A Two-Factor Model for Capturing Momentum and Mean Reversion in Stock Returns


Financial modeling has long sought to develop frameworks that accurately capture the complex dynamics of asset prices. Traditional models often focus on either momentum or mean reversion effects, struggling to incorporate both simultaneously. In this blog post, we introduce a two-factor model that aims to address this issue by integrating both momentum and mean reversion effects within the stochastic processes governing stock prices.

The development of the two-factor model is motivated by the empirical observation that financial markets exhibit periods of persistent trends (momentum) and reversion to historical means or intrinsic values (mean reversion). Capturing both effects within a single framework has been a challenge in financial econometrics. The proposed model seeks to tackle this challenge by incorporating momentum and mean reversion effects within a unified framework.

The two-factor model consists of two main components: a drift factor and a mean-reverting factor. The drift factor, denoted as d μ(t), represents the long-term trend or momentum of a stock’s price. It incorporates a constant drift parameter θ, reflecting the underlying direction driven by broader market forces or fundamental changes. The mean-reverting factor, denoted as d θt, captures the short-term deviations from the drift. It is characterized by a mean-reversion speed κ, which determines the rate at which prices revert to their long-term equilibrium following temporary fluctuations. These factors are influenced by their respective volatilities (σμ, σθ) and driven by correlated Wiener processes, allowing the model to reflect the interaction between momentum and mean reversion observed in markets

To demonstrate the model’s application, the research applies the two-factor framework to daily returns data of Coca-Cola (KO) and PepsiCo (PEP) over a twenty-year period. This empirical analysis explores the model’s potential for informing pairs trading strategies. The parameter estimation process employs a maximum likelihood estimation (MLE) technique, adapted to handle the specifics of fitting a two-factor model to real-world data. This approach aims to ensure accuracy and adaptability, enabling the model to capture the evolving dynamics of the market.

The introduction of the two-factor model contributes to the field of quantitative finance by providing a framework that incorporates both momentum and mean reversion effects. This approach can lead to a more comprehensive understanding of asset price dynamics, potentially benefiting risk management, asset allocation, and the development of trading strategies. The model’s insights may be particularly relevant for pairs trading, where identifying relative mispricings between related assets is important.

The two-factor model presented in this blog post offers a new approach to financial modeling by integrating momentum and mean reversion effects. The model’s empirical application to Coca-Cola and PepsiCo demonstrates its potential for informing trading strategies. As quantitative finance continues to evolve, the two-factor model may prove to be a useful tool for researchers, practitioners, and investors seeking to understand the dynamics of financial markets.

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Modeling Asset Processes

Introduction

Over the last twenty five years significant advances have been made in the theory of asset processes and there now exist a variety of mathematical models, many of them computationally tractable, that provide a reasonable representation of their defining characteristics.

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While the Geometric Brownian Motion model remains a staple of stochastic calculus theory, it is no longer the only game in town.  Other models, many more sophisticated, have been developed to address the shortcomings in the original.  There now exist models that provide a good explanation of some of the key characteristics of asset processes that lie beyond the scope of models couched in a simple Gaussian framework. Features such as mean reversion, long memory, stochastic volatility,  jumps and heavy tails are now readily handled by these more advanced tools.

In this post I review a critical selection of asset process models that belong in every financial engineer’s toolbox, point out their key features and limitations and give examples of some of their applications.


Modeling Asset Processes

Stochastic Calculus in Mathematica

Wolfram Research introduced random processes in version 9 of Mathematica and for the first time users were able to tackle more complex modeling challenges such as those arising in stochastic calculus.  The software’s capabilities in this area have grown and matured over the last two versions to a point where it is now feasible to teach stochastic calculus and the fundamentals of financial engineering entirely within the framework of the Wolfram Language.  In this post we take a lightening tour of some of the software’s core capabilities and give some examples of how it can be used to create the building blocks required for a complete exposition of the theory behind modern finance.

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Conclusion

Financial Engineering has long been a staple application of Mathematica, an area in which is capabilities in symbolic logic stand out.  But earlier versions of the software lacked the ability to work with Ito calculus and model stochastic processes, leaving the user to fill in the gaps by other means.  All that changed in version 9 and it is now possible to provide the complete framework of modern financial theory within the context of the Wolfram Language.

The advantages of this approach are considerable.  The capabilities of the language make it easy to provide interactive examples to illustrate theoretical concepts and develop the student’s understanding of them through experimentation.  Furthermore, the student is not limited merely to learning and applying complex formulae for derivative pricing and risk, but can fairly easily derive the results for themselves.  As a consequence, a course in stochastic calculus taught using Mathematica can be broader in scope and go deeper into the theory than is typically the case, while at the same time reinforcing understanding and learning by practical example and experimentation.