Question 4 [20 marks) 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 o=19.0620%, r=14.2872%, y=0% respectively. The nodes in the tree are as follows: 2 8 5 3 6 10 The tree of stock prices is as follows. $10.0000 $11.0000 $9.0909 $12.1000 $10.0000 $8.2645 $13.3100 $11.0000 $9.0909 $7.5131 a) Show that up factor should be a -1.100000 the down factor should be d = 0.909091 the risk neutral probability of an up jump in the stock price is a the stock prices of 13.31, 11.00 and 9.0909 at time step 3 are correct 3 marks 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. 3 marks c) Compute the forward price for the stock under a forward contract maturity in 9 months, from the information used to price the option 3 marks 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. 3 marks 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. 3 marks 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. 5 marks Question 4 [20 marks] 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 o=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. $12.1000 $13.3100 $11.0000 $11.0000 $10.0000 $9.0909 $10.0000 $8.2645 $9.0909 $7.5131 a) Show that up factor should be s = 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 2 2 the stock prices of 13.31, 11.00 and 9.0909 at time step 3 are correct 3 marks 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 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. 5 marks 5