Dimensional Analysis and Market Microstructure Invariance

Abstract

Physics researchers obtain powerful results by using dimensional analysis to reduce the dimensionality of problems. Quantitative asset managers often use dimensional analysis implicitly. Interest rates, risk premiums, and squared Sharpe ratios are measured per unit of time while discount factors, t-statistics, and R-squares are dimensionless quantities which do not have time units. Stock commissions, bid-ask spreads, and tick size are often measured in units of pennies while transaction costs and bid-ask spread costs may equivalently be measured in dimensionless basis points. Trading volume and order size can be measured either in shares or dollars.

This paper uses dimensional analysis to derive scaling laws which simplify quantitative models related to market microstructure. The invariance results generalize and refine the market microstructure invariance hypotheses of Kyle and Obizhaeva (2016). The scaling laws are based on four assumptions: (1) Investors are not confused by units of measurement. (2) The leverage-neutrality implications of the Modigliani-Miller theorem apply to market microstructure. (3) The cost of bets (decisions of institutional asset managers to take risky positions) is invariant across assets when measured in dollars per bet. (4) Quantities related to transaction costs and bet size are functions of only share volume, stock price, returns variance, and the dollar cost of a bet.

The result of these four assumptions is a non-obvious dimensionless measure of liquidity, denote L, which is proportional to the cube root of the ratio of (A) dollar volume to (B) the product of the cost of a bet and returns variance. These four assumptions imply that, when measured in basis points, transaction-cost related quantities like bid-ask spread costs and implementation shortfall are inversely proportional to the liquidity measure L. For quantitative asset managers, this proportionality relationship simplifies modeling of transactions costs by implying that only one constant needs to be estimated for all stocks instead of separate constants for each stock. In other words, it makes it possible to describe various microscopic properties of financial markets such as liquidity in simple macroscopic terms.

Dimensional analysis also implies that the size distribution of bets, orders, and trades -- when measured in dollars -- scale proportionally with L. The liquidity parameter lambda from the model of Kyle (1985) scales with the square of L. These results are based on the intuition that positions of large asset managers are limited by market liquidity. This is important for asset managers because it suggests an approach for optimizing the size of speculative positions across assets in the presence of transactions costs.

When market microstructure quantities are functions of other variables in addition to share volume, stock price, and returns volatility, dimensional analysis places specific constraints on the functional forms of derived relationships. This principle is illustrated using tick size and minimum lot size. This is important for asset managers because it potentially provides a method to better understand, for example, how much wider quoted bid-ask spreads will be when tick size or minimum lot size are large. The scaling laws also can be used to understand stock market crashes and to calibrate public policy proposals of interest to asset managers such as optimal tick size, optimal minimum lot size, position limits.

The paper illustrates the invariance principle by showing, using data from the Moscow Exchange, that bid-ask spreads and number of trades for Russian stocks are consistent with the scaling laws implied by invariance.

Keywords: Market microstructure, invariance, transaction costs, bid-ask spread

JEL Classification: G10, G12, G14, G20