User:Stevelihn/Lihn's law

Lihn's law states that the value of a financial market in equilibrium is proportional to the 3/2 power of the number of constituents (e.g. stocks) in the market (n3/2). This scaling law was first formulated by Stephen H. Lihn in 2009.

There are strong statistical reasons for the n3/2 law. First, the population distribution in the market must follow a quasi-normal distribution when measured in a canonical coordinate. Second, the constituent's influence to the market is exponential. Third, the population must adjust itself around a level of optimal volatility equilibrium in order to survive as a viable market. The statistical result based on these assumptions is the n3/2 law.

Population distribution
The first pillar of the n3/2 law is that the population distribution of the market constituents follows a normal distribution can be described by the probability density function

P(H) = \tfrac{N}{\sqrt{2\pi}\eta_c}\, e^{-\,(H-H_c)^2\!/2\eta_c^2}, $$ in which N is the number of constituents (e.g. stocks) in the market, H is the measurement coordinate, $$\eta_c^2$$ is the variance of the distribution, and $$H_c$$ is the mean of the distribution. In the case of the stock market, H is measured by the log-market capitalization. Obviously,

\int_{-\infty}^{\infty}P(H)\, dH=N. $$

Exponential influence
The second pillar of the n3/2 law is pretty straightforward. It assumes the influence of the constituents (that is, its contribution to the value of the market) is proportional to the exponential of its coordinate, $$e^H$$. Thus, the total value of the market is

Z \, = \, \int_{-\infty}^{\infty} e^H P(H) \, dH. $$ This integral can be carried out as

Z \, = \, N \, e^{H_c+\tfrac{1}{2}\eta_c^2}. $$ This is simply N times the expected value of a lognormal distribution.

The dynamic equilibrium of a growing population and the 3/2 power law
The third pillar of the n3/2 law is to define the dynamic equilibrium when the population is growing. It assumes the growing population tends to reach an equilibrium level when $$\eta_c^2$$ is expanding in $$log(N)$$. That is,

\eta_c^2 \, = \, \psi_c + log(N), $$ in which $$\psi_c$$ is a constant. Thus the total market value becomes

Z \, = \, N^{\tfrac{3}{2}} \, e^{H_c+\tfrac{1}{2}\psi_c}. $$ If $$H_c$$ is constant over time, then we obtain the n3/2 law
 * $$Z \sim N^{\tfrac{3}{2}}$$

Optimal volatility equilibrium
The assumption that $$\eta_c^2$$ is expanding in $$log(N)$$ is based on the stochastic calculus in lognormal cascade. The hypothesis is that a growing population will reach (or oscillate around) a dynamic equilibrium such that the volatility of the market value process (Z(t)) always stays in a reasonable range: What remains is that a healthy, long-running market will oscillate around an optimal level of volatility over the long term. Such condition is called the Optimal Volatility Equilibrium.
 * 1) The volatility doesn't depend on the number of constituents in the market.
 * 2) The volatility can't be very high since high volatility makes the market unstable, thus cannot survive.
 * 3) The volatility can't be very low. Market complacency entices people to make stupid decisions, thus is not sustainable either.

In the context of stochastic calculus, in which we only focus on the (geometric) Brownian motion terms and the growth rate terms are ignored, we have

d\, log\, Z(t)\, =\, \sum_{i=1}^n \,\mu_i(t) \,d\,log\, X_i(t), $$ where $$X_i$$ and $$\mu_i$$ are the value and weight processes of constituent i and $$\mu_i(t)=X_i(t)/Z(t)$$. Each constituent follows a simple Wiener process while the growth rate terms are ignored:

d\, log\, X_i(t)\, =\, \sigma_c\, dW(t) $$ where $$\sigma_c$$ is the characteristic volatility for the constituents. By the additive rule of normal distributions, we can obtain the formula for the market volatility

\sigma_{log\,Z}\, =\, \sigma_c \,\tfrac{1}{\sqrt{N}} \,e^{\tfrac{1}{2}\eta_c^2}. $$

Assume that $$\eta_c^2$$ is a time-dependent process in the form of

\eta_c^2(t)\, =\, \psi_c + log(N) + \psi(t), $$ we arrive at

\sigma_{log\,Z}\, =\, \sigma_c \,e^{\tfrac{1}{2}\psi_c} \,e^{\tfrac{1}{2}\psi(t)}. $$ Notice that $$\sigma_{log\,Z}$$ doesn't depend on N any more, that is, Requirement 1 is fulfilled. To fulfill Requirement 2 and 3, $$\psi(t)$$ must be a mean-reverting process that has bounded variance. Ornstein–Uhlenbeck process is one such process. Therefore, $$E[\eta_c^2(t)]\, =\, \psi_c + log(N)$$, and the log-return of the market value process Z follows a lognormal cascade distribution.

Long-term growth of the U.S. Stock Market
In 1929, there were about 700 stocks traded on the U.S. stock exchanges. By 1999, there were 7500 stocks (Section 5.1 of Fernholz 2002 ). During this period, DOW has risen from about 100 to 9000, an 90-fold increase (6.6% per year). The n3/2 law would predict a 35-fold increase in market size, which explains the bulk of DOW's increase in 70 years. This implies that the remaining 2.5-fold increase would come from the small increase in the mean log-market capitalization, which translates into 1.3% per year. The mean log-market capitalization is amazingly stable in the long term. Its rate of increase is much smaller than the long running inflation rate of 3%.