Paper: arXiv 2412.13669

Authors: Jintao Li, Shuaijie Qian

Abstract

We consider Merton’s problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries, the buy boundary and the sell boundary, between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical contribution to continuous-time portfolio selection with proportional transaction costs by proving the monotonicity of cost-adjusted trading boundaries via HJB PDEs and obstacle problems. It is entirely mathematical with no empirical data or backtesting, placing it squarely in the Lab Rats quadrant.

Research Flowchart

  flowchart TD
    A["Research Goal: Analyze impact of proportional<br>transaction costs on trading boundaries"] --> B["Inputs: Merton's problem parameters<br>(finite horizon, equities, cost rates)"]
    B --> C["Methodology: Formulate optimal portfolio selection<br>and characterize no-trading region"]
    C --> D["Comparative Statics Analysis:<br>Examine boundary movement vs. cost rates"]
    D --> E["Outcome 1: Cost-adjusted trading boundaries<br>are monotone in transaction costs"]
    D --> F["Outcome 2: Merton line lies in no-trading region<br>when positive, bounded by adjusted boundaries"]

Cited References (13)

  1. Optimum consumption and portfolio rules in a continuous-time model (1971)
  2. Portfolio selection with transactions costs (1976)
  3. Optimal Investment and Consumption with Transaction Costs (1994)
  4. Portfolio Selection with Transaction Costs (1990)
  5. Optimal Portfolio Selection with Transaction Costs and Finite Horizons (2002)
  6. Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem (2009)
  7. Finite Horizon Optimal Investment and Consumption with Transaction Costs (2009)
  8. Optimal consumption and investment under transaction costs (2019)
and 5 more
  1. Controlled Markov Processes and Viscosity Solutions (2006)
  2. A note on finite horizon optimal investment and consumption with transaction costs (2016)
  3. Penalty Methods for Continuous-Time Portfolio Selection with Proportional Transaction Costs (2010)
  4. Dynamic Mean-Variance Portfolio Selection with Transaction Costs (2024)
  5. Asymptotic analysis for optimal investment and consumption with transaction costs (2004)

0 of 13 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.