Title: Optimal Investment, Demand and Arbitrage under Price Impact,
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.