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A stock price is currently $50. Over each of the next two 3-month periods it is...

A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of $51? 4

Calculate the price of the put option in problem 3 if it was American.

Solutions

Expert Solution

A tree describing the behavior of the stock price is shown in below figure. The risk-neutral probability of an upward move, p, is given by

There is a payoff from the option of Cuu = 56.18-51= 5.18 for the highest final node (which corresponds to two up moves), zero in all other cases.

The value of the option is therefore,

This can also be calculated by working back through the tree as indicated in below Figure.

At each node, upper number is the stock price, next number is the call price, final number is put price.

We get a payoff of P uu =0, if the upper final node is reached,

Pud=51-50.35=0.65 ,if the middle final node is reached and,

a payoff of P dd=51-45.125=5.875, if the lowest final node is reached.

The value of the option is therefore

Using put-call parity too,

At node C the payoff from immediate exercise is 51-47.5=3.5 . This is greater than 2.8664, the option should be exercised at this node. The option should not be exercised at either node A or node B.


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