Continuous-time mean-variance hedging under different loan and deposit rates

Abstract

This paper investigates a continuous-time mean-variance hedging problem under different loan and deposit rates. The value function is shown to satisfy a fully nonlinear PDE and be of C 3,2 smooth by PDE method and verification theorem. We show that there are a borrowing and a saving boundary that divide the whole trading space into three regions: borrowing money region, no-trading region and saving money region. The optimal strategy is a mixture of the continuously trading strategy (as suggested by most continuous-time models) and discontinuously trading strategy (as suggested by models with transaction costs): one should put all her wealth in the stock in the middle no-trading region, and continuously trade the risky asset in the borrowing and saving money regions. Also one should never short sale the stock.