Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

Abstract

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.

Trading with small nonlinear price impact

“…9 Under Assumption 2, it follows from [15], Proposition 1.1, that the SDE has a unique strong solution on [0, T ] for all λ > 0, where Wφ ,Q is a Brownian motion under Q. 10 We can now define the trading strategy (…”

“…and define the stopping time (3.10) 10 In the notation of [15], take ε = λ 11 This happens if the trading cost becomes too large, or the target too volatile relative to the risky asset. 12 For proportional or fixed costs, this can be done by a single bulk trade at T , without affecting the asymptotic results at the leading order.…”

“…The probabilistic approach in this paper crucially exploits that the invariant distributions of scalar diffusion processes are readily available. In a multidimensional Markovian setting, a small-cost expansion of the value function is studied using stability results for viscosity solutions of the corresponding dynamic programming equations in[10] 8. As is well known[20], random endowments such as labour income or option positions can be readily absorbed into a change of measure.…”

Utility Indifference Pricing with High Risk Aversion and Small Linear Price Impact

“…In financial markets, trading moves prices against the trader: buying faster increases execution prices, and selling faster decreases them. This aspect of liquidity, known as market depth [3] or price-impact has recently received increasing attention (see, for instance, [2,4,6,8,9,11,12,13,14,15,16,17] and the references therein).…”