108156.pdf

Solutions 265

1.7a) The payoff of a call option with strike price $90 will be

C(1) =

{

30 if stock goes up,

0 if stock goes down.

The replicating investment intoxshares andybonds satisfies the system of

equations {

x120 +y110 = 30,

x80 +y110 = 0.

The solution isx=^34 andy=− 116. Hence the price of the option must be

C(0) =^34 × 100 − 116 × 100 ∼= 20. 45

dollars.

b) The payoff of a call option with strike price $110 will be

C(1) =

{

10 if stock goes up,

0 if stock goes down.

The replicating investment intoxshares andybonds satisfies

{

x120 +y110 = 10,

x80 +y110 = 0.

Solving this system of equations, we find thatx=^14 andy=− 112. Hence the

price of the option is

C(0) =^14 × 100 − 112 × 100 ∼= 6. 82

dollars.

1.8a) The replicating investment intoxshares andybonds satisfies the system

of equations {

x120 +y105 = 20,

x80 +y105 = 0,

which givesx=^12 andy=− 218. Hence

C(0) =^12 × 100 − 218 × 100 ∼= 11. 91

dollars.

b) In this case the replicating investment intoxshares andybonds satisfies

{

x120 +y115 = 20,

x80 +y115 = 0,

sox=^12 andy=− 238 .It follows that

C(0) =^12 × 100 − 238 × 100 ∼= 15. 22

dollars.