Behavioural and dynamical scenarios for contingent claims valuation in incomplete markets

Abstract

Abstract. We study the problem of determination of asset prices in an incomplete market proposing three different but related scenarios. One scenario uses a market game approach whereas the other two are based on risk sharing or regret minimizing considerations. Dynamical schemes modeling the convergence of the buyer's and of the seller's prices to a unique price are proposed. MSC2000 : 91B24, 91B26, 60H30, 37N99

A projected gradient dynamical system modelling the dynamics of bargaining

“…) is satisfied and trade takes place. The agents are willing to deviate as little as possible from their original beliefs [4] therefore it is natural to assume that (Q * A , Q * B ) is as close as possible to (Q A (0), Q B (0)). This remark indicates that the choice of (Q * A , Q * B ) can be obtained as the solution of a properly chosen optimization problem.…”

“…Note that the dual problem (2.3) corresponds to maximizing the difference of the reservation prices by the buyer and by the seller, thus ensuring the trade of the contingent claim, under the constraint that the agents beliefs do not change more than a fixed value as measured by the functions ψ A and ψ B [4]. As is well known, duality implies that the solutions of the two problems above are intimately related.…”

“…Let U ′ β , ψ ′ β , Φ ′ be Lipschitz continuous and assume that r β ∈ R β where R β is a bounded subset of R + 0 . Then, (i) The discrete-time dynamical system defined implicitly by (4.20) is well posed 4 . (ii) the functional L λ P has Lipschitz continuous partial derivatives; (iii) there exists α 0 > 0 such that for every 0 < α < min{1, α 0 } the map F : D → D implicitly defined by F (x) := (1 − α)x + αΠ D [x − ∇ 1 L λ P (F (x), x)] is Lipschitz continuous.…”

Contingent claim pricing through a continuous time variational bargaining scheme

“…The ultimate goal of such variational pricing scheme is to provide a novel behavioural explanation for the pricing of contingent claims and similar financial assets, traded in realistic setups leading to market incompleteness. Therefore, the approach developed here, extending previous work in Azevedo et al (2013), Boukas et al (2011), Pinheiro et al (2013), Xanthopoulos and Yannacopoulos (2008), is an alternative point of view to the pricing of contingent claims in incomplete markets, a relevant problem in financial mathematics. Recall that incompleteness of the market can arise from all sorts of market imperfections and, in particular, it may be due to the non-existence of a large enough number of assets in the market so that all contingent claims can be hedged, to lack of liquidity in the financial markets, to taxation rules and transaction costs, among other reasons.…”