Optimal Control of Execution Costs

Motivation(s)

A hypothetical portfolio can exhibit 20% in annual returns when trading costs are ignored, yet the realized gains can be reduced to a fraction (e.g. tenth) of the original when the strategy is implemented. This demonstrates that execution costs (e.g. commissions, bid/ask spreads, opportunity costs of waiting, price impact from trading) have a substantial impact on investment performance.

Since the typical institutional investor’s trades are so large, they tend to be broken up into smaller trades executed over the course of several days. Denoting these trade sequences as packages, recent studies on the transactions of large investment firms show that only one fifth of the market value of these packages are completed within one day and more than half are spread over four trading days or more. Since the act of trading affects price dynamics which in turn affects future trading costs, the best proprietary execution strategies cannot be defined in the context of a single transaction.

Proposed Solution(s)

The authors propose a new notion of best-execution strategies: given the price dynamics that capture market price impact, find the optimal sequence of trades that minimizes the expected cost of executing \(\bar{S}\) shares within \(T\) periods. Formally, this is

\[\begin{split}\begin{aligned} \min_{\{S_t\}} \quad \mathrm{E}\left[ \sum_{t = 1}^T P_t S_t \right] &\\ \text{subject to} \quad \sum_{t = 1}^T S_t &= \bar{S}\\ 0 &\leq S_t \quad t = 1, 2, \ldots, T \end{aligned}\end{split}\]

where the price dynamics \(P_t\) follow some “law of motion” such as

\[P_t = P_{t - 1} + \theta S_t + \epsilon_t, \quad \theta > 0, \quad \mathrm{E}\left[ \epsilon_t \mid S_t, P_{t - 1} \right] = 0.\]

They argue that the search for the best-execution strategies needs to be modeled as a dynamic optimization problem because trading takes time, the demand for financial securities is not perfectly elastic, and the price impact of current (possibly small) trades can affect the course of future prices. The common naive strategy — breaking up \(\bar{S}\) shares evenly into a package of \(T\) trades each of size \(\bar{S} / T\) — is a subset of the best-execution strategies and is optimal only when the price-impact function and price dynamics obey the proposed (naive) law of motion.

Evaluation(s)

The naive law of motion’s purpose is to demonstrate the basic approach to deriving the best-execution strategies. Since the objective function is convex, each decision is optimal for the remaining program and is attained when the derivative is zero.

The authors derived a closed-form model of linear price impact with information to better capture market conditions. Although the numerical examples illustrate that the best-execution strategies have lower expected execution costs compared to the naive strategy, the model does not guarantee the lowest execution costs. This stems from the law of motion’s linearity and the assumption that prices follow arithmetic random walks. The linearity forces the percentage price impact, as a percentage of execution price, to be a decreasing function of price level, which contradicts the fact that absolute price impact increases linearly with trade size. The arithmetic random walk could lead to negative prices and implies that both price impact and information have only permanent effects on prices, which differs from several empirical studies suggesting some combination of permanent and temporary effects.

To address the previous limitations, the authors propose a more plausible law of motion called linear-percentage temporary (LPT) and assume a geometric Brownian motion for price dynamics. One additional benefit is that the model is able to capture the implementation shortfall in executing \(\bar{S}\). Monte Carlo simulations indicate that this model reduces on average 25% to 40% of the naive strategy’s execution costs.

For all of the proposed models, the optimization is unconstrained and consists of a single security. Adding constraints require discretization of the state space or approximating the optimal-value function at each stage. The former yields an exact numerical solution while the latter gives an analytical approximate solution. Extending these models to a portfolio of securities is literally rewriting them in vector notation.

Despite the reduction in execution costs, the authors admit there are still many issues left to resolve:

  • The price impact model does not account for market versus limit orders.

  • The overall risk of best-execution strategies is due to the variability of prices, not to the variability of price impact.

  • There are no simple solutions to handle the uncertainty resulting from parameter estimation errors and parameter instabilities.

  • The maximization of expected utility with transaction costs is analytically and computationally intractable.

  • The proposed law of motion are unaffected by the investor’s trades.

Future Direction(s)

  • How to uncover a firm’s execution strategy using their published transactions history and earnings?

  • How should game theory be incorporated into the objective function?

  • How can deep learning be used to model an appropriate execution strategy?

Question(s)

  • How should the parameters of the proposed models be calibrated?

  • How accurate is the claim that the minimum variation of prices on most US stocks is $0.125?

  • The authors focused on convex solutions to ensure a global optimum, but is the desired objective function convex?

Analysis

All proposed trading strategies need to account for execution costs to avoid overoptimistic returns. Furthermore, the model extension to list trading is necessary to account for cross effects among the securities.

One glaring issue is the lack of backtesting the proposed best-execution strategies. It would have been much more interesting to see than simulations with carefully calibrated models. These results would make the claim in cost avoidance more believable. Likewise, the claim that the approach is still valid in the presence of adversarial strategies needs to be supported by evidence.

Nevertheless, the application of the Bellman equation and the derivations in the appendix are very insightful into future modeling. The concept of a law of motion for price dynamics enables future integration of features like illiquidity and style factors.

References

BL98

Dimitris Bertsimas and Andrew W Lo. Optimal control of execution costs. Journal of Financial Markets, 1(1):1–50, 1998.