CHAPTER 3: PRINCIPLES OF OPTION PRICING

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1. (Basic Notation and Terminology) The average of the bid and ask discounts is 8.22.

Discount = 8.22(68/360) = 1.5527 Price = 100 – 1.5527 = 98.4473 Yield = (100/98.4473) (365/68) – 1 = 0.0876

Note that even though the T-bill matured in 67 days, we must use 68 days since that is the option's time to expiration.

2. (Minimum Value of a Call) This would create an arbitrage opportunity. The call would be purchased and immediately exercised. For example, suppose S0 = 44, X = 40, and the call price is $3. Then an investor would buy the call and immediately exercise it. This would cost $3 for the call and $40 for the stock. Then the stock would be immediately sold for $44, netting a risk-free profit of $1. In other words, the investor could obtain a $44 stock for $43. Since everyone would do this, it would drive the price of the call up to at least $4. If the call were European, however, immediate exercise would not be possible (unless, of course, it was the expiration day), so the European call could technically sell for less than the intrinsic value of the American call. We saw, though, that the European call has a lower bound of the stock price minus the present value of the exercise price (assuming no dividends). Since this is greater than the intrinsic value, the European call would sell for more than the intrinsic value. Then at expiration, it would sell for the intrinsic value.

3. (Lower Bound of a European Call) The call is underpriced, so buy the call, sell short the stock, and buy risk-free bonds with face value of X. The cash received from the stock is greater than the cost of the call and bonds. Thus, there is a positive cash flow up front. The payoffs from the portfolio at expiration are as follows:

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