Consider a 3-step binomial option pricing model for valuing a put option over a stock with

S = $10

exercise price X=$10 and

maturity T=0.75. The volatility, risk free rate and dividend yield are =19.0620%, r =14.2872%, y=0% respectively.

The nodes in the tree are as follows:

7

4

2 8

1 5

3 9

6

10

The tree of stock prices is as follows.

$13.31000

$12.10000

$11 $11.000

$10 $10.000

$9.0909 $9.0909

$8.2645

$7.5131

a) Show that

up factor should be u =1.100000

the down factor should be d = 0.909091

the risk neutral probability of an up jump in the stock price is q=2/3

the stock prices of 13.31, 11.00 and 9.0909 at time step 3 are correct

The option price tree for a European at the money put option is as per the table below:

$0.00

$0.00

$0.09 $0.00

$0.26 $0.29

$0.63 $0.91

$1.38

$2.49

b) Show how the put option values at nodes 5, 8 and 9 are calculated and verify that the figures shown are correct to 2 decimal places.

c) Compute the forward price for the stock under a forward contract maturity in 9 months, from the information used to price the option

d) Compute the value of a long forward contract over the stock with a maturity of 9 months and with the same exercise price as the option.

e) Using the put call parity relationship, compute the price of a European call option over the stock from the value of the forward contract and the value of the put.

The option price tree for an American put option is as follows:

$0.00

$0.00

$0.09 $0.00

$0.35 $0.29

$0.91 $0.91

$1.74

$2.49

f) For which nodes are the American put option values different from the European put option values? Why is there a difference? Explain? Are the results for those nodes correct? Assume the results in the European option tree are correct in answering this.