# Part II. Numerical Problems: you are required to show steps/formulas/solutions to obtain credit, unl

Part II. Numerical Problems: you are required to show steps/formulas/solutions to obtain credit, unless stated otherwise. That is, I need to be able to follow how you get the answer; you cannot simply tell me that the answer is generated by computer.

2. Suppose that there are two calls on the same stock: one with exercise price K of \$30, the other \$35. The market value of the call with K = \$30 is \$2 while that for call with K=\$35 is \$1.5. What positions you need to take in each of the options to create a bullish call spread? Bearish call spread? Describe the payoffs at various stock prices with a set of equations or table, for each strategy. Show all work.

3. On November 21, 2007, Citigroup (C) stock was trading around \$30 a share. Its January call with exercise price of \$27.5 traded for \$4. The risk-free rate was 4%. Compute the theoretical option values at standard deviations of returns at

(a) 20%

(b) 40%

(c) 60%. Which is the above is closest to its implied volatility? What does implied volatility reflect?

4. A stock price is currently \$42. Its stock price will be either \$45 or \$38 one year from now. The risk-free rate is 5%. A one-year call on the stock has an exercise price of \$40.

(a) What are its intrinsic values at stock prices of \$45 and \$38, respectively?

(b) What should be the hedge ratio?

(c) What should be the value of the hedged portfolio at expiration?

(d) What is the value of the call today?

(e) Verify your answer using the risk-neutral approach—do not just say that you have the same answer; you will need to show the work that the two approaches give the same answer.

5. Suppose that an investor holds \$4,000 option that has delta of 0.6 and vega of 1.4. The investor wants to make his portfolio both delta and vega neutral. Suppose that he wants to use another option to achieve the objectives and suppose that this option has a delta of 0.9 and vega of 1. What is the required position (a complete answer requires a dollar amount and specifying long or short) in this option and in futures?