Partial Hedging in Financial Markets with a Large Agent

We investigate the partial hedging problem in financial markets with a large agent. An agent is said to be large if his/her trades influence the equilibrium price. We develop a stochastic differential equation (SDE) with a single large agent parameter to model such a market. We focus on minimizing the expected value of the size of the shortfall in the large agent model. A Bellman-type partial differential equation (PDE) is derived, and the Legendre transform is used to consider the dual shortfall function. An asymptotic analysis leads us to conclude that the shortfall function (expected loss) increases when there is a large agent, which means that one would need more capital to hedge away risk in the market with a large agent. This asymptotic analysis is confirmed by performing Monte Carlo simulations.