證明套利機會
連續複利利率為 $ r $ . 標的資產的目前價格為 $ S(0) $ 交貨時間為 1 年的遠期價格為 $ F(0,1) $ . 賣空股票需要保證金，金額為 $ fS(0) $ 對於一些 $ f \in (0,1) $ . 假設保證金產生利息 $ d $ 是不斷複合的。如果滿足以下條件，則證明存在套利機會。
$$ d > ln(e^r - \frac{e^rS(0)-F(0,1)}{fS(0)}) $$ 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?
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.