arXiv subjects

Robert Fernholz

Publications and source records attributed to Robert Fernholz.

15 recordsLinked to original sources

arXiv2023

Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing

This paper presents a synthesis of the theories of portfolio generating functions and option pricing. The theory of portfolio generation is extended to measure the value of portfolios generated by positive C^{2,1} functions of asset prices X_1,... , X_n directly, rather than with respect to a numeraire portfolio. If a portfolio generating function satisfies a specific partial differential equation, then the value of the portfolio generated by that function will replicate the value of the function. This differential equation is a general form of the Black-Scholes equation. Similar results apply to contingent claim functions, which are portfolio generating functions that are homogeneous of degree 1. With the addition of a riskless asset, an inhomogeneous portfolio generating function V : R^{+n} x [0, T] \to R^+ can be extended to an equivalent contingent claim function \hat{V} : R^+ x R^{+n} x [0, T] \to R^+ that generates the same portfolio and is replicable if and only if V is replicable. Several examples are presented.

q-fin.PR↗

arXiv2020

Permutation-Weighted Portfolios and the Efficiency of Commodity Futures Markets

A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally generated only for markets with two assets (except for the identity permutation). A reverse-weighted portfolio is a portfolio in which the asset with the greatest market weight is assigned the smallest market weight, the asset with the second-largest weight is assigned the second-smallest, and so forth. Although the reverse-weighted portfolio in a market with four or more assets is not functionally generated, it is still possible to characterize its long-term behavior using rank-based methods. This result is applied to a market of commodity futures, where we show that the reverse price-weighted portfolio substantially outperforms the price-weighted portfolio from 1977-2018.

q-fin.MF↗

arXiv2018

Diversification, Volatility, and Surprising Alpha

It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies. Indeed, in the U.S. stock market, equal-weighted portfolios, random-weighted portfolios, and other naive, non- optimized portfolios tend to outperform a capitalization-weighted index over the long term. This outperformance is generally attributed to beneficial factor exposures. Here, we provide a deeper, more general explanation of this phenomenon by decomposing portfolio log-returns into an average growth and an excess growth component. Using a rank-based empirical study we argue that the excess growth component plays the major role in explaining the outperformance of naive portfolios. In particular, individual stock growth rates are not as critical as is traditionally assumed.

q-fin.PM↗

arXiv2018

Numeraire markets

In a stock market, the numeraire portfolio, if it exists, is the portfolio with the highest expected logarithmic growth rate at all times. A numeraire market is a stock market for which the market portfolio is the numeraire portfolio. We study open markets, markets comprising the higher capitalization stocks within a broad equity universe. The stocks we consider are represented by continuous semimartingales, and we construct an example of a numeraire market that is asymptotically stable.

q-fin.MF↗

arXiv2017

Zipf's Law for Atlas Models

A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

econ.GN↗

arXiv2017

Stratonovich representation of semimartingale rank processes

Suppose that $X_1, \ldots , X_n$ are continuous semimartingales that are reversible and have nondegenerate crossings. Then the corresponding rank processes can be represented by generalized Stratonovich integrals, and this representation can be used to decompose the relative log-return of portfolios generated by functions of ranked market weights.

math.PR↗

arXiv2016

A new decomposition of portfolio return

For a functionally generated portfolio, there is a natural decomposition of the relative log-return into the log-change in the generating function and a drift process. In this note, this decomposition is extended to arbitrary stock portfolios by an application of Fisk-Stratonovich integration. With the extended methodology, the generating function is represented by a structural process, and the drift process is subsumed into a trading process that measures the profit and loss to the portfolio from trading.

q-fin.MF↗

arXiv2016

A Rank-Based Approach to Zipf's Law

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is -1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.

econ.GN↗

arXiv2015

Variations on an example of Karatzas and Ruf

Markets composed of stocks with capitalization processes represented by positive continuous semimartingales are studied under the condition that the market excess growth rate is bounded away from zero. The following examples of these markets are given: i) a market with a singular covariance matrix and instantaneous relative arbitrage; ii) a market with a singular covariance matrix and no arbitrage; iii) a market with a nonsingular covariance matrix and no arbitrage; iv) a market with a nonsingular covariance matrix and relative arbitrage over an arbitrary time horizon.

q-fin.MF↗

arXiv2015

An example of short-term relative arbitrage

Long-term relative arbitrage exists in markets where the excess growth rate of the market portfolio is bounded away from zero. Here it is shown that under a time-homogeneity hypothesis this condition will also imply the existence of relative arbitrage over arbitrarily short intervals.

q-fin.MF↗

arXiv2015

Identification of Atlas models

Atlas models are systems of Ito processes with parameters that depend on rank. We show that the parameters of a simple Atlas model can be identified by measuring the variance of the top-ranked process for different sampling intervals.

q-fin.MF↗

arXiv2013

A second-order stock market model

A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First- and second-order models exhibit stability properties that make them appropriate as a backdrop for the analysis of the idiosyncratic behavior of individual stocks. Methods for the estimation of the parameters of second-order models are developed in this paper.

q-fin.ST↗

arXiv2011

Hybrid Atlas models

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.

math.PR↗

arXiv2008

Diversity and relative arbitrage in equity markets

A financial market is called "diverse" if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Ito-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of "weak diversity" and "asymptotic weak diversity") in precise terms. We show that diversity is possible to achieve, but delicate. Several illustrative examples are provided, which demonstrate that weakly-diverse financial markets contain relative arbitrage opportunities: it is possible to outperform (or underperform) such markets over sufficiently long time-horizons, and to underperform them significantly over arbitrary time-horizons. The existence of such relative arbitrage does not interfere with the development of option pricing, and has interesting consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study.

q-fin.PM↗

arXiv2006

Atlas models of equity markets

Atlas-type models are constant-parameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this paper we study the basic properties of this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock.

math.PR↗