HullFund8eCh17ProblemSolutions

HullFund8eCh17ProblemSolutions - CHAPTER 17 The Greek...

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CHAPTER 17 The Greek Letters Practice Questions Problem 17.8. What does it mean to assert that the theta of an option position is –0.1 when time is measured in years? If a trader feels that neither a stock price nor its implied volatility will change, what type of option position is appropriate? A theta of 0 1   means that if t units of time pass with no change in either the stock price or its volatility, the value of the option declines by 0 1 t   . A trader who feels that neither the stock price nor its implied volatility will change should write an option to create as high a positive theta position as possible. Problem 17.9. The Black–Scholes–Merton price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has written the option plans to use a stop-loss strategy. The trader’s plan is to buy at $40.10 and to sell at $39.90. Estimate the expected number of times the stock will be bought or sold. The strategy costs the trader 0 10 each time the stock is bought or sold. The total expected cost of the strategy, in present value terms, must be $4. This means that the expected number of times the stock will be bought or sold is approximately 40. The expected number of times it will be bought is approximately 20 and the expected number of times it will be sold is also approximately 20. The buy and sell transactions can take place at any time during the life of the option. The above numbers are therefore only approximately correct because of the effects of discounting. Also the estimate is of the number of times the stock is bought or sold in the risk-neutral world, not the real world. Problem 17.10. Suppose that a stock price is currently $20 and that a call option with an exercise price of $25 is created synthetically using a continually changing position in the stock. Consider the following two scenarios: a) Stock price increases steadily from $20 to $35 during the life of the option. b) Stock price oscillates wildly, ending up at $35. Which scenario would make the synthetically created option more expensive? Explain your answer. The holding of the stock at any given time must be 1 ( ) N d . Hence the stock is bought just after the price has risen and sold just after the price has fallen. (This is the buy high sell low strategy referred to in the text.) In the first scenario the stock is continually bought. In second scenario the stock is bought, sold, bought again, sold again, etc. The final holding is the same in both scenarios. The buy, sell, buy, sell... situation clearly leads to higher costs than the buy, buy, buy... situation. This problem emphasizes one disadvantage of creating options synthetically. Whereas the cost of an option that is purchased is known up front and depends on the forecasted volatility, the cost of an option that is created synthetically is not known up
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front and depends on the volatility actually encountered.
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