In modern financial markets, one of the key concerns of large market participants is how to trade in significant size without creating an adverse effect on price. Solutions have included technical methods, such as algorithms that can slice and dice trades to feed them into the market over time, and also structural changes, including the creation of dark pools, which allow traders to execute block trades in private settings. Given the needs of institutional investors, sovereign wealth funds, hedge funds and other investors with AUM in the billions, it is useful to take a closer look at how market impact can be measured and evaluated.

Speaking at Bloomberg’s monthly quant seminar (BBQ) in September, Professor Walter Schachermayer of the University of Vienna explained how dimensional analysis provides a powerful way to examine both market impact and intraday trading effects. Dimensional analysis is a mathematical technique taken from and engineering that centers on finding similarities and relationships between various elements in a system. The method reduces complex physical problems into simpler forms to learn about interactions between forces or variables, particularly in cases where the equations and boundary conditions are not specified completely. In financial markets, we find numerous examples of such systems, including the behavior of stocks and trading flows, bond duration and ratios commonly used in finance, accounting, and economics. Studying the properties behind these processes and metrics can generate useful insights for trading, asset allocation and risk management.

The general approach taken with dimensional analysis can be demonstrated in a relatively simple example: the periodicity of a pendulum. In this case, three variables, length (*l*), mass (*m*), and gravity (*g*) comprise the dimensions of the system, under the basic assumption that they can fully explain the period of the pendulum. The analysis in this case rests on the fact that there are three linear equations in three unknowns, which leads to a relatively tidy conclusion.

Turning to the subject of market impact, let’s consider an agent who wishes to buy or sell a large amount of stock, but is aware that the transaction has the potential to move the market to the agent’s disadvantage. The goal is to minimize the market impact, even if it is not possible to eliminate it completely.

In this case, we have the size of the order (*Q*), the price of the stock (*P*), the traded volume of the stock (*V*), the squared volatility of the stock (σ²), and the market impact (*G*), which is a dimensionless quality. As with the pendulum example, the basic assumption is that the four variables, *Q*, *P*, *V*, and σ² fully explain the size of the market impact. Unlike the previous example, however, this leads to three linear equations in four unknowns, resulting in a solution with one degree of freedom. The question now becomes, “Can we find one more equation that will allow for a unique solution to the system?” According to recent work by Kyle and Obizhaeva (2016), the answer is yes.

The approach involves the concept of leverage neutrality and incorporates the Theorem of Modigliani–Miller (1958), using assets, debt, and equity to come up with a stochastic process for analyzing the value of the assets. We discover that the relative dynamics of the equity are proportional to the leverage. Thus, the value of the firm does not depend on its capital structure and this no-arbitrage-type condition produces an equation, *M* that measures the leverage of the company. Armed with *M*, we now have four equations in four unknowns, and after a series of operations, we find that the market impact *G* is proportional to the square root of the size *Q* of the meta-order.

So, this is quite satisfying from a theoretical perspective, but how does it relate to practice? Professor Schachermayer presented a series of empirical studies, consisting of actual orders, based on limit order book data provided by the LOBSTER database (https://lobsterdata.com). The sampling period covered January 2, 2015 through August 31, 2015 (167 trading days) and was constructed using 128 sufficiently liquid stocks with high market capitalizations. The models demonstrated intriguing results with regard to the linear relations between considered qualities, but questions remain, particularly concerning the dimension of volatility.

Naturally we have a definition of volatility at time T and examples of price processes that make use of it (e.g. Black Scholes). One of the challenges in developing useful quantitative methods is that when we are working with estimates of the true volatility, we often find that real world behavior may differ from what was projected. This has been an important subject of study for several decades; explanations for actual behavior include fractional Brownian motion (Mandelbrot), rough volatility (Bayer; Gatheral; Rosenbaum), and market microstructure effects (Bouchaud; Rosenbaum). Clearly this is a vibrant area for past and current research, with valuable insights for today’s uncertain markets. Recent work by Pohl, Ristig, and Tangpi (2018), for example, explores the relationship between the number of trades and the exchanged risk.

Further, Kyle and Obizhaeva (2017) have combined dimensional analysis, leverage neutrality, and a principle of market microstructure invariance to derive scaling laws expressing transaction costs functions, bid-ask spreads, bet sizes, number of bets, and other financial variables in terms of dollar trading volume and volatility. As they aptly point out, such research can help to produce practical metrics for risk managers and traders; develop scientific benchmarks for evaluating issues around high frequency trading, market crashes, and liquidity measurement; and design guidelines public policies and industry practices in the aftermath of the financial crisis.

As we continue to explore issues related to liquidity, market microstructure, institutional investors’ behavior, risk management, and the efforts of regulators to ensure fair and orderly markets, methods that shed light on underlying dynamics are worth deeper study. The combination of dimensional analysis and extensive trading data sets may hold some keys to the future.