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Unformatted text preview: swap price of $2.2044, and the forward prices of question 8.11., we can calculate the
implied loan amount. We can calculate the net position by subtracting the swap price from the
forward prices. The 1quarter implied forward rate is calculated from the zerocoupon bond
prices. The column implicit loan balance adds the net position each quarter and the implicit loan
balance plus interest of the previous quarter. Please note that the shape of the forward curve – we
are initially loaning money, because the swap price is lower than the forward price.
Quarter Forward Price Net Balance Forward interest rate Implicit loan balance
1
2.25
0.0456
1.0150
0.0456
2
2.60
0.3955
1.0156
0.4418
3
2.20
0.0042
1.0162
0.4447
4
1.90
0.3046
1.0168
0.1476
5
2.20
0.0043
1.0170
0.1458
6
2.50
0.2954
1.0172
0.4437
7
2.15
0.0543
1.0175
0.3971
8
1.80
0.4042
1.0178
0.0000 Question 8.14. From the given zerocoupon bond prices, we can calculate the onequarter forward interest rates.
They are:
Quarter
1
2
3
4
5
6
7
8 Forward interest rate
1.0150
1.0156
1.0162
1.0168
1.0170
1.0172
1.0175
1.0178 56 Chapter 8 Swaps Now, we can calculate the swap prices for 4 and 8 quarters according to the formula: ∑
X= n i =1 P0 (0, ti )r0 (ti −1 , ti ) ∑ n P (0, ti )
i =1 0 , where n = 4 or 8 This yields the following prices:
4quarter fixed swap price: 1.59015%
8quarter fixed swap price: 1.66096%
Question 8.16. The dollar zerocoupon bond prices for the three years are:
1
= 0.9434
1.06
1
P0, 2 =
= 0.8900
(1.06)2
1
P0,3 =
= 0.8396
(1.06)3
P0,1 = R * is 0.035, the Eurobond coupon rate. The current exchange rate is 0.9$/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 zerocoupon prices. Doing so yields a current
exchange rate of 0.90 $/Euro. R* is the 8quarter 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 putcallparity. 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 putcallparity 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 putcallparity 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 nonnegative. 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.
 Spring '14
 NguyenThiMaiLan
 Derivatives

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