Optimal Trading Strategies within the Hamilton-Jacobi-Bellman Equation - An Application to Statistical Arbitrage

We propose optimal trading strategies based on the Hamilton-Jacobi-Bellman equation when a trader is allowed to place only market orders with a particular focus on a statistical arbitrage. A process that a placed limit order is filled is often modeled as a stochastic point process. In case of a market order, however, it is filled with a hundred percent probability meaning that it is not a stochastic process and is rather a deterministic one. They are therefore represented using the Heaviside step function and their infinitesimal generators are just time derivative and give delta functions. An integral linear utility function with an inventory penalty and a discount enables us to obtain optimal number of market order with a simple calculus and those in a limit T → ∞, i.e. a stationary limit. We show that, at least within our formulation, we cannot generate a positive PnL by trading multiple set of assets whose noises are correlated but their drifts are zero which contradicts the common statement that "Given correlated multiple assets, we can exploit those correlations by trading an appropriate linear combination of them". However, we show that if multiple assets are cointegrated, we can create a mean-reverting portfolio and exploit them. In that case, an optimal number of market order which takes current status of market as well as a trader's preference into account is given in a simple but intuitive and reasonable form and it does generate positive PnL while controlling risk of the portfolio.