A note on jump Atlas models

Abstract

The market weight of a stock is its capitalization (cap) divided by the total market cap. Rank these weights from top to bottom. The capital distribution curve is a plot of weights versus ranks. For the US stock market, it is linear on a double logarithmic scale, and stable with respect to time (Stochastic Portfolio Theory (2002) Springer). This property has been captured by models with rank-dependent dynamics: Each stock’s cap logarithm is a Brownian motion with drift and diffusion coefficients depending on its current rank (Probability Theory and Related Fields 147 (2010) 123–159). However, short-term stock movements have heavy tails. One can add jumps to Brownian motions to capture this. Observed time stability follows from a long-term stability result, stated and proved here. Via simulations, we find which properties of continuous models are preserved after adding jumps.