Solving Forward Contract Problems

This document provides solutions to forward contract problems, including calculating the optimal hedge ratio and determining the effective price paid by the company. It includes step-by-step explanations and examples.

Question 1
(a) F0=¿ $928.17, r = 0.04, S0 = $980, and q =?
F0=S0 e (r−q ) T=980 e
( 0.01−q ) 1
928.17=970.25 e−q
q=ln ( 970.25
928.17 )=0.0443
coupon value = 0.0443 x $1,000 = $44.3.
(b) Given T = 6 months = 6
12=0.5, r = 0.04, S0 = $980, and q = 0.0443
F0=S0 e (r−q ) T=9 80 e
( 0.01−0.0443 ) x 0.5=$ 963.34
(c) The arbitrageur borrows $980 to purchase the bond and short a forward contract. To
compute the present value of the first coupon, we discount
44.3 e−0.0 4 x 3
12 =$ 43.86
The remaining $980 - $43.86 = $936.14 is borrowed at 4% annually for remaining 3
months
936.14 e−0.04 x 3
12 =$ 926.83.
The arbitrageur makes
$962.28 - $926.83 = $ 35.45 for 3 months. Therefore, the arbitrage strategy is action
now and borrow $980 for $43.86 for three months and $936.14 for 3 months. Buy one
unit of the asset and enter into forward contract to sell asset in 6 months for $962.28.
Next action in three months receive $43.86 of income on asset use $43.86 to repay
first loan with interest. Finally, action in 6 months. Sell asset for $962.28 use $926.83
to repay second loan with interest.

Question 2
(a) Here r = 0.01, S0 = 1458, T = 1, and q = 2ln (1+ 0.02
2 ) = 0.0199.
The future price F0 is therefore,
F0=S0 e (r−q ) T=1458 e( 0.01−0.0199 ) 1=1,443.64
(b) F0 = 1,432, r = 0.01, T = 1, q = 0.0199, and S0 =?
1,432=S0 e (0.01−0.0199 ) 1
S0 = 1,446.25 = K
The value of the contract is
f = ( F0−K ) e−rT = ( 1,432−1446 ) e−0.01=−0.010.
Therefore, this contract does not contain value its not worth investing in.
(c) F0 = 1,450, r = 0.01, T = 1, q = 0.0199, and S0 =?
1,4 50=S0 e( 0.01−0.0199 ) 1
S0 = 1,464 = K
The value of the contract is
f = ( F0−K ) e−rT = ( 1,4 50−146 4 ) e−0.01=−13.86.
Therefore, this contract does not contain value it is not worth investing in.
Question 3
(a) r = 0.04, T = 1, F0 = $14.72, S0 =?
F0= ( S0 +U ) erT
U =2 e−rT =2 e−0.04=1.922
14.72= ( S0+1.922 ) e0.04
14.72=S0 e0.04 +2
S0 =$ 12.22 spot price.
(b) The following are given r = 0.04, T = 0.5, S0 = $12.22 and

U =2 e−rT =2 e−0.04 x 0.5 =1.96
F0= ( S0 +U ) erT = ( 12.22+ 1.96 ) e0.04 x 0.5=$ 14.21
(c) Gain $20000 the gain per asset = $20000/10 = $2000/5000 = $0.40 per ounce.
Gain = F0 – quoted price implying F0 = 0.4 + 14.72 = $15.12.
Now, r = 0.04, T = 0.25, F0 = $15.12, S0 =?
F0= ( S0 +U ) erT
U =2 e−rT =2 e−0.04 x 0.25=1.9 8
1 5.12= ( S0 +1.9 8 ) e0.04 x0.25
15.12=S0 e0.04 x0.25 +2
S0 =$ 12.99 spot price.
Question 4
(a) The data is as shown in the tale
Month Future Prices Spot Price
1 20.70 21.40
2 21.80 21.80
3 20.80 21.10
4 22.00 21.70
5 21.80 21.90
6 22.10 22.20
7 22.40 22.30
8 21.50 22.70
9 22.20 22.00
10 22.80 21.70
11 22.00 21.60
12 21.40 21.90
13 21.60 22.50
14 22.50 22.40
15 22.40 22.30
16 22.60 21.40
17 21.30 21.80
18 22.00 22.50
19 22.50 22.30
20 22.40 22.70
21 22.50 22.90
22 23.20 22.50

23 22.50 22.70
24 23.20 22.70
The minimum variance hedge ratio is given as follows:
h=ρ . δs
δF
Where,
ρ – correlation between future price and spot price
δsand δF – standard deviation of the spot price and future price respectively.
From excel, ρ = 0.520259, δs=0. 492774 and δF =0. 648689.
h=0.520259 x 0. 492774
0.648689 =0.395213
(b) The company should take long contracts.
Optimal number of contracts = ¿ desired underlying portfolio
contract ¿ one futurecontract x h
Optimal contract = 125,000
1,000 x 0.395213=49 contracts.
(c) The effective price paid by the company is calculated as
Price = 21.4 + (23.2 – 20.9) = $23.7