Optimal Trading with Linear and (small) Non-Linear Costs

Abstract

We reconsider the problem of optimal trading in the presence of linear and quadratic (market impact) costs for arbitrary linear costs but in the limit where quadratic costs are small. Using matched asymptotic expansion techniques, we find that the trading speed vanishes inside a band that is narrower in the presence of market impact by an amount that scales as a cube root of the market impact parameter. Outside the band we find three regimes: a small boundary layer where the velocity vanishes linearly with the distance to the band, an intermediate region where the velocity behaves as a square-root and an asymptotic region where it becomes linear again. Our solution is consistent with available numerical results. We determine the conditions under which our expansion is useful in practical applications and generalize our solution to other forms of non-linear costs.