3.
a.
The two possible values of the index in the first period are:
uS0 = 1.06 50 = 53
dS0 = 0.95 50 = 47,5
The possible values of the index in the second period are:
uuS0 = (1.06)2 50 = 56,18
udS0 = 1.06 0.95 50 = 50,35
duS0 = 0.95 1.06 50 = 50,35
ddS0 = (0.95)2 50 = 45,125
b.
The call values in the second period are:
Cuu = 56,18 − 53 = 3,18
Cud = Cdu = Cdd = 0
Since Cud = Cdu = 0, then Cd = 0.
To compute Cu, first compute the hedge ratio:
Form a riskless portfolio by buying five shares of stock and writing nine calls.
The cost of the portfolio is: 5S – 9Cu = 5 x $53 – 9Cu
The payoff for the riskless portfolio equals $252:
Therefore, find the value of the call by solving:
5 x $53 – 9Cu = $252/1.0125 Cu = $ 1,79
To compute C, compute the hedge ratio:
Form a riskless portfolio by buying 0.3255 of a share and writing one call.
The cost of the portfolio is: 0.3255S – C = $16.275 – C
The payoff for the riskless portfolio equals $15.4615:
Therefore, find the value of the call by solving:
$16.275 – C = $15.4615/1.0125 Þ C = $1.00438
d.
The put values in the second period are:
Puu = 0
Pud = Pdu = 53 – 50,35 = 2,65
Pdd = 53 – 45,125 = 7,875
To compute Pu, first compute the hedge ratio:
