Relative wealth concerns with partial information and heterogeneous priors

Abstract

We establish a Nash equilibrium in a market with N agents with the performance criteria of relative wealth level when the market return is unobservable. Each investor has a random prior belief on the return rate of the risky asset. The investors can be heterogeneous in both the mean and variance of the prior. By a separation result and a martingale argument, we show that the optimal investment strategy under a stochastic return rate model can be characterized by a fully-coupled linear FBSDE. Two sets of deep neural networks are used for the numerical computation to first find each investor's estimate of the mean return rate and then solve the FBSDEs. We establish the existence and uniqueness result for the class of FBSDEs with stochastic coefficients and solve the utility game under partial information using deep neural network function approximators. We demonstrate the efficiency and accuracy by a base-case comparison with the solution from the finite difference scheme in the linear case and apply the algorithm to the general case of nonlinear hidden variable process. Simulations of investment strategies show a herd effect that investors trade more aggressively under relativeness concerns. Statistical properties of the investment strategies and the portfolio performance, including the Sharpe ratios and the Variance Risk ratios (VRRs) are examed. We observe that the agent with the most accurate prior estimate is likely to lead the herd, and the effect of competition on heterogeneous agents varies more with market characteristics compared to the homogeneous case.

Large Ranking Games with Diffusion Control

“…They characterize a Nash equilibrium in terms of a system of coupled backward stochastic differential equations (BSDEs). [10] establishes a Nash equilibrium in a market with N agents with the performance criteria of relative wealth level when the mean return rate is unobservable. Each investor has a heterogeneous prior belief on the return rate of the risky asset.…”

Optimal portfolio with relative performance and partial information: A mean‐field game approach

“…When this parameter degenerates to 0, relative performance concern disappears and it is the classic Merton model [20]. Competition among investors is well studied in the past few years; see previous studies [21][22][23][24].…”

Nash equilibria for relative investors via no-arbitrage arguments

“…Finally, [15] deal with a common Itô-diffusion market for all agents which may be incomplete in case of CARA utilities or complete in case of random risk tolerance coefficients. Further papers among others are [9] where the problem with partial information and heterogeneous priors is considered and [22] which discusses more economical questions like the structure of equilibria and the effect of additional agents.…”