compute a value for the option using the portfolio replication and the risk-neutral probability approach. Please turn in a brief answer to all the questions. Consider International Business Machine Corporation (IBM). The shares of IBM are currently trading at $83. Assume the yearly volatility of IBM is 30% for the first two months and then, after two months, changes: if the price of the sock has increased in the two months the annualized volatility will decrease to 25% (for the next two months), while if the stock price decreased the annualized volatility will increase to 35%. Using a two step binomial tree approach, we are going to price options with 4 months to maturity. The annualized risk-free rate is equal to 5%. Lets also assume that the company pays a $5 dividend in two months. Note that the change in volatility will affect the stock only after the dividend is paid.

1. Compute the price of the European call option with a strike price of 85 using the portfolio replication method

2. Compute the price of the European call option with a strike price of 85 using the risk-neutral probability method

3. Compute the price of the American call option with a strike price of 85 using the risk-neutral probability method

4. Compute the price of the European put option with a strike price of 90 using the risk-neutral probability method

5. Compute the price of the American put option with a strike price of 90 using the risk-neutral probability method