Prove arbitrage opportunity

Question

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 1 year is $F(0,1)$. Short selling of the stock requires a security deposit in the amount of $fS(0)$ for some $f \in (0,1)$. Assume that the security deposit incurs an interest $d$ that is compounded continuously. Prove there is an arbitrage opportunity if the following is satisfied.

$$ d > ln(e^r - \frac{e^rS(0)-F(0,1)}{fS(0)}) $$ My intuition is to simplify this inequality somehow to reflect a necessary and sufficient condition for an arbitrage opportunity. To do this, I think I need to set some of the quantities to the boundaries of their domains, but I don't know what they are.

I only know for sure $fS(0) \in (0,1)$ (please let me know if this is wrong!) can I assume $r >= 0$ and/or $d >= 0$?

Any other ideas on how to approach this problem?

Answer 1

Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now, $d$ is an interest rate, and the condition suggests that $d$ is too high. So you will want to receive $d$ in order to profit.

If you could, you would borrow money at $r$ and lend it to the stock broker or exchange to collect the interest rate differential (assuming that $d > r$). But you can't just lend money at $d$, it is available only to someone shorting the stock. So let us build the simplest portfolio which allows us to be paid that rate of interest.

Short one unit of stock, obtaining $S(0)$ in cash. Of that cash, exactly $fS(0)$ has to be lent to the broker as a deposit. The rest, $(1-f)S(0)$, gets put in the money market account at interest rate $r$. So far, the portfolio is self-funding until time $t=1$. At that point we will be short one stock and have $$ e^d fS(0) + e^r (1-f)S(0)$$ in cash. Now, we don't want the short stock position, so we should buy that stock forward immediately. This doesn't change the self-funding nature since no cash is paid upfront for a forward sale. Now, at $t=1$ we will be flat the stock and have $$ e^d fS(0) + e^r (1-f)S(0) - F(0,1) $$ cash (with certainty).

This is positive exactly when $$ e^d fS(0) > F(0,1) - e^r (1-f)S(0) $$ or when $$ d > \log\left(\frac{F(0,1) - e^r (1-f)S(0)}{fS(0)} \right)$$ which simplifies to your inequality.

Note that if it is negative, there is not necessarily an arbitrage by taking the exact opposite portfolio, since presumably buying the stock doesn't mean that you are lent money at the same rate of interest which shorts receive on their deposit.