Pricing securities and risky income streams by no arbitrage arguments has become the corner stone of modern asset pricing theory. No arbitrage arguments have also been an impressive practical success. The valuation techniques derived from them have become the daily tools and workhorses of thousands of practitioners and financial engineers worldwide. The idea of no arbitrage is simple. It requires that correctly priced securities should make it impossible to achieve by financial transactions a consumption bundle at zero costs that increases some investor,s utility. This idea ultimately relies on an equilibrium argument and has powerful implications for asset pricing formulas. A great deal of this power comes from the fact that the question whether or not security prices do allow arbitrage, can be inferred from observable data: the prices of actively traded securities and their payoff structure. We do not have to know the entire equilibrium. Moreover once the correct security prices have been found, the price of any risky income stream which can be generated by combinations of these securities is determined. Thus security pricing by no arbitrage leads to a general valuation technique for arbitrary contingent claims, which can be generated from securities traded on financial markets.
Yet the formulas are derived under highly idealized conditions. Among them, perfect competition and frictionless security trading are the two most important ones. Evidence as well as practical experience suggests that the assumption of price taking behavior is to a large extent fairly appropriate for financial markets for standard securities, such as options, futures, stocks and bonds. The assumption of frictionless trading however is surely inappropriate. Margin requirements, short selling restrictions, borrowing constraints and collateral requirements belong to the basic facts of (financial) life, even for the most competitive financial markets.
In this paper we ask whether and how we can transfer the power and the simplicity of pricing a risky income stream by no arbitrage arguments to a world where such constraints bind investors in their portfolio decisions. The answer we get is that this transfer is indeed possible but we have to introduce a new concept and we have to do some extra computational work. First of all it turns out that once portfolio constraints are taken into account the requirement that financial markets admit no arbitrage is too restrictive. We argue that the appropriate criterion we have to use in a world with frictions, is a concept which we call no unlimited arbitrage. Constraints can lead to situations where not all arbitrage opportunities are eliminated in equilibrium because the constraints prevent investors to fully take advantage of them. In parallel to the frictionless world we can characterize the requirement that financial markets admit no unlimited arbitrage by the existence of certain state prices. Unfortunately, contrary to the unconstrained world we are not ready for pricing income streams after we have obtained such a characterization of no unlimited arbitrage. We show that for any income stream that can be replicated we have to find among all the candidate state prices, the correct ones for a particular income stream. We propose a computationally simple procedure which is able to accomplish this task. It can be set up from the basic data of security prices, financial contracts with their payoff structure and the relevant constraints on feasible portfolio positions. We are thus able to present a pricing theory for arbitrary contingent claims that can be replicated by existing securities under constraints without requiring any particular knowledge of investor utility functions beyond some general assumptions on the structure of preferences.
Since it is our aim to analyze and clarify some of the conceptual questions that arise in transferring arguments in the spirit of no?arbitrage to a framework where investors are constrained in their potential portfolio holdings, we have decided to use a framework, which has the minimal structure that is able to address the issues in a meaningful way. The reader initiated to modern asset pricing theory and security pricing might thus perhaps miss the rich stochastic structure which has become a trade mark of this literature. We present our arguments in a framework that is stripped to the bare essentials to convey the basic logic of pricing contingent claims under constraints. Our results do however not depend on the simplified framework and can easily be generalized to richer setups.
The paper is organized as follows: Since at first sight all the different contributions to the pricing problem under constraints seem to offer their own (idiosyncratic) approach we have decided to start in section 2 with a discussion of the literature to put the papers including our own contribution into perspective. Section 3 gives an exposition of the model and introduces the formal description of constraints along with some examples. Section 4 describes the feasible income transfers in financial markets with portfolio constraints. Section 5 characterizes no unlimited arbitrage in terms of state prices. Section 6 demonstrates how this characterization can be used to price an arbitrary contingent claim by no unlimited arbitrage. Section 7 contains results which show the connections between the no arbitrage pricing approach and optimal investor decision problems. The final section 8 concludes. All proofs are in the appendix.
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Arbitrage and Optimal Portfolio Choice with Financial Constraints
