Calculation of Probability of Increase in Stock Price of a company

Question

Today (time t = 0) the stock price of company XYZ is S = 70. In 6 months (time t = 0:5) it changes (under the true probability measure P) with probability of 70% to S(u)0.5 = 90 and with probability 30% to S(d)0.5 = 60. From time t = 0:5 to time t = 1 the stock price may increase by 30% with a probability of 50% or decrease by 20% with a probability of 50%. The risk free interest rate is constant and equal to 5% (c.c.).
(a)  Draw a tree.
(b)  Use replicating portfolios to calculate the price of a European at-the-money call option with 1 year left to maturity.
(c) What is the price of an American at-the-money call option with 1 year left to maturity?
(d)  Suppose you own 10 stocks of company XYZ. At time t = 0, how many long or short positions do you have to take in European at-the-money call options with 1 year to maturity if you like to hedge your entire exposure to stock price fluctuations?

Summary

The question belongs to Statistics and it is about calculation of probability of increase in stock price of a company. By using American and European at the money call options have been calculated. These and other calculations about the stock prices of the company have been calculated.

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