Question

Problem 9. You are interested in pricing European and American put options on a stock using a four-period binomial model with notation and set-up as in Chapter 10 of the text. So = 30, u = 1.15, d= 0.78, h = 1,= 0, and r = 0.04 The European and American put options expire at the end of the 4th period and each has a strike price of 25. As in the text, r as a continuously compounded interest rate. Compute the price the European put option and the American put option.

Answer 1

American Put option . (A) 10.2); ba (0)60) (35.6) (0) +90-31 (30.95 | (8) 10-20 ton () s au 14) (34.52 1 0 : < (0) (0.43) 10:3N (E)/ 26.9) 20.99 23.4 10.7) 10/(1325) 7 (4:01) 0:3) (16.37) 10.74 13.07) 7 (8.63) (1:1) 0:39' (F) N (So=30) 10.3 Buy :"2838) 6. (18-252 yg (6.748) (14.24) o I(10.76) 10:37 og v 19.817 (13.9) ist period I period 111 period Ith period.

In American options, person has an option to exercise the option at any time, i.e. one does not have to wait till the maturity to exercise the option.

Firslty we have to calculate the value of nodes. For eg. Value at node B is 30*1.15 i.e. 34.5 and value at node C is 30*0.78 i.e. 23.4. We will calculate the value at each node.

Secondly, we have to calculate the probabilities at node A

Upper p = (1.06 - 0.78) / (1.15-0.78)

= 0.7

Lower p = 1- Upper p

= 0.3

Since the value of U and D are same, the value of upper p and lower p will be the same for all the nodes.

Then we will start from the end, ie. Node K

We will check whether this option will get exercises at node K or not. Since the value is greater than 25, it will be lapsed and will be exercised by the holder. Hence the payoff value becomes zero.

Same is the case with Node L.

For Node M, the payoff becomes 25-24.141

Fro Node G, we have to check 2 cases:

a. We will calculate the payoff. The formula is (0*0.7+0*0.3)/1.04 .

= 0

b. We will also check the payoff at that node.Since 45.63 is greater than 25, it will not get exercised and hence payoff will be zero.

We will chose that value which is greater. In this case both the values are 0 hence we will chose 0.

We will do this for all the remaining nodes and finally reach to the answer.

Therefore, the value of the American Put Option is 1.1

16.2 European put option $ 2.42 (0) 10.7 (45.63 (35.6 (39.675) (30.95 34.5 10:25) 103 24.14. LES (6.859) A co.85) (So=30 -7360-3523) 10.3) 32 20.47 Tea (2.122) 10.3 { b) 26.91 4.054 100) to lot 20.99 S 3 16.37 (367) (18.252 (4.896) 1.31 (14.24)

In European options, person do not have an option to exercise the option at any time, i.e. one has to wait till the maturity to exercise the option.

Firslty we have to calculate the value of nodes. For eg. Value at node B is 30*1.15 i.e. 34.5 and value at node C is 30*0.78 i.e. 23.4. We will calculate the value at each node.

Secondly, we have to calculate the probabilities at node A

Upper p = (1.06 - 0.78) / (1.15-0.78)

= 0.7

Lower p = 1- Upper p

= 0.3

Since the value of U and D are same, the value of upper p and lower p will be the same for all the nodes.

Then we will start from the end, ie. Node K

We will check whether this option will get exercises at node K or not. Since the value is greater than 25, it will be lapsed and will be exercised by the holder. Hence the payoff value becomes zero.

Same is the case with Node L.

For Node M, the payoff becomes 25-24.141

For Node I, we do not have to check 2 cases:

a. We will calculate the payoff. The formula is (0.859*0.7+8.63*0.3)/1.04 .

= 3.07

We do not have to compare it with its own payoff value of 4.01 which we have done in American Options.

We will do this for all the remaining nodes and finally reach to the answer.

Therefore, the value of the American Put Option is 0.85