# Prove arbitrage opportunity

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?

• I don't believe you can assume $f S(0) \in (0,1)$, because if $f=0.01$ and $S(0) \gt 100$ then the assumption is violated. For the purposes of an academic exercise, you can probably assume $r >= 0$ and $d >= 0$, but it's at least worth mentioning that there has been discussion in the news of negative rates and of how negative interest rates can go. – Jacob Amos Mar 2 '16 at 21:26

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.