The no-trade region is a fundamental feature of portfolio optimisation with proportional transaction costs. It defines a range of portfolio weights within which the agent does not trade, because the cost of rebalancing exceeds the utility gain. Trading occurs only at the boundaries of this region.
In the equilibrium model of Gonon, Muhle-Karbe, Shi (2020), proportional costs lead to a no-trade region [-l, l] for the state variable X_t (the deviation between agents’ actual and frictionless target positions). Within this region, positions evolve freely; at the boundaries, minimal trading (via local time) keeps X_t reflected.
The state variable follows a doubly-reflected Brownian motion (equation 4.5):
dX_t = (gamma^1 beta^1 - gamma^2 beta^2) / ((gamma^1 + gamma^2) sigma) dW_t + dL_t - dU_t
where L and U are the minimal increasing processes keeping X in [-l, l], with boundary width:
l = (3 lambda (gamma^1 beta^1 - gamma^2 beta^2)^2 / ((gamma^1 + gamma^2)^3 sigma^4))^{1/3}
This scales as lambda^{1/3}, consistent with the single-agent literature on optimal portfolio choice with small proportional costs.
Key Details
- Width scales as lambda^{1/3}: the classic cube-root scaling from the Merton problem with proportional costs
- Boundary trading: at the edges of the no-trade region, agents trade minimally to stay within the region (singular control / local time)
- Equilibrium: the equilibrium return with proportional costs takes the same form as with smooth costs (Theorem 4.2), but with the reflected Brownian motion replacing the mean-reverting diffusion
- Hahn-Jordan decomposition: trading strategies with proportional costs are described by their cumulative purchases phi^up and sales phi^down (finite variation processes)