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

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.     

TRANSIENT LINEAR PRICE IMPACT AND FREDHOLM INTEGRAL EQUATIONS

We consider the linear‐impact case in the continuous‐time market impact model with transient price impact proposed by Gatheral. In this model, the absence of price manipulation in the sense of Huberman and Stanzl can easily be characterized by means of Bochner’s theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure‐valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko on the nonexistence of transaction‐triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.     

Portfolio choice with small temporary and transient price impact

We study portfolio selection in a model with both temporary and transient price impact introduced by Garleanu and Pedersen. In the large‐liquidity limit where both frictions are small, we derive explicit formulas for the asymptotically optimal trading rate and the corresponding minimal leading‐order performance loss. We find that the losses are governed by the volatility of the frictionless target strategy, like in models with only temporary price impact. In contrast, the corresponding optimal portfolio not only tracks the frictionless optimizer, but also exploits the displacement of the market price from its unaffected level.     

DYNAMIC TRADING VOLUME

We derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a large investor, with a long horizon and constant relative risk aversion, trades a safe and a risky asset. Trading volume approximately follows a Gaussian, mean‐reverting diffusion, and increases with depth, volatility, and risk aversion. Unlike the frictionless theory, finite depth excludes leverage and short sales because such positions may not be solvent even with continuous trading.     

OPTION PRICING AND HEDGING WITH SMALL TRANSACTION COSTS

An investor with constant absolute risk aversion trades a risky asset with general Itô‐dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading‐order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.     

INVESTING WITH LIQUID AND ILLIQUID ASSETS

We find optimal trading policies for long‐term investors with constant relative risk aversion and constant investment opportunities, which include one safe asset, liquid risky assets, and an illiquid risky asset trading with proportional costs. Access to liquid assets creates a diversification motive, which reduces illiquid trading, and a hedging motive, which both reduces illiquid trading and increases liquid trading. A further tempering effect depresses the liquid asset's weight when the illiquid asset's weight is close to ideal, to keep it near that level by reducing its volatility. Multiple liquid assets lead to portfolio separation in four funds: the safe asset, the myopic portfolio, the illiquid asset, and its hedging portfolio.     

AN ONLINE PORTFOLIO SELECTION ALGORITHM WITH REGRET LOGARITHMIC IN PRICE VARIATION

We present a novel efficient algorithm for portfolio selection which theoretically attains two desirable properties:

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