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Relative Arbitrage Opportunities with Interactions among $N$ Investors

Tomoyuki Ichiba, Nicole Tianjiao Yang

Abstract

The relative arbitrage portfolio outperforms a benchmark portfolio over a given time-horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi-agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well-posed market dynamical system of McKean-Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guarantee relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.

Relative Arbitrage Opportunities with Interactions among $N$ Investors

Abstract

The relative arbitrage portfolio outperforms a benchmark portfolio over a given time-horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi-agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well-posed market dynamical system of McKean-Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guarantee relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.

Paper Structure

This paper contains 20 sections, 10 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Market Model
  3. Capitalizations, wealth and portfolios
  4. Optimization of relative arbitrage in finite systems
  5. Benchmark of the market and investors
  6. Optimization in relative arbitrage
  7. The modified subproblems
  8. PDE characterization of the best relative arbitrage
  9. Existence of Relative Arbitrage
  10. The N player game
  11. Construction of Nash equilibrium
  12. Optimal arbitrage opportunities and the corresponding strategies in N-player game
  13. Searching Nash equilibrium
  14. Fixed point problem
  15. The uniqueness of Nash equilibrium
  16. ...and 5 more sections

Key Result

Proposition 3.1

Benchmark $\mathcal{V}(t) = \delta X(t) + (1-\delta) \overline{V}(T)$ in benchmark can be generated from a strategy $\Pi(\cdot) := (\Pi_1(\cdot), \ldots, \Pi_n(\cdot)) \in \mathbb{A}$, where $\mathcal{Y}_i(t)$ is defined in eq: y.

Figures (1)

  • Figure 1: The formulation of the fixed point problems. Note that this chart works for the fixed-point problem on the space of the paths of strategies as well, i.e., $\pi = \Phi(\pi)$ in Section \ref{['sec4.2.1']}-\ref{['sec4.2.2']}, if we start the flow from $\{\phi^{\ell}_t(\mathbf{x},\mathbf{y})\}_{\ell=1}^N$.

Theorems & Definitions (38)

  • Definition 2.1: Investment strategy
  • Definition 3.1: Relative Arbitrage
  • Definition 3.2: Benchmark
  • Proposition 3.1
  • proof
  • Definition 3.3: Optimal arbitrage among agents
  • Proposition 3.2
  • proof
  • Remark 1
  • Proposition 3.3
  • ...and 28 more