CHAPTER 1
1. (a) Payo diagram at expiration:
20
short stock
short call
combined position
15
10
short stock
5 written call
0
−5 short stock + short call
−10
−15
−20
0 2 4 6 8 10 12 14 16 18 20
FIGURE 0.1 Payo diagram for both a short sale of stock and an at-the-money call.
i
, Payo diagram at expiration:
3.5
long put
long call
put + call
3
2.5
2
1.5
1
0.5
0
1 2 3 4 5 6 7 8 9 10
FIGURE 0.2 Payo diagram for a long put with strike K1 and a long call with strike K2 , K1 < K2 .
Payo diagram at expiration:
6
long put+short call
short put + long call
combined position
4
2
K1 K2
0
−2
−4
−6
0 2 4 6 8 10 12
FIGURE 0.3 Payo diagram for a (long put/short call) combination at K1 plus a (long call/short put) combination at K2 > K1 .
ii
, (b) Payo diagram before expiration:
10
short stock
short call
short stock combined position
5
0
−5 written call
−10
short stock + short call
−15
−20
−25
0 2 4 6 8 10 12 14 16 18 20
FIGURE 0.4 Pre-maturity payo diagram for both a short sale of stock and an at-the-money call.
Payo diagram before expiration:
4
long put
long call
put + call
3.5
3
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10
FIGURE 0.5 Pre-maturity payo diagram for a long put with strike K1 and a long call with strike K2 , K1 < K2 .
iii
, Payo diagram before expiration:
6
long put+short call
short put+long call
combined position
4
2
0
−2
−4
−6
0 1 2 3 4 5 6 7 8 9 10
FIGURE 0.6 Pre-maturity payo diagram for a (long put/short call) combination at K1 plus a (long call/short put) combination
at K2 > K1 .
2. (a) Let N denote the notional amount of the swap and L12 and L18 the USD Libor rate at 12 months
and 18 months respectively. The cash ows are given by
12 months 18 months 24 months
Floating leg +N +N L212 +N 1 + L218
Fixed leg N N :05
2 N 1 + :05
2
where the 1 in the 24 months column represents the notional amount.
(b) If one had a oating rate obligation and wished to pay a xed rate, , then enter into two FRA
contracts at rate with maturity 18 and 24 months. For example, at 18 months, if the oating rate
were above , then the FRA would be in-the-money by precisely the amount required to o set the
higher oating rate payment. Therefore, the total payment is at the rate .
(c) If one had a oating rate obligation and wished to pay a xed rate, a swap is not necessary as long
as the appropriate interest rate options are available. A long position in an interest rate cap at
rate and a short position in an interest rate oor at rate , both maturing on the oating rate
payment date, ensure that a xed rate of is paid. If the oating rate, say rT , is above at expiry,
a net payment at rate is required after taking into account the value of the cap, N (rT ).
If the oating rate is below at expiry, say rT , then a payment at rate rT must be made on the
oating rate obligation. However, the short position in the oor requires an additional payment of
N ( rT ). The result is a total payment at precisely rate .
3. (a) St (1 + r) Ft (St + c + s)(1 + r) where c is the annual storage cost for 1 ton of wheat, s is the
annual insurance cost for 1 ton of wheat, and r is the simple interest rate. If Ft > (St + c + s)(1 + r),
then construct the following arbitrage portfolio
iv
1. (a) Payo diagram at expiration:
20
short stock
short call
combined position
15
10
short stock
5 written call
0
−5 short stock + short call
−10
−15
−20
0 2 4 6 8 10 12 14 16 18 20
FIGURE 0.1 Payo diagram for both a short sale of stock and an at-the-money call.
i
, Payo diagram at expiration:
3.5
long put
long call
put + call
3
2.5
2
1.5
1
0.5
0
1 2 3 4 5 6 7 8 9 10
FIGURE 0.2 Payo diagram for a long put with strike K1 and a long call with strike K2 , K1 < K2 .
Payo diagram at expiration:
6
long put+short call
short put + long call
combined position
4
2
K1 K2
0
−2
−4
−6
0 2 4 6 8 10 12
FIGURE 0.3 Payo diagram for a (long put/short call) combination at K1 plus a (long call/short put) combination at K2 > K1 .
ii
, (b) Payo diagram before expiration:
10
short stock
short call
short stock combined position
5
0
−5 written call
−10
short stock + short call
−15
−20
−25
0 2 4 6 8 10 12 14 16 18 20
FIGURE 0.4 Pre-maturity payo diagram for both a short sale of stock and an at-the-money call.
Payo diagram before expiration:
4
long put
long call
put + call
3.5
3
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10
FIGURE 0.5 Pre-maturity payo diagram for a long put with strike K1 and a long call with strike K2 , K1 < K2 .
iii
, Payo diagram before expiration:
6
long put+short call
short put+long call
combined position
4
2
0
−2
−4
−6
0 1 2 3 4 5 6 7 8 9 10
FIGURE 0.6 Pre-maturity payo diagram for a (long put/short call) combination at K1 plus a (long call/short put) combination
at K2 > K1 .
2. (a) Let N denote the notional amount of the swap and L12 and L18 the USD Libor rate at 12 months
and 18 months respectively. The cash ows are given by
12 months 18 months 24 months
Floating leg +N +N L212 +N 1 + L218
Fixed leg N N :05
2 N 1 + :05
2
where the 1 in the 24 months column represents the notional amount.
(b) If one had a oating rate obligation and wished to pay a xed rate, , then enter into two FRA
contracts at rate with maturity 18 and 24 months. For example, at 18 months, if the oating rate
were above , then the FRA would be in-the-money by precisely the amount required to o set the
higher oating rate payment. Therefore, the total payment is at the rate .
(c) If one had a oating rate obligation and wished to pay a xed rate, a swap is not necessary as long
as the appropriate interest rate options are available. A long position in an interest rate cap at
rate and a short position in an interest rate oor at rate , both maturing on the oating rate
payment date, ensure that a xed rate of is paid. If the oating rate, say rT , is above at expiry,
a net payment at rate is required after taking into account the value of the cap, N (rT ).
If the oating rate is below at expiry, say rT , then a payment at rate rT must be made on the
oating rate obligation. However, the short position in the oor requires an additional payment of
N ( rT ). The result is a total payment at precisely rate .
3. (a) St (1 + r) Ft (St + c + s)(1 + r) where c is the annual storage cost for 1 ton of wheat, s is the
annual insurance cost for 1 ton of wheat, and r is the simple interest rate. If Ft > (St + c + s)(1 + r),
then construct the following arbitrage portfolio
iv
