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Unformatted text preview: $86.89 The stock price at node (C) is from a down move, or: Stock price (C) = $58(0.8170) Stock price (C) = $47.39 And the stock price at node (F) is two down moves, or: Stock price (F) = $58(0.8170)(0.8170) Stock price (F) = $38.72 Finally, the stock price at node (E) is from an up move followed by a down move, or a down move followed by an up move. Since the binomial tree recombines, both calculations yield the same result, which is: Stock price (E) = $58(1.2239)(0.8170) = $58(0.8170)(1.2239) Stock price (E) = $58.00 Now we can value the put option at the expiration nodes, namely (D), (E), and (F). The value of the put option at these nodes is the maximum of the strike price minus the stock price, or zero. So: Put value (D) = Max($65 – 86.89, $0) Put value (D) = $0 Put value (E) = Max($65 – 58, $0) Put value (E) = $7 Put value (F) = Max($65 – 38.72, $0) Put value (F) = $26.28 The value of the put at node (B) is the present value of the expected value. We find the expected value by using the value of the put at nodes (D) and (E) since those are the only two possible stock prices after node (B). So, the value of the put at node (B) is: Put value (B) = [.4599($0) + .5401($7)] / 1.0042 Put value (B) = $3.77 473 Similarly, the value of the put at node (C) is the present value of the expected value of the put at nodes (E) and (F) since those are the only two possible stock prices after node (C). So, the value of the put at node (C) is: Put value (C) = [.4599($7) + .5401($26.28)] / 1.0042 Put value (C) = $17.34 Using the put values at nodes (B) and (C), we can now find the value of the put today, which is: Put value (A) = [.4599($3.77) + .5401($17.34)] / 1.0042 Put value (A) = $11.05 Challenge 9. Since the exercise style is now American, the option can be exercised prior to expiration. At node (B), we would not want to exercise the put option since it would be out of the money at that stock price. However, if the stock price falls next month, the value of the put option if exercised is: Value if exercised = $65 – 47.39 Value if exercised = $17.61 This is greater then the present value of waiting one month, so the option will be exercised early in one month if the stock price falls. This is the value of the put option at node (C). Using this put value, we can now find the value of the put today, which is: Put value (A) = [.4599($3.77) + .5401($17.61)] / 1.0042 Put value (A) = $11.20 This is slightly higher than the value of the same option with a European exercise style. An American option must be worth at least as much as a European option, and can be worth more. Remember, an option always has value until it is exercised. The option to exercise early in an American option is an option itself, therefore it can often has some value. 10. Using the binomial mode, we will find the value of u and d, which are: u=e
σ/ n u = e.30/ 1/2 u = 1.2363 d=1/u d = 1 / 1.2363 d = 0.8089 This implies the percentage increase is if the stock price increases will be 24 percent, and the percentage decrease if the stock price falls will be 19 percent. The six month interest rate is: Six month interest rate = 0.06/2 Six month interest rate = 0.03 474 Next, we need to find the risk neutral probability of a price increase or decrease, which will be: 0.03 = 0.24(Probability of rise) + –0.19(1 – Probability of rise) Probability of rise = 0.5173 And the probability of a price decrease is: Probability of decrease = 1 – 0.5173 Probability of decrease = 0.4827 The following figure shows the stock price and call price for each possible move over each of the six month steps: Value (D) Call price Value prepayment Value postpayment (B) Call price $76,423,258 $24,423,258 $61,815,555 $61,165,555 $12,267,307
Value (E) Call price $49,474,242 $0 Stock price(A) Call price $50,000,000 $6,161,619 Value (F) Call price Value prepayment Value postpayment (C) Call price $49,196,398 $0 $40,442,895 $39,792,895 $0 Value (G) Call price $32,186,797 $0 First, we need to find the building value at every step along the binomial tree. The building value at node (A) is the current building value. The building value at node (B) is from an up move, which means: Building value (B) = $50,000,000(1.2363) Building value (B) = $61,815,555 475 At node (B), 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 (B) is: Building value (B) after payment = $61,815,555 – 650,000 Building value (B) after payment = $61,165,555 To find the building value at node (D), we multiply the afterpayment building value at node (B) by the up move, or: Building value (D) = $61,165,555(1.2363) Building value (D) = $76,423,258 To find the building value at node (E), we multiply the afterpayment building value at node (B) by the down move, or: Building value (E) = $61,165,555(0.8089) Building value (E) = $49,474,242 The building value at node (C) is from a down move, which means the building value will be...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of TexasTyler.
 Spring '10
 eshmalwi
 Finance, Corporate Finance

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