Problem A-1 Consider a European Call and a European Put option on a stock trading at a price of A1. The exercise price of either option is A2 and the time to maturity is A3 months. The stock's volatility (sigma) is A4 per annum, and the risk-free interest rate is A5 percent per annum, continuously compounded. Use a five step binomial model to find the current fair values of the call and put options.

1. Write the six risk-neutral probability weights from top to bottom of stock price realizations at maturity:

2. Write the six values (cash flows) of the Call option from top to bottom at maturity: 7.

3. Write the six values (cash flows) of the Put option from top to bottom at maturity: 13.

4. Write the current fair value of the European Call

5. Write the current fair value of the European Put

6. Write the value of "u"

7. Write the value of "p":

Problem A-2. Consider an American Call and an American Put option on a stock trading at a price of A1. The exercise price of either option is A2 and the time to maturity is A3 months. The stock's volatility (sigma) is A4 per annum, and the risk-free interest rate is A5 percent per annum, continuously compounded. Use a five step binomial model to find the current fair values of the call and put options.

8. Write the current fair value of the American Call

9. Write the current fair value of the American Put

Problem B. The price of a stock is currently B1. The stock price by the end of the next three-month period is expected to be up by 10 percent or down by 10 percent. The risk-free interest rate is B2 percent per annum with continuous compounding. What is the current value of a three-month American call option with strike price of B3 using a single-step binomial tree? How will you trade involving one call option to make arbitrage profits if the call option's current market price is B4.

1. What is the value of p?

2. What are the two values (cash flows) of the call option at maturity from top to bottom of the tree?

3. Cash flows at the top of the tree.

4. Cash flows at the bottom of the tree.

5. What is the current fair value of the call option?

6. What is the value of delta at time 0?

7. Write 1 if your answer is to take a long position in the call option and 0 if it is to take a short position in the call option in an arbitrage trading strategy at time zero.

8. Write the net cash position (if borrowed write with a negative sign) at time zero. Write the two net cash positions from top to bottom at the end of one step (maturity) in your trading strategy:

9. Write the net cash made as of maturity from the arbitrage trading strategy.

Problem C. A stock's price is currently C1. Over each of the next two three-month periods it is expected to go up by 10 percent or down by 10 percent. The risk-free interest rate is C2 percent per annum with continuous compounding. What is the current value of a six-month European Put option with strike price of C3 using a two-step binomial tree? How will you trade to make profits if the put option's current market price is C4, using one option in trading? Answer the following.

1. Current fair value of put option.

2. Value of delta at time zero.

3. Value of delta at three months when the stock price is up.

4. Value of delta at three months when the stock price is down.

5. Net cash position (bank balance) at time zero.

6. Net cash position (bank balance) at upper branch at time 3 months.

7. Net cash position (bank balance) at lower branch at time 3 months.

8. Net cash position (bank balance) at upper branch at maturity.

9. Net cash position (bank balance) at middle branch at maturity.

10. Net cash position (bank balance) at lower branch at maturity.

Problem D. A stock is currently trading at D1. The annual volatility is D2. The risk-free interest rate is D3 percent per annum with continuous compounding. What is the current value of a six-month American Put option with strike price of D4 using a five-step binomial tree? Find the following.

1. Value of "u".

2. Value of "d"

3. Value of risk-neutral probability "p."

4. Current fair value of American Put.

5. Write the value of the stock on the tree where pre-mature exercise of the American put is optimal. Report "0" if premature exercise of the option is not optimal.