Financial/Actuarial Mathematics Seminar

Given an equity market with n stocks, a pseudo-arbitrage is an investment strategy (i.e. a portfolio map) which outperforms the market portfolio (i.e. the buy-and-hold option) almost surely in the long run. When the market weights evolve via some unknown discrete time process, Fernholz proved that such portfolio maps exist, under mild and realistic assumptions. Recently, Pal and Wong showed that the problem of finding pseudo-arbitrages is equivalent to solving a certain Monge-Kantorovich optimal transport problem where the cost function is given by the so-called "diversification return," which is closely related to the free energy in statistical physics. In our work, we study the regularity theory for these maps. In other words, we consider the question "If the market conditions change slightly, does the investment portfolio also change in a continuous way?" By addressing this problem, an unexpected connection to Kähler geometry emerges. This provides a new geometric interpretation for the regularity theory of optimal transport. Speaker(s): Gabriel Khan (UM)