Mathematical Finance Seminar – Fall 2019

Schedule for Fall 2019

Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Location: Columbia University, 903 SSW (1255 Amsterdam Ave, between 121st and 122nd Street)

Organizers: Ruimeng Hu, Ioannis Karatzas, Marcel Nutz, Philip Protter

MAFN Seminar Archive


Ioannis Karatzas (Columbia)

Arbitrage Theory via Numeraires

We develop a mathematical theory for finance based on the following “viability” principle: That it should not be possible to fund a non-trivial liability starting with arbitrarily small initial capital. In the context of continuous asset prices modeled by semimartingales, we show that proscribing such egregious forms of what is commonly called “arbitrage” (but allowing for the possibility that one portfolio might outperform another), turns out to be equivalent to any one of the following conditions: (i) a portfolio with the local martingale numeraire property exists, (ii) a growth-optimal portfolio exists, (iii) a portfolio with the log-optimality property exists, (iv) a local martingale deflator exists, (v) the market has locally finite maximal growth.

We assign precise meaning to these terms, and show that the above equivalent conditions can be formulated entirely, in fact very simply, in terms of the local characteristics (the drifts and covariations) of the underlying asset prices. Full-fledged theories for hedging and for portfolio/consumption optimization can be developed in such a setting, as can the important notion of “market completeness”.

The semimartingale property of asset prices is shown to be necessary and sufficient for viability, when investment is constrained to occur only along a discrete-time schedule, and to be long-only. When a strictly positive martingale (as opposed to local martingale) deflator exists, so does an equivalent martingale measure (EMM) on each time-horizon of finite length. We show that EMMs need not exist in a viable market, and that this notion is highly normative: Two markets may have the exact same local characteristics, while one of them admits such an EMM and the other does not.

(Joint work—book in progress by the same title—with Constantinos Kardaras.)