William Brayer, George Mason University


Bifurcation and Continuation Analysis of Equilibria of the Diblock Copolymer Equation in One Dimension

Abstract: Standard financial models are inadequate because they make very strong assumptions about rationality and efficiency that imply a Gaussian distribution of price changes. Yet bubbles and crashes frequently occur, with the distributions of price changes having a power-law decay ("fat tails"): thus conventional models severely underestimate the risk of extreme events in financial markets. The standard models use geometric Brownian motion to simulate the evolution of an asset price p over a timestep h via

p(n + 1) = p(n) exp( √h η - h/2),

where η ~ N(0; 1). We replace this pricing formula with

p(n + 1) = p(n) exp(√ h η -h/2 + k Δσ (n)),

where the additional term k Δσ(n) corresponds to changes in demand for the asset amongst M agents who are simulated directly.
We will describe simple rules governing the behavior of the agents that can mimic both the eects of (irrational) human psychology as well as systemic market defects that result in rational-but-perverse behavior. Using such rules, the market generates internal dynamics that correspond to the gradual formation of price bubbles followed by rapid crashes that closely resemble those seen in real financial markets (note that such internal eects are explicitly ruled out in the standard model).
We use this model to test a variety of trading rules germane to technical analysis. Surprisingly, we show that technical analysis can produce excess returns and that herding may account for this phenomenon.

Mentors: Harbir Lamba and Timothy Sauer (George Mason University)