Question

A company in the oil refining business issues some zero coupon bonds indexed to the price of crude oil, which mature after one year.

• These securities repay the face value of $1,000 on the maturity date.
• They also pay after one year an additional positive amount which equals 20[50 - S(T)] where S(T) is the spot price of crude oil at maturity (time T).
• The extra payment is 0 if crude price is over $50 (per barrel).        

5) a) Draw the payoff to this security with S(T) along the horizontal axis.  

Answer 1

Maturity of ZCB = 1 year

Cash flow at maturity, CF:

• CF = 1000 + [ 20 * {50 - S(T) } ] if Spot price at maturity S(T) < $50
• CF = 1000 if S(T) > 50 (as the additional component is not paid, only face value is received)

If S(T) = 45, then CF = 1000 + [ 20 * { 50 - 45 } ] = 1000 + [ 20 * 5] = 1100

Similar calculation for other values of S(T), If S(T) = 46, then CF = 1080

• If S(T) = 51, then CF = 1000 + 0 = 1000. S(T) = 52, CF = 1000

Below is the table:

S(T)	Amount paid
45	1100
46	1080
47	1060
48	1040
49	1020
50	1000
51	$1,000
52	$1,000
53	$1,000
54	$1,000
55	$1,000
56	$1,000
57	$1,000
58	$1,000
59	$1,000
60	$1,000