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Unformatted text preview: : Building value (E) = $50,000,000(0.8089) Building value (E) = $40,442,895 At node (C), the accrued rent payment will be made, so the value of the building after the payment will be reduced by the amount of the payment, which means the building value at node (C) is: Building value (C) after payment = $40,442,895 – 650,000 Building value (C) after payment = $39,792,895 To find the building value at node (F), we multiply the after-payment building value at node (C) by the down move, or: Building value (F) = $39,792,895(1.2363) Building value (F) = $49,196,398 Finally, the building value at node (G) is from a down move from node (C), so the building value is: Building value (G) = $39,792,895(0.8089) Building value (G) = $32,186,797 Note that because of the accrued rent payment in six months, the binomial tree does not recombine during the next step. This occurs whenever a fixed payment is made during a binomial tree. For example, when using a binomial tree for a stock option, a fixed dividend payment will mean that the tree does not recombine. With the expiration values, we can value the call option at the expiration nodes, namely (D), (E), (F), and (G). The value of the call option at these nodes is the maximum of the building value minus the strike price, or zero. We do not need to account for the value of the building after the accrued rent payments in this case since if the option is exercised, you will receive the rent payment. So: Call value (D) = Max($76,423,258 – 52,000,000, $0) 476 Call value (D) = $24,423,258 Call value (E) = Max($49,474,242 – 52,000,000, $0) Call value (E) = $0 Call value (F) = Max($49,196,398 – 52,000,000, $0) Call value (F) = $0 Call value (G) = Max($32,186,797 – 52,000,000, $0) Call value (G) = $0 The value of the call at node (B) is the present value of the expected value. We find the expected value by using the value of the call at nodes (D) and (E) since those are the only two possible building values after node (B). So, the value of the call at node (B) is: Call value (B) = [.5173($24,423,258) + .4827 ($0)] / 1.03 Call value (B) = $12,267,307 Note that you would not want to exercise the option early at node (B). The value of the option at node (B) if exercised is the value of the building including the accrued rent payment minus the strike price, or: Option value at node (B) if exercised = $61,165,555 – 52,000,000 Option value at node (B) if exercised = $9,165,555 Since this is less than the value of the option if it left “alive”, the option will not be exercised. With a call option, unless a large cash payment (dividend) is made, it is generally not valuable to exercise the call option early. The reason is that the potential gain is unlimited. In contrast, the potential gain on a put option is limited by the strike price, so it may be valuable to exercise an American put option early if it is deep in the money. We can value the call at node (C), which will be the present value of the expected value of the call at nodes (F) and (G) since those are the only two possible building values after node (C). Since neither node has a value greater than zero, obviously the value of the option at node (C) will also be zero. Now we need to find the value of the option today, which is: Call value (A) = [.5173($12,267,307) + .4827($0)] / 1.04 Call value (A) = $6,161,619 477 CHAPTER 24 WARRANTS AND CONVERTIBLES
Answers to Concepts Review and Critical Thinking Questions 1. A warrant is issued by the company, and when a warrant is exercised, the number of shares increases. A call option is a contract between investors and does not affect the number of shares of the firm. a. If the stock price is less than the exercise price of the warrant at expiration, the warrant is worthless. Prior to expiration, however, the warrant will have value as long as there is some probability that the stock price will rise above the exercise price in the time remaining until expiration. Therefore, if the stock price is below the exercise price of the warrant, the lower bound on the price of the warrant is zero. If the stock price is above the exercise price of the warrant, the warrant must be worth at least the difference between these two prices. If warrants were selling for less than the difference between the current stock price and the exercise price, an investor could earn an arbitrage profit (i.e. an immediate cash inflow) by purchasing warrants, exercising them immediately, and selling the stock. If the warrant is selling for more than the stock, it would be cheaper to purchase the stock than to purchase the warrant, which gives its owner the right to buy the stock. Therefore, an upper bound on the price of any warrant is the firm’s current stock price. 2. b. c. 3. 4. An increase in the stock price volatility increases the bond price. If the stock price becomes more volatile, the conversion option on the stock becomes more valuable. The two components of the value of a convertible bond are the straight bond value and the option value. An increase in...
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