• Consider a stock, XYZ, whose current price is $100 with volatility equal to 60% per annum. XYZ does not pay dividends. The annual (not continuously compounded) risk free interest rate is 12%.

Consider a standard European option on XYZ with 3 months to maturity and exercise price . What is the Black-Scholes value of the call option?

What is the value of a put on XYZ with the same E and same time to maturity as the call?

The CBOE has recently approved the creation of a special European option on XYZ, which they are calling a "High-Five". This high-five, on expiration, gives the holder the right to sell the underlying stock (XYZ) for the highest stock price that occurred during the life of the option (inclusive of the date on which it was issued) minus $5.00.

Assuming this option has a maturity of 3 months and there are four `trading periods' in this time period. What is the price of this High-Five? (Hint use the Binomial model and remember that the Binomial model uses the per period interest rate calculated on a simple compounding basis).

If you were to attempt to replicate this option with a position in the XYZ stock and risk free debt. what position would you initially construct? How would it change if the up state were first observed (context of this question is still the four period binomial model.

• You are the senior assistant to the Chairman of the Board of Hi-Teck Inc. In order to attract the chief operating officer of his choice from another firm, the Chairman offers the following incentive supplement to her base salary package:

2% of any price appreciation in the price of the stock up to $60 per share;

3% of any price appreciation above $60 but less than $75 per share;

4% of any price appreciation above $75 per share;

The price appreciation is to be computed with respect to the share price exactly 3 years from now, and there is no early payment of the supplement. The aggregate value of the supplement is based on the price appreciation of 10 million shares. The annual (not continuously compounded) risk free rate is 10% and the stock volatility is 50% annualized. The current price of the stock is $55.

Use the Black Scholes formula to value the supplement. Should you be concerned in your calculations that the new chief operating officer might increase the expected rate of price appreciation by wise actions?

• Suppose calls were traded with the indicated exercise prices where

and

for all i and j (equally spaced). Draw the payoff diagram of the following portfolio:

{write 1 call, buy 1 call, buy 1 call, write 1 call, write 1 call, buy 1 call}

₁, _{2 3 4 5 6}

1. Consider a stock, XYZ, whose current price is $100 with volatility equal to 60% per annum. XYZ does not pay dividends. The annual (not continuously compounded) risk free interest rate is 12%. (i) (ii) (iii) Consider a standard European option on XYZ with 3 months to maturity and exercise price E=100. What is the Black-Scholes value of the call option? What is the value of a put on XYZ with the same E and same time to maturity as the call? The CBOE has recently approved the creation of a special European option on XYZ, which they are calling a "High-Five". This high-five, on expiration, gives the holder the right to sell the underlying stock (XYZ) for the highest stock price that occurred during the life of the option (inclusive of the date on which it was issued) minus $5.00. Assuming this option has a maturity of 3 months and there are four `trading periods' in this time period. What is the price of this High-Five? (Hint use the Binomial model and remember that the Binomial model uses the per period interest rate calculated on a simple compounding basis). (iv) If you were to attempt to replicate this option with a position in the XYZ stock and risk free debt. what position would you initially construct? How would it change if the up state were first observed (context of this question is still the four period binomial model. 2. You are the senior assistant to the Chairman of the Board of Hi-Teck Inc. In order to attract the chief operating officer of his choice from another firm, the Chairman offers the following incentive supplement to her base salary package: (i) 2% of any price appreciation in the price of the stock up to $60 per share; (ii) 3% of any price appreciation above $60 but less than $75 per share; (iii) 4% of any price appreciation above $75 per share; The price appreciation is to be computed with respect to the share price exactly 3 years from now, and there is no early payment of the supplement. The aggregate value of the supplement is based on the price appreciation of 10 million shares. The annual (not continuously compounded) risk free rate is 10% and the stock volatility is 50% annualized. The current price of the stock is $55. Use the Black Scholes formula to value the supplement. Should you be concerned in your calculations that the new chief operating officer might increase the expected rate of price appreciation by wise actions? 3. Suppose calls were traded with the indicated exercise prices where E1 E 2 E 3 E 4 E 5 E 6 and E i 1 E i E j 1 E j for all i and j (equally spaced). Draw the payoff diagram of the following portfolio: {write 1 call, buy 1 call, buy 1 call, write 1 call, write 1 call, buy 1 call} E=E1, E=E2 E=E3 E=E4 E=E5 E=E6