Optimal Execution with a Price Limiter

Agents often wish to limit the price they pay for an asset. If they are acquiring a large number of shares, they must balance the risk of trading slowly (to limit price impact) with the risk of future uncertainty in prices. Here, we address the optimal acquisition problem for an agent who is unwilling to pay more than a specified price for an asset while they are subject to market impact and price uncertainty. The problem is posed as an optimal stochastic control and we provide an analytical closed form solution for the perpetual case as well as a dimensionally reduced PDE for the general case. The optimal speed of trading is found to no longer be deterministic and instead depends on the fundamental price of the asset. Moreover, we demonstrate that a price limiter constraint significantly reduces the conditional tail expectation of the securities costs.