TEST BANK FOR An Introduction to the Mathematics of Financial Derivatives 2nd Ed By Salih N. Neftci

1. (a) Payo diagram at expiration:
0 2 4 6 8 10 12 14 16 18 20
−20
−15
−10
−5
0
5
10
15
20
short stock
short call
combined position
written call
short stock
short stock + short call
FIGURE 0.1 Payo diagram for both a short sale of stock and an at-the-money call.
i
Payo diagram at expiration:
1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
long put
long call
put + call
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:
0 2 4 6 8 10 12
−6
−4
−2
0
2
4
6
long put+short call
short put + long call
combined position
K1
K2
FIGURE 0.3 Payo diagram for a (long put/short call) combination at K1 plus a (long call/short put) combination at K2 >K1.
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(b) Payo diagram before expiration:
0 2 4 6 8 10 12 14 16 18 20
−25
−20
−15
−10
−5
0
5
10
short stock
short call
short stock combined position
written call
short stock + short call
FIGURE 0.4 Pre-maturity payo diagram for both a short sale of stock and an at-the-money call.
Payo diagram before expiration:
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
long put
long call
put + call
FIGURE 0.5 Pre-maturity payo diagram for a long put with strike K1 and a long call with strike K2, K1 < K2.
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Payo diagram before expiration:
0 1 2 3 4 5 6 7 8 9 10
−6
−4
−2
0
2
4
6
long put+short call
short put+long call
combined position
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 L12
2 +N 􀀀1 + L18
2 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 oset 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
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Position Payo at t Payo at T
Short futures 0 Ft 􀀀 ST
Borrow St + c + s +(St + c + s) 􀀀(St + c + s)(1 + r)
Buy wheat and pay storage, insurance costs 􀀀(St + c + s) ST
Total 0 Ft 􀀀 (St + c + s)(1 + r) > 0
Thus, Ft (St + c + s)(1 + r). If Ft < (St + c + s)(1 + r), one cannot immediately reverse the
holdings in the above portfolio to create another arbitrage portfolio. A problem arises since wheat
is not typically held as an investment asset. If one sells wheat, it is not reasonable to assume that
one is entitled to receive the storage and insurance. Therefore, a weaker condition ensues with
Ft St(1+r) but not Ft (St+c+s)(1+r). If the asset were of a nancial nature or a commodity
held for investment such as gold, one could sell the asset and save on the storage and insurance costs.
These assets produce an exact relationship, Ft = (St +c+s)(1+r). Holding an asset such as wheat
has value since it may be consumed. For instance, a large bakery requires wheat for production and
maintains an inventory. These companies would be reluctant to substitute a futures contract for
the actual underlying. Hence, the price of a futures is allowed to be less than (St + c + s)(1 + r).
However, if Ft < St(1 + r), then construct the following arbitrage portfolio
Position Payo at t Payo at T
Buy futures 0 ST 􀀀 Ft
Invest St 􀀀St +St(1 + r)
Sell wheat +St 􀀀ST
Total 0 St(1 + r) 􀀀 Ft > 0
Thus, Ft St(1 + r) and combining the two inequalities implies
St(1 + r) Ft (St + c + s)(1 + r)
(b) Ft = $1; 500 < $1; 543:50 = (1; 470)(1+:05) = St(1+r). This violates the above inequality. To take
advantage of this arbitrage opportunity, follow the second arbitrage strategy outlined above.
(c) Prot / Loss = 1,543.50 - 1,500 = $43.50.
4. (a) Ft = St(1 + r)(T􀀀t) = $105 where T 􀀀 t = 1 year.
(b) Ft = 101. Consider the following arbitrage portfolio
Position Payo at t Payo at T
Long forward $0 ST 􀀀 $101
Short stock +$100 􀀀ST
Invest at risk - free rate 􀀀$100 +$105
Total $0 $4
or the following arbitrage portfolio
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Position Payo at t Payo at T
Short stock +$100 􀀀ST
Long Call 􀀀$3:0 max(ST 􀀀 $100; 0)
Short Put +$3:5 min(ST 􀀀 $100; 0)
Invest PV (100) at risk - free rate 􀀀$ 100
1:05 $100
Total $5:26 $0
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CHAPTER2
1. (a) p = r􀀀d
u􀀀d = 1+:05􀀀260
280
320
280􀀀260
280
= :3917
(b) Value of the call option.
Ct =
1
(1 + :05)
E~p(Ct+)
=
1
(1 + :05)
(320 􀀀 280) p
= $15:47
(c) Normalize by St. The elements of the state price vector must be solved. Consider the following two
equations
1 = (1 + r) u + (1 + r) d
St = Su
t+1 u + Sd
t+1 d
and after dividing the second equation by St
1 = (1 + r) u + (1+r) d
1 =
Su
t+1
St
 u +
Sd
t+1
St
 d
Substitute in the values for r, Su
t+1, Sd
t+1 and express these equations as
1:0125 1:0125
320
280
260
280  u
 d = 1
1
Solving this system gives  u = :3868 and  d = :6008. The rst equation
1 = (1 + r) u + (1+r) d
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