80x115x1.15 2 20 XL.IS 80 >0.24 sed). So = 80 uu Please do all calculations and report your answers using four decimal points, e.g. if r = 0.08; and T = 0.5, then erT = e(0.08x0.5) 1.0408 1. The binomial model can be used to price European and Ainerican op- tions. Consider a two-period (At = 1, T = 2), two-state world. Let the current stock price be $80, and the risk-free rate be 4%. Each pe- riod the stock price can go up by 15% or down by 15%, 1.e. u = = 1.15, d=0.85. Consider options with strike price $80. (a) (15%) Please find the stock price sequence; ( So, S, sd, szusua S," = 92.5 d = 68 , S = 105.8. Soud = 78.2. (b) (10%) Please compute the current price of a European call op- tion that expires in two periods' time (i.e. T = 2). (C) (10%) Please compute the current price of a European put op- tion that expires in two periods' time (i.e. T = 2). (d) (10%) Please compute the current price of an American put option that expires in two periods' time. (e) (15%) For a European call option, please compute the initial hedge ratio at t = 0 (40), and the two possible hedge ratios at the end of the first period (i.e. A and A). (f) (20%) Please construct a riskless hedge portfolio (with call options and stocks) at time t = 0, node 0, and show that your hedge works at t = 1. (p.s. For question (f), (1) you need to construct the hedge by combining a position of short 10,000 calls and some units of stocks (long or short) at time t = 0, and the transaction amount of stocks should be reported in integers. (2) You need to show that your hedge works by (i) calculating the value of the hedge portfolio on each node at t = 1; and (ii) calculating the return of your constructed hedge portfolio from t = 0 to t = 1 for nodes u and d) 80x115x1.15 2 20 XL.IS 80 >0.24 sed). So = 80 uu Please do all calculations and report your answers using four decimal points, e.g. if r = 0.08; and T = 0.5, then erT = e(0.08x0.5) 1.0408 1. The binomial model can be used to price European and Ainerican op- tions. Consider a two-period (At = 1, T = 2), two-state world. Let the current stock price be $80, and the risk-free rate be 4%. Each pe- riod the stock price can go up by 15% or down by 15%, 1.e. u = = 1.15, d=0.85. Consider options with strike price $80. (a) (15%) Please find the stock price sequence; ( So, S, sd, szusua S," = 92.5 d = 68 , S = 105.8. Soud = 78.2. (b) (10%) Please compute the current price of a European call op- tion that expires in two periods' time (i.e. T = 2). (C) (10%) Please compute the current price of a European put op- tion that expires in two periods' time (i.e. T = 2). (d) (10%) Please compute the current price of an American put option that expires in two periods' time. (e) (15%) For a European call option, please compute the initial hedge ratio at t = 0 (40), and the two possible hedge ratios at the end of the first period (i.e. A and A). (f) (20%) Please construct a riskless hedge portfolio (with call options and stocks) at time t = 0, node 0, and show that your hedge works at t = 1. (p.s. For question (f), (1) you need to construct the hedge by combining a position of short 10,000 calls and some units of stocks (long or short) at time t = 0, and the transaction amount of stocks should be reported in integers. (2) You need to show that your hedge works by (i) calculating the value of the hedge portfolio on each node at t = 1; and (ii) calculating the return of your constructed hedge portfolio from t = 0 to t = 1 for nodes u and d)