As a firm founded on research, INTECH is committed to furthering knowledge through insightful thought leadership and academic papers. Our multidisciplinary team provides insightful and diverse research and commentary.

As a firm founded on research, INTECH is committed to furthering knowledge through insightful thought leadership and academic papers. Our multidisciplinary team provides insightful and diverse research and commentary.

**Semimartingales on Rays, Walsh Diffusions, and Related Problems of Control and Stopping**

Ioannis Karatzas, Ph.D. and Minghan Yan, Ph.D.

September 12, 2016

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of functions. Special cases of such planar processes are diffusions which choose, once they reach the origin, the rays for the subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results up to an explosion time for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of control with discretionary stopping involving Walsh diffusions.

E. Robert Fernholz, Ph.D., Ioannis Karatzas, Ph.D. and Johannes Ruf, Ph.D.

August 23, 2016

At least since Fernholz (2002), it has been known that volatility in a stock market can generate arbitrage, or at least *relative arbitrage* between a specific portfolio and the market portfolio. However, the questions of exactly what level of volatility is required, and how long it might take, for this arbitrage to be realized, have never been fully answered. Here we hope to shed some light on these questions and come to an understanding about what might represent adequate volatility, and over which time-frame relative arbitrage might be achieved.

**Trading Strategies Generated by Lyapunov Functions**

Ioannis Karatzas, Ph.D. and Johannes Ruf, Ph.D.

July 21, 2016

Functional portfolio generation, initiated by E.R. Fernholz almost twenty years ago, is a methodology for constructing trading strategies with controlled behavior. It is based on very weak and descriptive assumptions on the covariation structure of the underlying market model, and needs no estimation of model parameters. From a probabilistic point of view, the present paper yields results concerning the interplay of stochastic discount factors and concave transformations of semimartingales on compact domains.

Ioannis Karatzas, Ph.D. and Johannes Ruf, Ph.D.

February 4, 2016

We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in pathwise sense.

**Portfolio Optimization in the Stochastic Portfolio Theory Framework**

Vassilios Papathanakos, Ph.D.

January 29, 2016

Portfolio optimization is a fundamental concept in investing, but it presents many technical challenges. The two most important issues are the imperfect nature of the estimates of the market characteristics, and the ambiguity of the optimization objective. This paper discusses some theoretical results with a view to motivate some practical choices in portfolio optimization.

**Trading-Profit Attribution for the Size Factor**

Vassilios Papathanakos, Ph.D.

January 29, 2016

In 2015, INTECH Investment Management LLC introduced a novel approach to attribution that focuses on the estimation of the trading profit captured through systematic rebalancing. The applicability of this approach extends to a broad range of portfolios, potentially including the most diversified strategies exhibiting regular reconstitution and rebalancing. In this paper, we extend the trading-profit attribution methodology to analyze the size factor.

**Variations on an Example of Karatzas and Ruf**

Robert Fernholz, Ph.D.

December 9, 2015

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.

**An Example of Short-term Relative Arbitrage**

Robert Fernholz, Ph.D.

October 9, 2015

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.

**Stochastic Integral Equations for Walsh Semimartingales**

Tomoyuki Ichiba, Ph.D., Ioannis Karatzas, Ph.D., Vilmos Prokaj, Ph.D. and Minghan Yan, Ph.D.

May 11, 2015

We construct a class of planar semimartingales which includes the Walsh Brownian motion as a special case, and derive stochastic integral equations and a change-of-variable formula for these so-called “Walsh semimartingales.”

**Optional Decomposition for Continuous Semimartingales Under Arbitrary Filtrations**

Ioannis Karatzas, Ph.D. and Constantinos Kardaras, Ph.D.

2015

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations.

**Diversity-Weighted Portfolios with Negative Parameter**

Alexander Vervuurt, Ph.D. and Ioannis Karatzas, Ph.D.

2015

We analyze a negative-parameter variant of the diversity-weighted portfolio studied by Fernholz, Karatzas, and Kardaras (Finance Stoch 9(1):1-27, 2005), which invests in each company a fraction of wealth inversely proportional to the company’s *market weight *(the ratio of its capitalization to that of the entire market).

**Distribution of the Time to Explosion for One-Dimensional Diffusions**

Ioannis Karatzas, Ph.D. and Johannes Ruf, Ph.D.

October 2014

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales.

**Diverse Market Models of Competing Brownian Particles with Splits and Mergers**

Ioannis Karatzas, Ph.D. and Andrey Sarantsev, Ph.D.

We study models of regulatory breakup, in the spirit of Strong and Fouque (2011) but with a fluctuating number of companies.

**On the One-Sided Tanaka Equation with Drift**

Ioannis Karatzas, Ph.D., Albert N. Shiryaev, Ph.D., Mykhaylo Shkolnikov, Ph.D.

We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift.

E. Robert Fernholz, Ph.D., Ioannis Karatzas, Ph.D., Tomoyuki Ichiba, Ph.D., and Vilmos Prokaj, Ph.D.

We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that these involve singularly continuous components governed by local time.

**Strong Solutions of Stochastic Equations with Rank-Based Coefficients**

Tomoyuki Ichiba, Ph.D., Ioannis Karatzas, Ph.D. and Mykhaylo Shkolnikov, Ph.D.

We study finite and countably infinite systems of stochastic differential equations, in which the drift and diffusion coefficients of each component (particle) are determined by its rank in the vector of all components of the solution.

**Systems of Brownian Particles with Asymmetric Collisions**

Ioannis Karatzas, Ph.D., Soumik Pal, Ph.D., and Mykhaylo Shkolnikov, Ph.D.

We study systems of Brownian particles on the real line which interact by splitting the local times of collisions among themselves in an asymmetric manner.

**Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions**

E. Robert Fernholz, Ph.D., Tomoyuki Ichiba, Ph.D., and Ioannis Karatzas, Ph.D.

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients,and with a full range of patterns of behavior upon collision that range from totally frictionless interaction,to elastic collision, to perfect reflection of one particle on the other.

** A Second-Order Stock Market Model**

E. Robert Fernholz, Ph.D., Tomoyuki Ichibay, Ph.D., and Ioannis Karatzas, Ph.D.

February 2012

First-order and second-order stock market models are relatively simple stochastic models that manifest some of the stable properties of actual stock market behavior.

**Diffusions with Rank-based Characteristics and Values in the Nonnegative Quadrant**

Tomoyuki Ichiba, Ph.D., Ioannis Karatzas, Ph.D. and Vilmos Prokaj, Ph.D.

February 2012

We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of couple Skorokhod reflection equations, then “unfolding” the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. **79** (2009) 534-536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited.

**Optimal Arbitrage Under Model Uncertainty**

Ioannis Karatzas, Ph.D. and Daniel Fernholz Ph.D.

December 2010

In an equity market model with “Knightian” uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon and under any admissible configuration of model parameters that might materialize.

**Probabilistic Aspects of Arbitrage**

Daniel Fernholz, Ph.D. and Ioannis Karatzas, Ph.D.

November 2009

The pioneering work of Fernholz demonstrated that, under appropriate conditions, it is possible to systematically outperform a market portfolio over sufficiently long-term horizons.

Tomoyuki Ichiba, Ph.D., Vassilios Papathanakos, Ph.D., Adrian Banner, Ph.D., Ioannis Karatzas, Ph.D., and Robert Fernholz, Ph.D.

November 2009

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stability of the capital distribution.

**The Effect of Value Estimation Errors On Portfolio Growth Rates**

Robert Ferguson, Ph.D., Dean Leistikow, Ph.D., Joel Rentzler, Ph.D., and Susana Yu, Ph.D.

September 2009

This article examines how value estimation errors affect the growth rates and relative growth rates for the following four portfolio weighting methods: capitalization weights, estimation error independent weights, fundamental weights and diversity weights.

Daniel Fernholz, Ph.D. and Ioannis Karatzas, Ph.D.

June 2008

In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using non-anticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model.

**Stochastic Portfolio Theory: An Overview**

Robert Fernholz, Ph.D. and Ioannis Karatzas, Ph.D.

November 2006

Stochastic Portfolio Theory is a flexible framework for analyzing portfolio behavior and equity market structure.

**Short-Term Relative Arbitrage in Volatility-Stabilized Markets**

Adrian Banner, Ph.D. and Daniel Fernholz

September 2006

We answer in the affirmative the following open question posed in Fernholz & Karatzas (2005): Do there exist relative arbitrage opportunities over arbitrarily short time horizons in the context of certain volatility-stabilized market models?

** A Forecasting Model for Stock Market Diversity**

Francesco Audrino, Ph.D., Robert Fernholz, Ph.D., and Roberto G. Ferretti, Ph.D.

March 2006

We apply the recently introduced general tree-structured (GTS) model to the analysis and forecast of stock market diversity.

**Portfolio Growth Rates In the Presence of Value Estimation Error**

Robert Ferguson, Ph.D., Dean Leistikow, Ph.D., Joel Rentzler, Ph.D., and Susana Yu, Ph.D.

January 2006 (Revised November 2008)

This paper analyzes the impact of value estimation error on portfolios’ growth rates.

**Local Times Of Ranked Continuous Semimartingales**

Adrian Banner, Ph.D. and Raouf Ghomrasni, Ph.D.

December 2005

Given a finite collection of continuous semimartingales, we derive a semimartingale decomposition of the corresponding ranked (order-statistics) processes. We apply the decomposition to extend the theory of equity portfolios generated by ranked market weights to the case where the stock values admit triple points.

**The Implied Liquidity Premium for Equities**

Robert Fernholz, Ph.D. and Ioannis Karatzas, Ph.D.

May 2005

Over the long term, the returns on smaller stocks are likely to be higher than the returns on larger stocks. This phenomenon has been called the size effect, and a number of explanations have been proposed to account for it.

Robert Fernholz, Ph.D.

January 2005

Stock market diversity, first considered in Fernholz (1999), is a measure of the distribution of capital in an equity market.

**Relative Arbitrage in Volatility-Stabilized Markets**

Robert Fernholz and Ioannis Karatzas

October 2004

We provide simple, easy-to-test criteria for the existence of relative arbitrage in equity markets.

**Diversity and Relative Arbitrage in Equity Markets**

Robert Fernholz, Ioannis Karatzas, Constantinos Kardaras

January 2004

An equity market is called “diverse” if no single stock is ever allowed to dominate the entire market in terms of relative capitalization.

Robert Fernholz, Ioannis Karatzas

December 2003

Miller and Modigliani (1961) showed that the value of a company as an investment should be independent of whether or not the company paid dividends.

**Stable Models for the Distribution of Equity Capital**

Robert Fernholz

February 2001

The distribution of capital, or equivalently, the distribution of firm size, is studied in the context of equity markets modeled in continuous time.

**Equity Portfolios Generated by Functions of Ranked Market Weights**

Robert Fernholz

December 1998

Revised July 2000

Dynamic equity portfolios can be generated by positive twice continuously differentiable functions of the ranked capitalization weights of an equity market.

**The Size Factor In Equity Returns**

Robert Fernholz

February 2000

Company size is known to be an important factor affecting equity returns, but statistical estimation of the impact that this factor has on returns is complicated by its unstable and nonlinear nature.

**Factorization of Equity Returns**

Robert Fernholz

August 1999

The relative return of an equity portfolio with respect to the market is factored into three components.

**Manager Performance and the Diversity Cycle**

Robert Fernholz and Robert Garvy

1999

The S&P 500 Index is frequently used as a benchmark for equity managers’ performance.

Robert Fernholz

1998

Diversity is a measure of the spread of capital across an equity index or market.

**Internal Estimation of Leakage **

Robert Fernholz

October 1998

Leakage in a large-cap equity index occurs when some of the stocks in the index decline in capitalization and are subsequently replaced by larger capitalization stocks from the ambient stock universe.

Robert Fernholz

March 1998

Suppose that an equity market is composed of stocks that do not pay dividends.

**Antitrust and the No-Arbitrage Hypothesis**

Robert Fernholz

December 1997

Revised January 1998

Suppose that a continuously traded equity market is compliant with a weak form of antitrust regulation that prevents the concentration of practically all the market capital into a single company.

**Turnover in the INTECH Diversity Index**

Robert Fernholz

November 1996

The INTECH Diversity IndexSM is a functionally generated portfolio of the stocks in the S&P 500® Index.

**Leakage in Diversity Weighted Index Portfolios**

Robert Fernholz

November 1996

The fundamental theorem on mathematically generated portfolios provides a relationship between changes in the value of the generating function of a portfolio and the portfolio’s performance relative to the capitalization weighted market portfolio on which it is based.

Robert Fernholz

November 1996

Measures of diversity generate diversity-weighted indices.

**Portfolio Generating Functions**

Robert Fernholz

December 1995

Revised June 1998

A general method is presented for constructing dynamic equity portfolios through the use of mathematical generating functions.

**Stochastic Portfolio Theory and Stock Market Equilibrium**

Robert Fernholz, Brian Shay

May 1982

Harry Markowitz [1952, 1956, 1959] developed a theory of portfolio selection based on the optimization of a quadratic function subject to linear constraints.