A Price-Based Framework for Stochastic Portfolio Theory

2026-10-03

AI Summary

This study addresses the challenges of weight discontinuities and benchmark evaluation induced by stock splits within stochastic portfolio theory. It constructs an analytical framework based on nominal price weights, handling adjustment events through a modified self-financing condition to establish rigorous correspondence with market-capitalization frameworks. Methodologically, this work proposes a jump-corrected wealth decomposition technique, employing semimartingale market models, entropy methods, and generating functions for theoretical derivation alongside empirical validation on NYSE data. The findings reveal the failure mechanism of relative arbitrage under repeated splits, establish strategy-independent criteria for benchmark conversion, and elucidate the fundamental distinctions between price-weighted and capitalization-weighted benchmarks, thereby offering novel perspectives for long-term investment strategies.

Abstract

We develop a price-based framework for stochastic portfolio theory in which trading strategies are generated from nominal price weights and evaluated relative to a price-weighted benchmark. Stock splits and reverse splits induce jumps in the generating weights without changing the value of existing investments. In a semimartingale market with predictable adjustment events, we incorporate the corresponding share adjustments into the self-financing condition and construct additively and multiplicatively generated strategies with explicit jump-corrected wealth decompositions. For additive generation, an entropy-based example exhibits relative wealth tending to $-\infty$ almost surely under repeated splits, demonstrating why the usual argument for long-horizon relative arbitrage does not extend directly. We also show that advance adjustment information alone provides no model-free guarantee of improved performance for functionally generated portfolios. For multiplicative generation, wealth-preserving restarts retain the classical portfolio allocation evaluated at the current price weights. We also establish a correspondence with the capitalization-based framework that preserves absolute wealth and yields a strategy-independent benchmark conversion. We illustrate the framework using daily NYSE data, comparing price- and capitalization-weighted benchmarks and the corresponding diversity-weighted portfolios across price- and capitalization-selected universes.

Problem

• Stochastic Portfolio Theory
• Price-Based Framework
• Stock Splits
• Functionally Generated Portfolios
• Self-Financing Condition

Innovation

• Stochastic Portfolio Theory
• Price-Based Framework
• Functionally Generated Portfolios
• Jump-Corrected Wealth Decomposition
• Semimartingale Market