This homework is intended to show how to extend the multi-period binomial tree to price an option that pays dividends and has constant volatility. You are required to 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%. Let's 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 portfolio replication method
2. Compute the price of the European risk-neutral probability method
3. Compute the price of the American risk-neutral probability method
4. Compute the price of the European risk-neutral probability method
5. Compute the price of the American risk-neutral probability method

HOMEWORK 3: Multi-period binomial tree Due before class. This homework is intended to show how to extend the multi-period binomial tree to price an option that pays dividends and has constant volatility. You are required to 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%. Let's 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 1