The continuously compounded interest rate is $r$. The current price of the underlying asset is $S(0)$, and the forward price with delivery time in one year is $F(0,1)$. Short selling of the stock requires a security deposit in the amount of $fS(0)$ for some $f∈(0,1)$. Assume that the security deposit earns an interest $d$ that is compounded continuously. Prove that there is an arbitrage opportunity if the following inequality is satisfied $$d \gt ln\left(e^r-\frac{e^rS(0)-F(0, 1)}{fS(0)}\right)$$
I am not entirely sure of what I have to do here, I know I have to show that the price in one year after interest is less than the current price $S(0)$, but I don't know how to do that?
Do I get $F(0, 1)$ alone on one side of the inequality?
The only trade we have to look at is: buy the forward position at the forward rate and short the position through to the forward date. "arbitrage free" would require that this be a zero-sum.
To achieve the short position I sell the stock today, netting $S(0)$. I then invest the proceeds at $r$. I also must borrow $fS(0)$ at $r$, but at least I earn $d$ on it. Thus arbitrage-free implies $$F(0,1)=e^rS(0)-e^rfS(0)+e^dfS(0)$$ rearranging this says that $$e^d=e^r -\frac {e^rS(0)-F(0,1)}{fS(0)}$$
From which the desired claim follows.