Stochastic Portfolio Theory Reading Seminar

Fall 2021


Welcome to the Stochastic Portfolio Theory Reading Seminar, run by the students of Columbia University.

This Fall we we will be studying Portfolio Theory and Arbitrage: A Course in Mathematical Finance by I. Karatzas and C. Kardaras. Our talks will be held in hybrid form over Zoom and in Columbia University on Fridays from 4p.m. to 5p.m. EDT.

This seminar is the continuation of the same seminar held in Summer - SPT Seminar, Summer 2021.

If you would like to come to our seminars or to be added on the mailing list, please email ggaitsgori@math.columbia.edu.

Past Seminars

Date and time Speaker Title and abstract
Friday, October 1, 4:00p.m. EDT Richard Groenewald Functionally Generated Trading Strategies in Markets of General Continuous Semimartingales

We discuss generalizing the approach taken in Fernholz's Stochastic Portfolio Theory text to markets whose stock price processes are arbitrary continuous semimartingales, and outline an alternative "additive" portfolio generation. We will also introduce sufficient conditions for both long and short-term arbitrage with respect to the market portfolio, and review some specific examples.
Friday, October 8, 4:00p.m. EDT Richard Groenewald Arbitrage and a New Class of Portfolio Generating Functions

We continue to discuss sufficient conditions for both long and short term arbitrage with respect to the market portfolio in markets whose stock price processes are general continuous semimartingales. We will also develop some notions of portfolio generating functions which take inputs other than the market weights alone.
Friday, October 15, 4:00p.m. EDT Georgy Gaitsgori Stochastic Portfolio Theory - New and old definitions (PTA, Chapter 1)

We start discussing the book "Portfolio Theory and Arbitrage" by I.Karatzas and C.Kardaras. In particular, we will try to cover Chapter 1 of this book. We will revisit old definitions of stock prices, portfolios and functionally generated portfolios. We will also generalize these notions and introduce new notions of wealth process, numeraire, and others.
Friday, October 22, 4:00p.m. EDT Georgy Gaitsgori Stochastic Portfolio Theory - New and old definitions, Part 2 (PTA, Chapters 1-2.1)

We continue discussing Chapter 1 of the PTA book. We introduce new notions of wealth process, admissibility, numeraires and others. We also start Chapter 2 introducing and discussing the notion of Supermartingale numeraires.
Friday, October 29, 4:00p.m. EDT Richard Groenewald Supermartingale numeraires and market viability (PTA, Chapters 2.1-2.2)

We follow sections 2.1-2.3 of Kardaras and Karatzas (2021). We discuss the proof that a market is "viable" in the sense of not being able to fund a consumption stream starting with 0 initial capital if and only if a supermartingale numeraire exists.
Friday, November 5, 4:00p.m. EDT Hindy Drillick and Sid Mane Machine Learning for Functionally Generated Portfolios

We will discuss how machine learning methods can be used to generate portfolios that will outperform the market.
Friday, November 12, 4:00p.m. EDT Walter Schachermayer (University of Vienna) The duality theory of stretched Brownian motion

We continue the investigation of martingale transports from a measure $\mu$ to a measure $\nu$ on $\R^d$. The notion of stretched Brownian motion, as introduced by Backhoff, Beiglbock, Huesmann, and Kallblad, can be defined as the solution to this problem which is "closest to Brownian motion". We develop further the duality theory attached to this concept. As an application we show that in the case of a single invariant deMarch-Touzi component the stretched Brownian motion is of standard type. Joint work with Backhoff, Beiglbock, and Tschiderer.
Friday, November 19, 4:00p.m. EDT Richard Groenewald More on Supermartingale Numeraires and Market Viability

We follow sections 2.1-2.3 of Kardaras and Karatzas (2021). We continue to discuss the proof that a market is "viable" in the sense of not being able to fund a consumption stream starting with 0 initial capital if and only if a supermartingale numeraire exists. We further establish the uniqueness of the supermartingale numeraire and some of its optimality properties.
Friday, November 26, 4:00p.m. EDT No seminar

Friday, December 3, 4:00p.m. EDT Richard Groenewald Optimal Portfolio Generating Functions in a Discrete Time Setting

We follow the analysis of stochastic portfolio theory in discrete time by Campbell and Wong (2021) and focus on the optimal selection of a generating function over a specific class. We discuss results on empirical data and some ideas for extending this methodology.
Friday, December 10, 4:00p.m. EDT Richard Groenewald Optimality Properties of the Supermartingale Numeraire

We discuss the growth optimality, long run growth optimality and relative log optimality properties of the supermartingale numeraire, and (time permitting) the capital asset pricing model in the framework of Karatzas and Kardaras (2021).