At what stock price at maturity will you break even

Problem 1. Currently, a stock price is \$47. Over each of the next 2 6-month periods it is expected to go up by 15% or down by 15%. The risk-free rate is 6% per annum with continuous compounding. What is the value of a 1-year European call option with a strike price of \$50?

Problem 2. Suppose that put options on a stock with strike prices \$35 and \$45 cost \$1 and \$5, respectively. Use these options to create a bear spread. At what stock price at maturity will you break even?

Problem 3. A stock price is currently \$70. Over each of the next two three-month periods it is expected to go up by 8% or down by 6%. The risk-free rate is 4% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of \$72?

Problem 4. Currently, a stock price is \$100. It is known that at the end of 3 months it will be either \$95 or \$120. The risk-free rate is 13% per annum with continuous compounding. What is the value of a 3-month European put option with a strike price of \$107?

Problem 5. A manager in charge of a portfolio worth \$500,000 is concerned that the market might decline rapidly during the next 3 months. He would like to use index options as a hedge such that his hedged value is constant if S&P 500 falls below 1000. The S&P 500 is standing at 1200 and his portfolio beta is 1.2. The risk-free rate is 8% per annum and the dividend yield on both the index and the portfolio is 2% per annum. His choice of option is put option on S&P 500 with a strike price of 1000. How many put option contracts does he have to buy and what is the lowest hedged value that the manager can expect in 3 months?

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