Partial hedging in financial markets with a large agent.

摘要

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. In any illiquid market almost all traders are large, and their trading strategies influence the market price. We develop a stochastic differential equation with a single large agent parameter to model such a market.;Partial hedging is a strategy that may be employed by an agent who is unwilling or unable to put up the premium needed to completely hedge the risk of having sold an option. We focus on minimizing the expected value of the size of the shortfall in the market with a large agent.;A Bellman type partial differential equation is derived for the shortfall function. The PDE is highly nonlinear, so we use the Legendre transform and consider the dual shortfall function. Its governing PDE is still nonlinear, but has more manageable properties. For the analysis of the dual Bellman PDE, we assume that the large agent parameter is small enough so that the dual PDE can be considered a small perturbation to the case when there is no large agent.;In the latter case, the PDE admits an explicit solution. An asymptotic analysis shows that the first order perturbation is positive. It 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.;Finally we estimate the large agent parameter using historical option price data from CBOE (Chicago Board Options Exchange). The parameter is positive but small, indicating presence of the large agent effect and justifying the validity of the asymptotic analysis.