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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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## Q : Unbiased estimate of the future spot rate

If the forward rate was expected to be an unbiased estimate of the future spot rate, and interest rate parity holds, then: