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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?