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Derivatives- option pricing

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1. A stock currently sells for \$50 and pays no dividends. There is a call option (striking price = \$55) on this security that expires in 61 days. At present U.S. Treasury bills are yielding 3.3 percent per year. You estimate the past volatility of the stock returns to be 39 percent (i.e., standard deviation). Using Black-Scholes option-pricing model, calculate the value of the call.

2. Using the arbitrage argument (i.e., put-call parity) for the value you derived for problem 1, calculate the implied value of the put on the same stock.

3. XYZ Corporation stock sells for \$35 per share. The AUG option series has exactly six months until expiration. At the moment, the AUG 35 call sells for \$3 and the AUG 35 put sells for \$1 3/8. What is the annual interest rate implied in these prices?

4. Consider a two-period, two-state world. Let the current stock price be \$45 and the risk-free rate 5 percent. Each period the stock price can either go up or down by 10 percent. A call option expiring at the end of the second period has an exercise price of \$ 40.

a. What is the initial hedge ratio?
b. What are the two possible hedge ratios at the end of the first period?
c. Illustrate by single example that the hedge portfolio works and earns the risk-free rate over the two periods.
Hint:
Given: S = \$45, r = 5%, u = 10%., d = -10% and E = \$40 where
'S' is stock price, 'r' is interest rate, 'u' is up factor, 'd' is down factor and 'E' is the exercise price.

5. You are going to use Black-Scholes option-pricing model on a six-month call option that has the following expected dividend stream; short-term interest rate is 8 percent.

Time until dividend 3 months 6 months

Expected dividend \$1.50 \$1.75

If the current stock price is \$95, what is the adjusted stock price you would use as n input for the Black-Scholes option-pricing model?

Solution Summary

Answers to 5 questions dealing with call and put options. The following are calculated:
1. Using Black-Scholes option-pricing model, value of the call
2. The implied value of the put on the same stock
3. Annual interest rate implied in call and put prices
4. Hedge ratio

\$2.19

Derivative problems: Option Pricing

1) Option pricing principles are now often used in corporate finance applications. It is common to look at equity in a corporation as an option. What type of option is equity similar to? Discuss what factors would affect the price of equity and in what direction based on this option pricing view.

2) Given recent events Yahoo options have become very popular. Answer remaining questions based on the following information regarding July 2008 (162 day) options.
Stock Price = \$29
Interest rate = 4%
X Call Put
25 1.20
30 2.04 2.85
35 6.10
a. Compute the value of the \$35 call based on put/call parity.
b. Verify that the \$30 call and put are selling at prices above their lower bounds.

3. The price of the \$25 call is \$4.30. Describe an arbitrage opportunity. Detail all your transactions and compute the minimum and maximum

Compute the breakeven points, maximum gain and maximum loss to the following positions:
a. A covered call with exercise price = \$30.
b. A bull spread: Long the put with X=\$35 and short the put with X=\$25
In a two-period pricing model (each period consists of 81 days) u = 1.16 and d = 83. Compute the value of an American put with exercise price = \$30.

4. Use the Black/Scholes model to price a put with X=\$30. (You can either use the equation for a put or calculate the value of a call and find the value of a put using put/call

5. You own 100 shares of Yahoo in your retirement account and would like to hedge against any loss in value (locking in the recent price increase). You want to use options to do so.
a. Based on the delta you calculated in number 6 what position should you take in the option market to hedge your position?
b. Subsequently, stock price decreases \$5. Compute your percentage gain or loss over the 162 day period on your total portfolio (stock and options

6. You own 100 shares of Yahoo in your retirement account and would like to hedge against any loss in value (locking in the recent price increase). You want to use options to do so.
a. Based on the delta you calculated in number 6 what position should you take in the option market to hedge your position?
b. Subsequently, stock price decreases \$5. Compute your percentage gain or loss over the 162 day period on your total portfolio (stock and options).

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