Solution of Derivative

25 00456 10150 00456 2 260 03955 10156 04418 3 220

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Unformatted text preview: swap price of $2.2044, and the forward prices of question 8.11., we can calculate the implied loan amount. We can calculate the net position by subtracting the swap price from the forward prices. The 1-quarter implied forward rate is calculated from the zero-coupon bond prices. The column implicit loan balance adds the net position each quarter and the implicit loan balance plus interest of the previous quarter. Please note that the shape of the forward curve – we are initially loaning money, because the swap price is lower than the forward price. Quarter Forward Price Net Balance Forward interest rate Implicit loan balance 1 2.25 0.0456 1.0150 0.0456 2 2.60 0.3955 1.0156 0.4418 3 2.20 -0.0042 1.0162 0.4447 4 1.90 -0.3046 1.0168 0.1476 5 2.20 -0.0043 1.0170 0.1458 6 2.50 0.2954 1.0172 0.4437 7 2.15 -0.0543 1.0175 0.3971 8 1.80 -0.4042 1.0178 0.0000 Question 8.14. From the given zero-coupon bond prices, we can calculate the one-quarter forward interest rates. They are: Quarter 1 2 3 4 5 6 7 8 Forward interest rate 1.0150 1.0156 1.0162 1.0168 1.0170 1.0172 1.0175 1.0178 56 Chapter 8 Swaps Now, we can calculate the swap prices for 4 and 8 quarters according to the formula: ∑ X= n i =1 P0 (0, ti )r0 (ti −1 , ti ) ∑ n P (0, ti ) i =1 0 , where n = 4 or 8 This yields the following prices: 4-quarter fixed swap price: 1.59015% 8-quarter fixed swap price: 1.66096% Question 8.16. The dollar zero-coupon bond prices for the three years are: 1 = 0.9434 1.06 1 P0, 2 = = 0.8900 (1.06)2 1 P0,3 = = 0.8396 (1.06)3 P0,1 = R * is 0.035, the Euro-bond coupon rate. The current exchange rate is 0.9$/E. Plugging all the above variables into formula (8.9) indeed yields 0.06, the dollar coupon rate: ∑ R= 3 i =1 ( ) P0 (0, t i )R * F0,ti / x 0 + P0,3 F0,tn / x 0 − 1 ∑ 3 i =1 P0 (0, t i ) = 0.098055 + 0.062317 = 0.060 2.673 Question 8.18. We can use equation (8.9), but there is a complication: We do not have the current spot exchange rate. However, it is possible to back it out by using the methodology of the previous chapters: We know that the following relation must hold: * F0,1 = X 0 e (r − r ) , where the interest rates are already on a quarterly level. We can back out the interest rates from the given zero-coupon prices. Doing so yields a current exchange rate of 0.90 $/Euro. R* is the 8-quarter fixed swap price payment of 0.0094572. By plugging in all the relevant variables into equation 8.9, we can indeed see that this yields a swap rate of 1.66%, which is the same rate that we calculated in exercise 8.14. 57 Chapter 9 Options Question 9.2. This problem requires the application of put-call-parity. We have: S 0 − C (30,0.5) + P(30,0.5) − e − rT 30 = PV (dividends ) ⇔ PV (dividends ) = 32 − 4.29 + 2.64 − 29.406 = $0.944 Question 9.4. We can make use of the put-call-parity for currency options: + P ( K , T ) = −e ⇔ − rf T x0 + C (K , T ) + e − rT K P(K , T ) = −e −0.04 0.95 + 0.0571 + e −0.06 .93 = −0.91275 + 0.0571 + 0.87584 = 0.0202 A $0.93 strike European put option has a value of $0.0202 Question 9.6. a) We can use put-call-parity to determine the forward price: +C (K , T ) − P (K , T ) = PV ( forward price ) − PV (strike) = e − rT F0,T − Ke − rT [ [ ⇔ F0,T = e rT + C (K , T ) − P (K , T ) + Ke − rT = e 0.05*0.5 $0.0404 − $0.0141 + $0.9e −0.05*0.5 ⇔ F0,T = $0.92697 b) Given the forward price from above and the pricing formula for the forward price, we can find the current spot rate: F0,T = x0 e ⇔ x0 = F0,T e (r − rf )T ( ) − r − rf T = $0.92697e −(0.05−0.035)0.5 = $0.92 58 Chapter 9 Parity and Other Option Relationships Question 9.8. Both equations (9.13) and (9.14) are violated. We use a call bull spread and a put bear spread to profit from these arbitrage opportunities. Transaction Buy 50 strike call Sell 55 strike call TOTAL Transaction Buy 55 strike put Sell 50 strike put TOTAL t=0 -9 +10 +1 Expiration or Exercise ST < 50 50 ≤ ST ≤ 55 ST > 55 0 ST − 50 ST − 50 0 0 55 − ST 0 5>0 ST − 50 > 0 t=0 -6 7 +1 Expiration or Exercise ST < 50 50 ≤ ST ≤ 55 55 − ST 55 − ST 0 ST − 50 5>0 55 − ST > 0 ST > 55 0 0 0 Please note that we initially receive money, and that at expiration the profit is non-negative. We have found arbitrage opportunities. Question 9.10. Both equations (9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values. We have: C (K1 ) − C (K 2 ) 18 − 14 = = 0 .8 55 − 50 K 2 − K1 and C (K 2 ) − C (K 3 ) 14 − 9.50 = = 0 .9 , 60 − 55 K3 − K2 and P (K 3 ) − P (K 2 ) 14.45 − 10.75 = = 0.74 , K3 − K2 60 − 55 which violates equation 9.17. and P (K 2 ) − P (K1 ) 10.75 − 7 = = 0.75 K 2 − K1 55 − 50 which violates equation 9.18. We calculate lambda in order to know how many options to buy and sell when we construct the butterfly spread that exploits this form of mispricing. Because the strike prices are symmetric around 55, lambda is equal to 0....
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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