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Optimal Investment under the Influence of Decision-changing Imitation

Huisheng Wang, H. Vicky Zhao

TL;DR

This paper investigates how decision-changing imitation from a leading expert influences a retail investor's dynamic investment decisions in a two-agent, continuous-time market with a risk-free and a risky asset. It introduces an integral disparity $D(\\dot{P}_1,\\dot{\bar{P}}_2)$ to penalize differences in decision-changing rates and solves the retail investor's problem via a variational approach, yielding a general solution for $P_1^*(t)$ expressed through special functions and integral constants. Two boundary-condition cases are analyzed (fixed endpoints vs. derivative-boundary), with an iterative procedure to compute the integral constants, and asymptotic results are derived for $\theta\to\infty$ to characterize the influence of imitation. Numerical experiments on real stock data validate the theory and illustrate how imitation shifts the retail investor's risk posture depending on the relative risk aversions $\alpha_1$ and $\alpha_2$, offering insights into how to guide investor decisions in practice.

Abstract

Decision-changing imitation is a prevalent phenomenon in financial markets, where investors imitate others' decision-changing rates when making their own investment decisions. In this work, we study the optimal investment problem under the influence of decision-changing imitation involving one leading expert and one retail investor whose decisions are unilaterally influenced by the leading expert. In the objective functional of the optimal investment problem, we propose the integral disparity to quantify the distance between the two investors' decision-changing rates. Due to the underdetermination of the optimal investment problem, we first derive its general solution using the variational method and find the retail investor's optimal decisions under two special cases of the boundary conditions. We theoretically analyze the asymptotic properties of the optimal decision as the influence of decision-changing imitation approaches infinity, and investigate the impact of decision-changing imitation on the optimal decision. Our analysis is validated using numerical experiments on real stock data. This study is essential to comprehend decision-changing imitation and devise effective mechanisms to guide investors' decisions.

Optimal Investment under the Influence of Decision-changing Imitation

TL;DR

This paper investigates how decision-changing imitation from a leading expert influences a retail investor's dynamic investment decisions in a two-agent, continuous-time market with a risk-free and a risky asset. It introduces an integral disparity to penalize differences in decision-changing rates and solves the retail investor's problem via a variational approach, yielding a general solution for expressed through special functions and integral constants. Two boundary-condition cases are analyzed (fixed endpoints vs. derivative-boundary), with an iterative procedure to compute the integral constants, and asymptotic results are derived for to characterize the influence of imitation. Numerical experiments on real stock data validate the theory and illustrate how imitation shifts the retail investor's risk posture depending on the relative risk aversions and , offering insights into how to guide investor decisions in practice.

Abstract

Decision-changing imitation is a prevalent phenomenon in financial markets, where investors imitate others' decision-changing rates when making their own investment decisions. In this work, we study the optimal investment problem under the influence of decision-changing imitation involving one leading expert and one retail investor whose decisions are unilaterally influenced by the leading expert. In the objective functional of the optimal investment problem, we propose the integral disparity to quantify the distance between the two investors' decision-changing rates. Due to the underdetermination of the optimal investment problem, we first derive its general solution using the variational method and find the retail investor's optimal decisions under two special cases of the boundary conditions. We theoretically analyze the asymptotic properties of the optimal decision as the influence of decision-changing imitation approaches infinity, and investigate the impact of decision-changing imitation on the optimal decision. Our analysis is validated using numerical experiments on real stock data. This study is essential to comprehend decision-changing imitation and devise effective mechanisms to guide investors' decisions.
Paper Structure (20 sections, 7 theorems, 24 equations, 2 figures, 1 algorithm)

This paper contains 20 sections, 7 theorems, 24 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Optimal Investment Problem for the Leading Expert
  4. Optimal Investment Problem for the Retail Investor
  5. Solution to the Optimal Investment Problem
  6. General solution to the Optimal Decision
  7. Two Special Cases of the Boundary Conditions
  8. Case 1
  9. Case 2
  10. Integral Constants
  11. Summary
  12. Asymptotic Properties of the Optimal Decision
  13. Case 1
  14. Case 2
  15. Numerical Experiments
  16. ...and 5 more sections

Key Result

Theorem 1

The general solution of Problem 2 is for $t\in\mathcal{T}$, where are two integral constants, $\mathrm{I}_0(\cdot)$ and $\mathrm{K}_0(\cdot)$ represent the zeroth-order modified Bessel and Neumann functions, In eq:optimal-decision, $\gamma_1,\gamma_2\in\mathbf{R}$ are two parameters satisfying where $\{\delta P_1^*(t)\}_{t\in\mathcal{T}}$ refers to the variation with respect to $\{P_1^*(t)\}_{

Figures (2)

  • Figure 1: Optimal decisions $\{P_1^*(t)\}_{t\in\mathcal{T}}$, rational decisions $\{\bar{P}_1(t)\}_{t\in\mathcal{T}}$ and $\{\bar{P}_2(t)\}_{t\in\mathcal{T}}$, and asymptotic decisions $\{P_{1\infty}^*(t)\}_{t\in\mathcal{T}}$ in Case 1.
  • Figure 2: Optimal decisions $\{P_1^*(t)\}_{t\in\mathcal{T}}$, rational decisions $\{\bar{P}_1(t)\}_{t\in\mathcal{T}}$ and $\{\bar{P}_2(t)\}_{t\in\mathcal{T}}$, and asymptotic decisions $\{P_{1\infty}^*(t)\}_{t\in\mathcal{T}}$ in Case 2.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7