[4] 191 [6] Question 5 [33 marks). A share price is modelled via a two-period binomial model with initial stock price S = 40, up/down multiplication factors u = 3/2 and d=1/2, and interest rate per time-period r = 4%. (a) Verify that the no-arbitrage assumption is valid in this model. (b) Show that the risk-neutral probabilities of up and down movements in the share price are Pu= 0.54 and Pa = 0.46. (e) Find the no-arbitrage price of a European call option on the share with strike K = 60 and expiry T = 2. (d) Use the delta-hedging formula to find the replicating portfolio (Y(0),(0)) at 1 = 0 for the option in part (e). Your stock broker advises you to consider buying an American call option. (e) Explain the difference between a European and American call option. (1) State, without explanation, whether you would expect the price of an American call option on the share with strike K = 60 and expiry T = 2 to be higher, lower, or the same as the corresponding European call option priced in part (e). The Black-Scholes formula for a European call option with strike K and expiryT written on a stock with current price S, drift us and volatility o is C = 50(1) - Ke-T0(-OVT), where In + rT OT (8) Using put-call parity, or any other method, prove that the Black-Scholes formula for a European put option with strike K and expiry T written on the same stock is P = Ke PloVT - 1) - S0(-4). [3] [3] +2OVT. [4] [4] 191 [6] Question 5 [33 marks). A share price is modelled via a two-period binomial model with initial stock price S = 40, up/down multiplication factors u = 3/2 and d=1/2, and interest rate per time-period r = 4%. (a) Verify that the no-arbitrage assumption is valid in this model. (b) Show that the risk-neutral probabilities of up and down movements in the share price are Pu= 0.54 and Pa = 0.46. (e) Find the no-arbitrage price of a European call option on the share with strike K = 60 and expiry T = 2. (d) Use the delta-hedging formula to find the replicating portfolio (Y(0),(0)) at 1 = 0 for the option in part (e). Your stock broker advises you to consider buying an American call option. (e) Explain the difference between a European and American call option. (1) State, without explanation, whether you would expect the price of an American call option on the share with strike K = 60 and expiry T = 2 to be higher, lower, or the same as the corresponding European call option priced in part (e). The Black-Scholes formula for a European call option with strike K and expiryT written on a stock with current price S, drift us and volatility o is C = 50(1) - Ke-T0(-OVT), where In + rT OT (8) Using put-call parity, or any other method, prove that the Black-Scholes formula for a European put option with strike K and expiry T written on the same stock is P = Ke PloVT - 1) - S0(-4). [3] [3] +2OVT. [4]