Problem $4$ You are given that the annual borrowing and lending rates are $r_B$ and $r_L$

respectively, both compounded continuously, with $r_B>r_L.$ We are interested in a forward

contract with delivery date $T,$ and the underlying asset is a $($non$-$dividend paying$)$ stock

whose current price is $S_0.$ Suppose that a fixed transaction cost of $A$ is imposed at the time

when one goes long or short a forward contract, and an additional transaction cost of $B$

is charged at time $T$ when the contract is settled. Find the no$-$arbitrage interval for the

forward price $F_0,$ and verify that arbitrage exists outside the interval. $($You may assume

that $A+B.$