Optimal execution comparison across risks and dynamics, with solutions for displaced diffusions

TL;DR

This paper compares optimal trade execution strategies under various risk criteria when the asset price follows a displaced diffusion, highlighting how the choice of risk measure and asset dynamics influence the optimal solutions.

Contribution

It introduces and compares optimal execution strategies under multiple risk criteria for displaced diffusion models, a class that interpolates between ABM and GBM, with numerical illustrations.

Findings

Optimal strategies vary significantly with risk aversion levels.

Displaced diffusion models allow for flexible support of asset prices.

Differences in strategies are notable for high risk aversion and low market impact assets.

Abstract

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called "squared asset expectation" (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.