Hedging with a Correlated Asset: An Insurance Approach
Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in an imperfect hedge. The residual risk from hedging with a correlated asset is priced using an actuarial standard deviation principle in infinitesmal time, which gives rise to a nonlinear partial differential equation (PDE). A fully implicit, monotone discretization method is developed for solving the pricing PDE. This method is shown to converge to the viscosity solution. Certain grid conditions are required to guarantee monotonicity. An algorithm is derived which, given an initial grid, inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied. At each timestep, the nonlinear discretized algebraic equations are solved using an iterative algorithm, which is shown to be globally convergent. Monte Carlo hedging examples are given, which show the standard deviation of the profit and loss at the expiry of the option.