  This paper studies the n-player game and the mean field game under the CRRA
relative performance on terminal wealth, in which the interaction occurs by
peer competition. In the model with n agents, the price dynamics of underlying
risky assets depend on a common noise and contagious jump risk modelled by a
multi-dimensional nonlinear Hawkes process. With a continuum of agents, we
formulate the MFG problem and characterize a deterministic mean field
equilibrium in an analytical form under some conditions, allowing us to
investigate some impacts of model parameters in the limiting model and discuss
some financial implications. Moreover, based on the mean field equilibrium, we
construct an approximate Nash equilibrium for the n-player game when n is
sufficiently large. The explicit order of the approximation error is also
derived.
