When executing a large trade across multiple assets, price impact doesn't stay in its lane. Selling one stock moves the price of correlated stocks. This cross-impact is well known and usually treated as a cost — executing in asset A affects the price you'll get in asset B, so you need to coordinate.
Ackermann, Kruse, and Urusov (arXiv:2503.05594) find a stranger consequence. In their multi-asset optimal execution framework with stochastic cross-effects, it can be optimal to trade in an asset you don't own and have no target position in. The optimal execution of your actual portfolio includes phantom trades — positions opened and closed in assets you never intended to hold, purely because the cross-impact dynamics make this cheaper than executing directly.
The mechanism works through the matrix structure of price impact and resilience. When impact is a matrix-valued stochastic process, the off-diagonal elements create coupling between assets. Trading in asset C to manipulate the impact experienced by asset A during execution isn't arbitrage — it's cost minimization. The phantom trade absorbs some of the price impact that would otherwise fall on your real execution, then unwinds as resilience restores the affected prices.
The result extends the Obizhaeva-Wang framework from single-asset to multi-asset execution. In the single-asset case, the optimal strategy balances urgency (trading quickly to reduce exposure) against impact (trading slowly to reduce cost). With multiple assets, a third dimension appears: cross-asset impact management, where the optimal strategy includes actions in assets that exist only to reshape the impact landscape for the assets you actually care about.
The structural insight is that in coupled systems, the optimal intervention set is larger than the target set. You act on things you don't care about to improve outcomes on things you do. The phantom trade has no direct value. Its value is entirely in how it changes the cost structure for everything else.