Convergence and optimality of BS-type discrete hedging strategy under stochastic interest rate

We focus on the asymptotic convergence behavior of the hedging errors of European stock option due to discrete hedging under stochastic interest rates. There are two kinds of BS-type discrete hedging differ in hedging instruments: one is the portfolio of underlying stock, zero coupon bond, and the money market account (Strategy BSI); the other is the underlying stock, zero coupon bond (Strategy BSII). Similar to the results of the deterministic interest rate case, we show that convergence speed of the discounted hedging errors is 1/2-order of trading frequency for both strategies. Then, we prove each of the BS-type strategy is not only locally optimal, but also globally optimal under the corresponding measure. Finally, we give some numerical examples to illustrate the results. All the discussion is based on non-arbitrage condition and zero transaction cost.