Market volatility

Here we show that our model has nonvanishing volatility as the number of participants goes to infinity, unlike some simulation models, which specify an exogenously fixed numbers of agents.

Let E(x) and E(y) be the expected values of the largest two fractions, x and y. Watterson (1976) shows that

where B₁∕₂ is an incomplete beta function; see Abramovitz and Stegun (1968, 26.5). Using this formula, E(y) ≈ 0.16 with θ = 0.4; hence E(x) + E(y) ≈ 0.95. Similarly, we have E(x + y) = 0.97 and 0.92 for θ = 0.3 and 0.5, res­pectively. We may therefore suppose θ is about 0.4. With θ = 0.4, the ex­pected numbers of clusters are E (K₁₀) = 2.1, E (K₁₀₀) = 3.0, E (K₁₀₀₀) = 4.0, E(K₁₀5) = 5.8, and E(K₁₀7) = 7.7. These figures indicate that there are several small fractions in addition to the two large ones when the number of participants is n ≥ 100.

Watterson also has bounds for other moments with k and l nonnegative integers:

with G = E(x)∕θΓ(θ)e^γθ, and where B₁∕₂(a, b) is the incomplete beta func­tion. The inequality comes from an approximation he used to evaluate some integrals. Abramovitz and Stegun have some series expansions for the incom­plete beta functions. Unfortunately, the bounds are not sharp enough to give precise bounds on the variances of x. If we use y ≈ 0.95 - x, then

may be used to estimate

In other words, the standard deviation of the largest fraction is about 1 /5 of its mean. See Watterson (1976) and Watterson and Guess (1977) for more precise calculation procedures.

What is most remarkable about the patterns of clusters of agents when n is large is that some small subsets of configurations account for a majority of possible patterns. That is, some small number of configurations are most likely to be realized or observed. This feature has been noticed in other contexts as well. Mekjian (1991) compares genetics and physics examples.

11.4