Question

Consider a binomial model with three dates t = 0, 1, 2, two events “up” and “down” at date 1, and two date-2 successors (“up” and “down”) of each date-1 event. The price of the risky asset is S0 = 50 at date 0 and moves up by factor u = 1.3 or down by factor d = 1 each period. One-period risk-free rate of return isr = 0.2, the same at dates 1 and 2.

(i) Find strictly positive state prices and risk-neutral probabilities.

(ii) Find date-0 price of European call option on the risky asset with expiration date 2 and exercise price 60. Find date 1 prices of the option in events “up”

and “down.”

(iii) Suppose that the call option is American so that it can be exercised at date

1. Is it better to exercise the option or sell it at date 1 in event “up”? Do the same comparison for event “down.” Would the price of the American option at date 0 be different from the price of the European option? Explain.

Answer 1

Given the inputs in the questions, we can build the following table or formula representations for different periods/time

Today, time = 0                                 time = 1                                                time = 2

1.                                                            2.                                                         4.

Stock Price = X                     Stock Price = X*u                                Stock Price = X*u*u

Option Price = c                    Option Payoff =                         Option Payoff =

                                                 

                                                                                                          Stock Price = X*u*d

                                                                                                        Option Payoff =

                                                                                                    5. The above is same as:

                                                               3.                                         Stock Price = X*d*u

                                                Stock Price = x*d                            Option Payoff =

                                          Option Payoff =                                             6.

                                                                                                           Stock Price = X*d*d

                                                                                                        Option Payoff =

Here, u = 1.3 and d = 1, X or S0 = 50, t = 0, 1, 2

c = [e(-rt)/u-d] * [(e(-rt)-d * Pup + (u - e(-rt)) * Pdn]

Or, Taking q = (e(-rt)-d) / (u-d)

the above equation becomes:

C = e(-rt) * (q * Pup + (1-q)Pdn) = 1.05127*50 = 52.5635