hw3_solution - 4.Have/3F$ at$back D$ 2 Sell toGet back 1...

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3. Get back 2. Sell F$ at to get D$ 4.Have/ to pay back. 1. borrow F$ DERIVATIVE SECURITIES SOLUTION TO HOMEWORK NUMBER 3 3.1 A $ 30 stock pays a dividend of $1 every 3 months, with the first dividend coming 3 months from today. The continuously compounded interest rate is 3%. (a) What is the forward price of this stock with expiry 1 year from today? (The forward is to purchase the stock ex-dividend, i.e. the purchaser under the forward does not receive the dividend being paid at expiry.) The cash flow of the stock’s dividend: 0 3 months 6 months 9 months 12 months (Maturity) Cash Flow 0 1 1 1 1= Total Cash Flow +++ = = 30.91 = 26.868$ (b) Suppose you were able to go long or short this forward for a forward price $ 1 less than your answer in part (a). Describe in detail the arbitrage that could be done. Here the forward price is 26.8681= 25.868 ( underpriced!) So to get the arbitrage profit, I have to make a reverse cash- and- carry, i.e. long the underpriced forward, and then make a synthetic short forward ( do the short-selling at time 0 and lend out the short-selling proceed) Notice: Because this stock pays a dividend of $1 every 3 months, a short-seller needs to pay the dividend to the stock lender at the dividend issuance date. The short-seller can borrow to finance the dividend payment. Time 0 3 months 6 months 9 months 12 months (Maturity) Short sell the stock at 50$ +30 -
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Lend out 50$ -30 +30* Borrow to finance the dividend payment +1 +1 +1 +1 Dividend payment -1 -1 -1 -1 Payment to the borrowing -(+++) Long forward -25.868 Total 50* -(+++) -25.868 =26.868-25.868=1$ 3.2 Repeat Question 3.1, replacing the stock by a foreign currency, with spot price $ 7.8, and which pays interest at 3%, continuously compounded. Lend out money at Exvhange rate at Time 0: Exchange rate at Time T: Borrow money at Because we borrow F$ at , at time T we will need to pay back *=1F$ Thus when /=1, there is no arbitrage opportunity. Here: --we used reverse cash and carry to get the formula of exchange rate forward pricing. Even if we use the cash and carry, i.e. borrow domestic currency to buy foreign currency, and short the exchange rate forward at time 0, we can also get the same formula. The only difference is that the cash flow would go in another direction in this circle.
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