The purpose of this paper is to develop a model for the inclusion of liquidity risk into arbitrage pricing theory that incorporates the impact of differing trade sizes on the price. The approach is consistent with price inelasticities. It is done by hypothesizing the existence of stochastic supply curve for a security price as a function of a trade size, for which traders act as price takers. This condition implies that the investor’s trading strategy has no lasting impact on the price process. The authors study conditions on the supply curve such that there is no arbitrage opportunity in the economy (appropriately defined). Given an arbitrage free evolution, complete markets and pricing of derivatives are discussed. This structure can also be viewed as an extension of the model where the traded securities have two distinct price processes, a selling price (the bid) and the buying price (the ask).
The paper provides a new perspective on classical arbitrage pricing theory. The first fundamental theorem of asset pricing appropriately generalized, holds in this new setting, while the second fundamental theorem fails. Herein, a martingale measure (appropriately defined) can be unique, and markets still be incomplete. However, a weakening of the second fundamental theorem holds. It is shown in an approximately complete market that by using trading strategies that are continuous and of finite variation, all liquidity costs can be avoided. Consequently, the arbitrage-free price of any derivative is shown to be equal to the expected value of its payoff under the risk-neutral measure.