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Relative Arbitrage Opportunities in an Extended Mean Field System

Nicole Tianjiao Yang, Tomoyuki Ichiba

Abstract

This paper studies relative arbitrage opportunities in a market with competitive investors through stochastic differential games in the limit as the number of players tends to infinity. With common noises introduced by the stock capitalization processes, we establish a conditional McKean-Vlasov system to study the market dynamics coupled to the expected trading volume of investors. We show that optimal arbitrage can be characterized as a solution of a Cauchy PDE constructed by the volatility terms in the market model. The structure of the market dynamics can be relaxed, and we provide a theoretical framework to study a general mean-field system, where the interaction is characterized by a joint distribution of wealth and strategies. In this setting, the optimal relative arbitrage constitutes the strong equilibrium of an extended mean-field game. We provide conditions for the existence and uniqueness of the mean-field equilibrium. We further prove the propagation of chaos result for the finite-player game counterpart, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity.

Relative Arbitrage Opportunities in an Extended Mean Field System

Abstract

This paper studies relative arbitrage opportunities in a market with competitive investors through stochastic differential games in the limit as the number of players tends to infinity. With common noises introduced by the stock capitalization processes, we establish a conditional McKean-Vlasov system to study the market dynamics coupled to the expected trading volume of investors. We show that optimal arbitrage can be characterized as a solution of a Cauchy PDE constructed by the volatility terms in the market model. The structure of the market dynamics can be relaxed, and we provide a theoretical framework to study a general mean-field system, where the interaction is characterized by a joint distribution of wealth and strategies. In this setting, the optimal relative arbitrage constitutes the strong equilibrium of an extended mean-field game. We provide conditions for the existence and uniqueness of the mean-field equilibrium. We further prove the propagation of chaos result for the finite-player game counterpart, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity.
Paper Structure (25 sections, 12 theorems, 108 equations, 1 figure, 1 table)

This paper contains 25 sections, 12 theorems, 108 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Organization
  4. Notations
  5. The Market Model
  6. Capitalizations
  7. Investors
  8. Interacting market system
  9. Relative arbitrage opportunities for competitive investors
  10. Relative arbitrage benchmark: competing with the market and the continuum of players
  11. Optimization of relative arbitrage opportunities
  12. PDE characterization of the best relative arbitrage
  13. Mean Field Games of relative arbitrage optimization
  14. Mean Field Games and Equilibrium
  15. Searching optimal strategies through fixed-point problems
  16. ...and 10 more sections

Key Result

Proposition 2.1

Suppose that the initial capitalization and trading volume $(\mathbf{x}, \mathbf{z})$ are independent of the Brownian motion $B (\cdot)$ on $(\Omega, \mathcal{F}, \mathbb F, \mathbb P)$, and they satisfy $\mathbb{E}[\|(\mathbf{x}, \mathbf{z})\|^2] \leq \infty$. Then, under Assumption xv, the McKean-

Figures (1)

  • Figure 1: The formulation of the fixed point problems. Note that to go from $(\mathcal{X}, \mathcal{Z})$ to $u(T-\cdot, \mathbf{x}, \mathbf{z})$, only the volatility of the market dynamics $(s(\cdot), \tau(\cdot))$ is needed to solve \ref{['inequ']}-\ref{['eqsol']}. For the arrow from $u(T-\cdot, \mathbf{x}, \mathbf{z})$ to get back to $\pi(\cdot)$, we elaborate in Section \ref{['sec: solveNE']} how to develop a consistency condition to check if the optimal strategy coincide with $\pi(\cdot)$ we started. The fixed point problem can start from any node (i.e., $\pi(\cdot)$, $u(T-\cdot, \mathbf{x}, \mathbf{z})$, or $(\mathcal{X}, \mathcal{Z})$) of this diagram by freezing the quantity of the node, compute the rest of the quantities sequentially according to the arrow of the diagram until reaching back to the node we started from. The fixed point solution corresponds to the quantity of the node that satisfies the consistency condition -- the frozen quantity as a starting point of the flow coincides with the quantity of the same node that is computed through the whole diagram.

Theorems & Definitions (31)

  • Definition 2.1: Investment strategy
  • Definition 2.2: Market system
  • Proposition 2.1
  • Proposition 3.1
  • Remark 1
  • Definition 3.1: Optimal arbitrage
  • Proposition 3.2
  • Remark 2: $\mathcal{F}^{B}$ in the market dynamics
  • Remark 3: $\mathcal{F}^{MF}$ in the optimization
  • Proposition 3.3
  • ...and 21 more