# 012 - 12 Bounds on Option Prices Answers to Questions and...

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90 Answers to Questions and Problems 1. What is the maximum theoretical value for a call? Under what conditions does a call reach this maximum value? Explain. The highest price theoretically possible for a call option is to equal the value of the underlying stock. This happens only for a call option that has a zero exercise price and an infinite time until expiration. With such a call, the option can be instantaneously and costlessly exchanged for the stock at any time. Therefore, the call must have at least the value of the stock itself. Yet it cannot be worth more than the stock, because the option merely gives access to the stock itself. As a consequence, the call must have the same price as the stock. 2. What is the maximum theoretical value for an American put? When does it reach this maximum? Explain. The maximum value of a put equals the potential inflow of the exercise price minus the associated outflow of the stock price. The maximum value for this quantity occurs when the stock price is zero. At that time, the value of the put will equal the exercise price. In this situation, the put gives immediate potential access to the exercise price because it is an American option. 3. Answer question 2 for a European put. As with the American put, the European put attains its maximum value when the stock price is zero. However, before expiration, the put cannot be exercised. Therefore, the maximum price for a European put is the present value of the exercise price, when the exercise price is discounted at the risk-free rate from expiration to the present. This discounting reflects the fact that the owner of a European put cannot exercise now and collect the exercise price. Instead, he or she must wait until the option expires. 4. Explain the difference in the theoretical maximum values for an American and a European put. The exercise value of a put option equals the exercise price (an inflow) minus the value of the stock at the time of exercise (an outflow). In our notation, this exercise value is X–S . For any put, the maxi- mum value occurs when the stock is worthless, S 5 0. The American and European puts have different maximum theoretical values because of the different rules governing early exercise. Because an American put can be exercised at any time, its maximum theoretical value equals the exercise price, X . If the stock price is zero at any time, an American put gives its owner immediate access to amount X through exer- cise. This is not true of a European put, which can be exercised only at expiration. If the option has time remaining until expiration and the stock is worthless, the European put holder must wait until expiration to exercise. With the stock worthless, the exercise will yield X to the European put holder.

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012 - 12 Bounds on Option Prices Answers to Questions and...

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