Behavioural and Dynamical Scenarios for Contingent Claims Valuation in Incomplete Markets

Realistic markets are incomplete, due to either the existence of frictions or to the nonexistence of the necessary assets needed to achieve the complete replication of any contingent claim by linear combinations of available (traded) assets. Incomplete markets is a very interesting field of economic theory and finance, which through seminal studies (see e.g. [7], [9] and [11]) has led to important results that have helped the community to reach a deeper understanding of the function of financial markets.

In this work we study the problem of determination of prices in an incomplete market. While there is a rich literature on this field, the majority of these works focuses on the determination of bounds on the prices that are consistent with general equilibrium considerations. It is well known for instance, that in an incomplete market set up there is no longer a unique pricing kernel (martingale measure) and the existence of more than one pricing kernel may at best point out a whole band of prices that do not allow for arbitrage opportunities. The determination of a single price, at which an asset will eventually be traded in this market, out of this whole band, is still an open problem. There exists an extensive and very interesting literature on the subject, focusing on the determination of the upper and lower hedging prices (see e.g. [9]) as well as a number of suggestions on the price selected by the market (e.g. Kuhlback-Leibner or related entropy minimization criteria [8], [14]) which lead to interesting implications, some of which are testable with real market data, however, a complete theory of price selection in incomplete markets is still missing.

The aim of the present paper is to contribute to this literature providing types of scenarios on price selection in incomplete markets. The scenarios are based on behavioural considerations and lead to interesting results. We present these scenarios within a one period model setting, so that the basic concepts and underlying ideas are made clear and technical details are kept to a minimum. Then the passage to a multiperiod model should follow without any major difficulty and we plan to present it in future work together with the case of the continuous model.

The rest of the paper is organized as follows. In Section 2 we discuss the determination of the upper and lower prices of a contingent claim in an incomplete market for risk averse agents via utility pricing. Although this is not a new issue (see e.g. [6]), we include it here to fix ideas and notation. In Section 3 we address the problem of the determination of one commonly accepted price for the contingent claim with the use of three different scenarios. The first approach is a market game approach, the second is a risk sharing approach whereas the third approach is one in which the agents update their beliefs about the possible states of the world, in a way which is consistent with the minimization of total regret. In Section 4 we propose dynamic mechanisms that may describe the bargaining procedure between the agents, leading to prices consistent with any of the above scenarios, and we study their stability. Furthermore, we discuss the robustness of the previously mentioned dynamical mechanisms with respect to uncertainty. In Section 5 we summarize and conclude.

We consider a one period market model with N tradable assets a 1 , …, a N and K states of the world ω 1 , …, ω K . One of the assets is a riskless asset with return rate r and without loss of generality we may assume that this asset is a 1 . Let p = (p 1 , …, p N ) denote the price vector of the N assets at time t = 0, i.e. for each i = 1, …, N we denote by p i the price at time t = 0 of the asset a i . Let also D = (d ij ) i=1,…,K j=1,…,N denote the K × N matrix of the assets payoffs at time t = 1, that is, d ij denotes the value of the asset a j at time t = 1 when the prevailing state of the world is ω i . In the sequel we may use the notation d j (ω i ) instead of d ij and we may write d j to indicate the j-th column vector of D without these causing any confusion. We also assume that the value of the riskless asset at time t = 1 is equal to 1 no matter what the prevailing state of the world is, i.e. d 1 (ω i ) = 1 for all i = 1, …, K (clearly, p 1 = 1/(1 + r)). Moreover, without loss of generality, we assume that none of the assets a 1 , …, a N is redundant, that is, none of the columns of the payoff matrix D can be expressed as a linear combination of the remaining columns of the matrix. Furthermore we assume that N < K, that is, we face an incomplete markets situation where there are more states of the world than assets and thus the existing assets are not enough to reproduce every possible contingent claim at time t = 1.

As it is known from the theory of mathematical finance, the incompleteness of the market leads to the existence of an infinity of risk neutral measures, all of which are consistent with the absen

…(Full text truncated)…

This content is AI-processed based on ArXiv data.