Optimal Investment, Demand and Arbitrage under Price Impact

TL;DR

This paper analyzes how price impact influences optimal investment, pricing, and arbitrage opportunities in a market with market makers, revealing effects like constrained trading and non-linear hedging costs.

Contribution

It introduces a comprehensive framework for understanding price impact effects on investment strategies, pricing rules, and arbitrage in markets with competitive market makers.

Findings

Price impact causes constrained trading and non-linear hedging costs.

Three notions of arbitrage-free prices are identified, differing with impact presence.

Arbitrage opportunities are limited by position size and can be endogenous in equilibrium.

Abstract

This paper studies the optimal investment problem with random endowment in an inventory-based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading, and non-linear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, non-linear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.