Assume that the current price of stock share is S(0) = 120, and we are interested to compute the price of an European call option to buy that stock at strike price X = 120 and exercise time T = 2, using the binomial tree model with step size ?t = 1 and N = 2 steps. Assume that the risk free interest rate is R = 0.1, and that at each step the price of the stock can go up at a rate of U = 0.2 or can go down at a rate D = −0.1 .
(a) Compute the price of the European call option using the self-?nancing strategy as below (a.i) Starting with S(1) = Su and the scenarios (Suu −X)+ and (Sud − X)+, by replicating the option with x-units of stock and y-units of $1
bond, ?nd the price Cu(1) of the option at time t = 1 (you will need to solve a linear system in x and y); then, starting with S(1) = Sd and the scenarios (Sud − X)+ and (Sdd − X)+, by replicating the option with x-units of stock and y-units of $1 bond, ?nd the price Cd(1) of the option at time t = 1 (you will need to solve a linear system in x and y); (a.ii) Starting with S(0) and the scenarios Cu(1) and Cd(1) from above, by replicating the option with x-units of stock and y-units of $1 bond, ?nd the price CE = C(0) of the option at time t = 0 (you will need to solve a linear system in x and y).
(b) Compute the price of the European call option by using the ‘Cox-RossRubinstein’ formula as follows: (b.i) Compute the risk-neutral probability p = R−D U−D ; (b.iii) Generate a new N-step binomial tree model for the stock price S(T), with N = 2, where at each step the price of the stock can go up at a rate of U with probability p or can go down at a rate D with probability 1−p. (b.ii) Compute the price of the option using the formula CE = E∗((1 + R)−N(S(2) − X)+), where the expectation is computed relative to the risk-neutral probability p;