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Value of a call option on a stock

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B&R, a reputable East Coast Bank, has created a portfolio of assets that is selling quite well on the market. This particular portfolio is a series of call options. In particular, the portfolio contains one call option for every year from now into the infinite future. Each call option gives the owner the right to buy 1 share of Frod-Motors at $50 per share in a given year. The first call option can only be exercised immediately; the second call option can only be exercised one year from now; the next one can only be exercised two years from now; etc. for every future year. To recap: anyone who owns this portfolio can exercise one call option each year from now into the infinite future.

Suppose that the interest rate is 10% for the entirety of this problem.

a) Draw a graph depicting the value of the year t call option as a function of Frod-Motors stock price in that same year t. Indicate the minimum price at which the call option would be exercised.

b) Prices for Frod-Motors are forecast to remain constant at $50/share for the foreseeable future. Given this, what is the price per share of the derivative asset?

c) How does your answer to part (b) change if in any given year (including the present year), Frod-Motor share prices are expected to be $30 with probability ½ and $80 with probability ½?

d) How does your answer to part (b) change if today Frod-Motor sells at $50/share, but is expected to grow at 5% per year afterwards? Be as precise as you can, and make sure that your reasoning is clear.

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This solution shows how to calculate the value of a call option on a stock given the stock price and the premium paid for the option.

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a) The interest rate represents the premium we pay for the call option. In this case the premium is 10% of $50, or $5. In a year that we exercise the option and buy the stock, we pay a total of $55. The value of the call option is therefore the difference between the stock price (the price for which we could ...

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