Prove Proposition 6.15. Proposition 6.15 Given the price-dividend couple of processes (S,D), the follow- ing are equivalent:

Prove Proposition 6.15.

Transcribed Image Text:

Proposition 6.15 Given the price-dividend couple of processes (S,D), the follow- ing are equivalent: (i) the market is dynamically complete; (ii) for every t = 0,1,..., T-1 and A, E F, it holds that rank (P+1(A+)) V(A); (iii) all the v, Arrow securities are attainable. Part (ii) of this proposition makes clear that market completeness is strongly related to the temporal resolution of uncertainty. Indeed, if the uncertainty is gradually resolved over time (i.e., the number of distinct branches leaving each node of the tree is small), then it suffices to trade in a small number of long-lived securities in order to achieve market completeness. On the contrary, if the uncertainty is resolved in a more abrupt way (i.e., there exist nodes from which many distinct branches of the tree originate), then many securities are needed to reach market completeness. This also reflects the fact that in a dynamic setting agents learn over time that the true state of the world belongs to a refining sequence of events and, hence, adjust their strategies according to a learning process. However, unlike in the case of the single-period economy considered in Chap. 4, dynamic completeness does not only depend on the asset dividends but also on their (endogenous) price processes.