衍生产品市场 课后答案 第八章

Chapter 8SwapsQuestion 8.1We first solve for the present value of the cost per two barrels:2$22$2341.033.1.06(1.065)+= We then obtain the swap price per barrel by solving:241.033(1.065)1.0622.483,x x x +=⇒=which was to be shown.Question 8.21. We first solve for the present value of the cost per three barrels, based on the forward prices:23$20$21$2255.3413.1.06(1.065)(1.07)++= Hence we could spend $55.3413 today to receive 1 barrel in each of the next three years. We thenobtain the swap price per barrel by solving:2355.34131.06(1.065)(1.07)20.9519x x x x ++=⇒=2. We first solve for the present value of the cost per two barrels (Year 2 and Year 3):23$21$2236.473.(1.065)(1.07)+= Hence we could spend $36.473 today and receive 1 barrel of oil in Year 2 and Year 3. We obtain theswap price per barrel by finding two equal payments we would make in Years 2 and 3 that have the same present value:2336.473(1.065)(1.07)21.481x x x +=⇒=94 M cDonald • Fundamentals of Derivatives MarketsQuestion 8.3Since the dealer is paying fixed and receiving floating, each year she has a cash flow $20.9519.T S − She can hedge this risk by selling 1 barrel forward (i.e., short one forward) in each of the three years. Her payoffs from the swap, the short forward contracts, and the net are summarized in the following table:Year Net Swap PaymentShort ForwardsNet Position 1$20.9519S − 1$20S − −0.9519 2 2$20.9519S −2$21S −+0.0481 33$20.9519S − 3$22S −+1.0481We need to discount the net cash flows to year zero. We have:230.95190.0481 1.0481PV(net CF)0.1.06(1.065)(1.07)−=++= Indeed, the present value of the net cash flow is zero.Question 8.4The fair swap rate was determined to be $20.9519. Therefore, compared to the forward curve price of$20 in one year, we are overpaying $0.9519. In year two, with interest, this overpayment increases to $0.9519 1.070024$1.01853,×= where we used the appropriate forward rate to calculate the interest payment.In year two, we underpay by $0.0481, so that our total accumulative underpayment is $1.01856 − $0.0481 = $0.97042. In year three, using the appropriate 1-year forward rate of 8.007%, this net overpaymentincreases to $0.97046 1.08007$1.0481.×=However, in year three, we receive a fixed payment of 20.9519, which underpays relative to the forward curve price of $22 by $22$20.9519$1.0481.−=Therefore, our cumulative balance is indeed zero, which was to be shown.Question 8.5Question 8.3 showed that, at the initial yield curve, the swap has a zero present value. To look at how the yield curve affects the value of the dealer’s swap position, we repeat the hedge and then look at the present value of the hedged cash flows under the new yield curve.As in Question 8.3, the dealer is paying fixed and receiving floating; hence each year she has a cashflow $20.9519.T S − She can hedge this risk by selling 1 barrel forward (i.e., short one forward) in each of the three years. Her payoffs from the swap, the short forward contracts, and the net are summarized in the following table:Year Net Swap Payment Short ForwardsNet Position 1 1$20.9519S − 1$20S − −0.9519 2 2$20.9519S − 2$21S − +0.0481 33$20.9519S −3$22S −+1.0481Chapter 8 Swaps 95We need to discount the net positions to year zero, taking into account the uniform shift of the termstructure. We have:230.95190.0481 1.0481PV(net CF)0.0081.1.065(1.07)(1.075)−=++=−The present value of the net cash flow is negative; the dealer never recovers from the increased interestrate he faces on the overpayment of the first swap payment. When the interest rates fall, the value of the swap becomes:230.95190.0481 1.0481PV(net CF)0.00831.055(1.06)(1.065)−=++=+The present value of the net cash flow is positive. The dealer makes money, because he gets a favorableinterest rate on the loan he needs to take to finance the first overpayment.The dealer could have tried to hedge his exposure with a forward rate agreement or any other derivative protecting against interest rate risk.Question 8.6In order to answer this question, we use Equation (8.13) of the main text with the substitution: 00,().i i t f t F = With an effective quarterly interest of 1.5%, the zero-coupon price for quarter i is 1.015i . Using this result, the zero-coupon prices are:QuarterZero-coupon price1 0.9852 20.970730.956340.942250.928360.914570.901080.8877Using Equation (8.13), the 4-quarter swap price is:.985221.970721.1.956320.8.942220.520.8533.9852.9707.9563.9422R ×+×+×+×==+++A similar calculation will yield an 8-quarter swap price of $20.4284.The total costs of prepaid 4- and 8-quarter swaps are the present values of the payment obligations.Note that a simple algebra exercise shows this is just the numerator in Equation (8.13). For the 4-quarter swap, we find this is equal to:4-quarter prepaid swap price 20.8533(.9852.9707.9563.9422)$80.3768=×+++=A similar calculation will yield an 8-quarter prepaid swap price of $152.9256.96 M cDonald • Fundamentals of Derivatives MarketsQuestion 8.7In order to answer this question, we use Equation (8.13) of the main text with the substitution: 00,().i i t f t F = For example, the 3-quarter swap price is3.985221.970121.1.954620.820.9677.9852.9701.9546R ×+×+×==++Similar calculations lead to the following swap prices:Quarter Swap price1 21.00002 21.04963 20.96774 20.85365 20.72726 20.61107 20.51458 20.4304Question 8.8We use Formula (8.4), and replace the forward interest rate with the forward oil prices. In particular, we calculate:60,363(0,).954620.8.938820.5.923120.2.90752020.3807.9546.9388.9231.9075(0,)iit i ii P t FR P t ==×+×+×+×===+++∑∑Therefore, the swap price of a 4-quarter oil swap with the first settlement occurring in the third quarter is $20.3807.Question 8.9We use Equation (8.6) of the main text to answer this question:80,1801(0,),where [1,2,1,2,1,2,1,2](0,)iiit it i t ii Q P t FR Q Q P t ====∑∑After plugging in the relevant variables given in the exercise, we obtain a value of $20.4099 for the swap price. Note that it is probably easiest to set up a spreadsheet with 0(0,)i t i Q P t × as one column and 0,i t F as another column. Then use the SUMPRODUCT function for the numerator and SUM function for the denominator.Chapter 8 Swaps 97Question 8.10We are now asked to invert our equations. The swap prices are given, and we want to back out the forwardprices. We do so recursively. For a one-quarter swap, the swap price and the forward price are identical. Given the one-quarter forward price, we can find the second quarter forward price, etc. For example,0,10,220,120,2(0,1)(0,2)[(0,1)(0,2)](0,1)(0,1)(0,2)(0,2)P F P F R P P P F R F P P P ++−=⇒=+Hence0,2 2.4236 1.9553.9852 2.252.60.9701F ×−×==Doing a similar calculation for Years 3, 4, etc. yields the following forward prices:Quarter Forward price1 2.252 2.603 2.204 1.905 2.206 2.507 2.158 1.80Question 8.11From the given zero-coupon bond prices, we can calculate the one-quarter forward interest rates(i.e., 01(,)i i r t t +). For instance, the forward interest rate factor for Quarter 4 is0(0,3)0.9546(3,4)11 1.6830%(0,4)0.9388P r P ===−=The forward interest factors for the quarters are:QuarterForward interest rate1 1.5022%2 1.5565%3 1.6237%4 1.6830%5 1.7008%6 1.7190%7 1.7491%8 1.7802%98 M cDonald • Fundamentals of Derivatives MarketsNow, we can calculate the deferred swap price according to the formula:61262(0,)(,)1.6553%(0,)i i i i ii P t r tt X P t −====∑∑Question 8.12From the given zero-coupon bond prices, we can calculate the one-quarter forward interest rates (i.e., 01(,)i i r t t +). For instance, the forward interest rate factor for Quarter 4 is0(0,3)0.9546(3,4)11 1.6830%(0,4)0.9388P r P ===−=The forward interest factors for the quarters are:QuarterForward interest rate1 1.5022%2 1.5565%3 1.6237%4 1.6830%5 1.7008%6 1.7190%7 1.7491%8 1.7802%Now, we can calculate the swap prices for 4 and 8 quarters according to the formula:111(0,)(,),where 4 or 8(0,)ni i i i nii P t r tt X n P t −====∑∑This yields the following prices:4-quarter fixed swap price: 1.59015% 8-quarter fixed swap price: 1.66096%Note that it is probably easiest to set up a spreadsheet with 0(0,)i P t as one column and 01(,)i i r t t − as another column. Then use the SUMPRODUCT function for the numerator and SUM function for the denominator.Chapter 8 Swaps 99Question 8.13We can calculate the value of an 8-quarter dollar annuity that is equivalent to an 8-quarter Euro annuity byusing Equation (8.8) of the main text. We have:8*0,1*81(0,)where1 euro (0,),iit i ii P t R FX R P t ====∑∑Plugging in the forward price for one unit of Euros delivered at time t i , which are given in the price table, yields a dollar annuity value of $0.9277.Question 8.14The dollar zero-coupon bond prices for the three years are:0,10,220,3310.94341.0610.8900(1.06)10.8396(1.06)P P P ======R * is 0.035, the Euro-bond coupon rate. The current exchange rate is 0.9$/€. Plugging all the above variables into Formula (8.9) indeed yields 0.06, the dollar coupon rate:3*0,0,00,30,0131/(/1)0.0980550.0623170.0602.673(0,)ii n t t t i ii PR F x P F x R P t ==+−+===∑∑Question 8.15We can use the standard swap price formula for this exercise, but we must use the euro zero-coupon bonds and the derived quarterly forward interest rates. From the given euro zero-coupon bond prices, we can calculate the one-quarter forward interest rates. For example, the quarterly euro forward rate with maturity in Quarter 4 is0(0,3)0.9735(3,4)110.9541%(0,4)0.9643P r P =−=−=100 M cDonald • Fundamentals of Derivatives MarketsThe euro denominated quarterly forward rates are:QuarterEuro denominated impliedforward interest rate1 0.8776%2 0.8957%3 0.9245%4 0.9541%5 0.9632%6 0.9726%7 0.9822%8 1.0028%Now, we can calculate the swap prices for 4 and 8 quarters according to the formula:111(0,)(,)where 4 or 8(0,),ni i i i nii P t r tt R n P t −====∑∑Using the euro forward rates in the table above with the euro zero coupon bond prices in the formula yields the following euro fixed swap rates:4-quarter fixed swap rate:0.91267%8-quarter fixed swap rate:0.94572%Question 8.16Equation (8.9) requires the following ingredients: • The euro fixed swap rate, which from Question 8.15 is *0.94572%R =.• The forward exchange rates in dollars per euro; i.e., 0,.i t F These are found in the Table 8.9. • The dollar denominated zero coupon bond prices. These are found in the Table 8.9. •The spot exchange rate, which is given as 00.9x =. Again this is in dollars per euro.With these substitutions, we have:8*0,0,0/0.0726,ii t t iPR F x =∑80,0,0(/1)0.05102,i t t P F x ×−=and80,7.4475it iP=∑Hence (.0726.05102)/7.4475 1.66%R =+= which agrees with our previous dollar denominated 8-quarter fixed swap rate.。