VJE Seminar: Alain Chateauneuf (Université Paris 1)
Financial markets with hedging complements
Abstract: This paper considers arbitrage-free financial markets with bid/ask spreads. Our first contribution concerns hedging prices. It shows that for a large class of markets said to be normalized namely those with a list of independent marketed assets (IND), and another one being frictionless and satisfying a condition (CC), the hedging price under a weak arbitrage-free assumption (PAF), can be written as the sum of two terms. The first one is a generalized convex Choquet integral, in the sense that, up to an isomorphism V it is a (standard) convex Choquet integral with a tractable explicit formula. The second term is modular since it is separable in each of its variables. Let us add that generically the modular term equals zero in other words that the hedging price is a generalized convex Choquet integral.
It happens that the notion of generalized Choquet integral coincides with the notion of Choquet integral on Riesz space introduced by Cerreia-Vioglio, Maccheroni, Marinacci, Montrucchio 2015, this is proved in the Appendix. This equivalence will allow us to deduce the Put-Call parity property of the hedging price as a direct consequence.
Another contribution concerns the fact that in our framework the marketed securities are hedging complements which can be regarded related to assets in the same way as perfect complementarity on preferences and utility functions, which dates back to Fisher, Pareto, and Edgeworth, thus according to Samuelson (1974) are usually assumed to illustrate super modularity.
A related property is that if the hedging price is written as a function of its coordinates on the independent securities it tuns out to be meaningfully submodular, but a counterexample shows that if independence is deleted, sub-modularity is no longer guaranteed.
Our second contribution proves that in fact a for a much larger class of bid/ask markets that the same hedging formula as above can be applied. In actual we show that if the initial market consists of a frictionless security and a list of independent securities, but CC is not satisfied one can obtain an indistinguishable market that is with the same discount factors hence the same hedging price but also satisfying CC, furthermore since the corresponding securities will be independent and indeed PAF will be satisfied, consequently our first contribution applies.
Our third contribution concerns superhedging prices. For these cost functions under some additional properties we obtain bounds in terms of Generalized Choquet integrals.
Finally in the particular case where the independent marketed assets are disjoint event securities both the hedging and superhedging prices are proved to boil down to tractable explicit regular convex Choquet integrals.