under which such economies have rational expectations equilibria.
We illustrate the effects of local interactions when agents are rational by studying in detail
the equilibrium properties of a simple economy with quadratic preferences which captures, in
turn, local preferences for conformity, habit persistence, and preferences for status or adherence
to aggregate norms of behavior.
We give sufficient conditions for a non-zero sum discounted stochastic game with
compact and convex action spaces and with norm-continuous transition probabilities,
but with possibly unbounded state space, to have a Nash equilibrium in homogeneous
Markov strategies that depends in a Lipschitz continuous manner on the current state. If
the underlying state space is compact this yields the existence of a stationary equilibrium.
Stochastic games with weakly interacting players provide a probabilistic framework within
which to study strategic behavior in models of non-market interactions.