User:Jonhanke496/AtlasModels

Atlas models

An Atlas Model is a stochastic model for a market of $n$ stocks where the stock with the smallest total capitalization is given a large positive growth rate of $ng,$ while all other stocks are assumed to have zero growth rate. The name "atlas model" stems from the fact that the growth of the market is supported entirely by the growth of the smallest stock, by analogy with the mythological Atlas. Atlas models are useful as a way of understanding the stability of the capital distribution curve. Their mathematical simplicity allows one to prove that in such a market the capital distribution curve is stable and given by Pareto distribution. The fact that even these simple models have a capital distribution curve that fairly accurately approximates its observed shape, shows that the stability of the capital distribution curve is a fairly universal market phenomenon.

More mathematically, the atlas model with parameters $g,\sigma$ is defined by the stochastic differential equation

$d\log(X_{i}(t))=\gamma _{i}(t)\,dt+\sigma \,dW_{i}(t)$

where for all $1\leq i\leq n$ the $W_{i}(t)$ are independent Brownian motions, and the positive real parameters $g,\sigma >0$ are respectively called the growth rate and variance of the Atlas model.

Asymptotic stability

To understand the existence and stability of the capital distribution curve in a general stochastic market it is necessary to make some reasonable assumptions about its behavior. A useful class of markets to conisder are those which are asymptotically stable, which is defined in [[[SPT Book, p100]]] by the technical conditions that the difference of the adjacent stocks log market weights (when ranked by their capitalization in decreasing order) have local time and variance with asymptotically constant slope. In more mathematical language, asymptotic stability requires that the parameters $\mathbb {L} _{k,k+1}$ and $\mathbb {V} _{k:k+1}$ defined by

1. $\mathbb {L} _{k,k+1}:=\lim _{t\rightarrow \infty }{\tfrac {1}{t}}\Lambda _{\log(\mu _{(k)})-\log(\mu _{(k+1)})}(t)$

1. $\mathbb {V} _{k:k+1}:=\lim _{t\rightarrow \infty }{\tfrac {1}{t}}\langle \log(\mu _{(k)})-\log(\mu _{(k+1)})\rangle _{t}$

exist for all $1\leq k\leq n-1$. (Here $\Lambda _{X}(t)$ and $\langle X\rangle _{t}$ respectively denote the local time and variance process of a given continuous semimartingale process $X(t),$ and $\mu _{(k)}$ denotes the $k^{\mathrm {th} }$ ranked market weight process.)

Misc fragments

Fun with bold characters: ${\boldsymbol {\sigma }}$ $\sigma$ ${\boldsymbol {\lambda }}$ $\lambda$

it is useful to require that the market has the property of Asymptotic stability.

Even these simple models have a capital distribution curve that fairly accurately approximates its observed shape,

as a first-order approximation to market

ranked by their total capitalization