This preview shows page 1. Sign up to view the full content.
Unformatted text preview: using traditional cash flow techniques, and then real options are applied to the resulting cash flows. Insurance is a put option. Consider your homeowner’s insurance. If your house were to burn down, you would receive the value of the policy from your insurer. In essence, you are selling your burned house (“putting”) to the insurance company for the value of the policy (the strike price). 2. 3. 4. 5. 6. 7. 8. 9. 10. In a market with competitors, you must realize that the competitors have real options as well. The decisions made by these competitors may often change the payoffs for your company’s options. For example, the first entrant into a market can often be rewarded with a larger market share because the name can become synonymous with the product (think of Q-tips and Kleenex). Thus, the option to become the first entrant can be valuable. However, we must also consider that it may be better to be a later entrant in the market. Either way, we must realize that the competitors’ actions will affect our options as well. Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance (σ 2) of the underlying asset, and the continuously-compounded risk-free interest rate (R). Since these options were granted at-the-money, the strike price of each option is equal to the current value of one share, or $55. We can use Black-Scholes to solve for the option price. Doing so, we find: d1 = [ln(S/K) + (R + σ 2/2)(t) ] / (σ 2t)1/2 d1 = [ln($55/$55) + (.06 + .452/2) × (5)] / (.45 × d2 = .8013 – (.45 ×
5 ) = –.2050 5 ) = .8013 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(0.8013) = 0.7885 N(d2) = N(–0.2050) = 0.4188 Now we can find the value of each option, which will be: C = SN(d1) – Ke-–RtN(d2) C = $55(0.7885) – ($55e–.06(5))(0.4188) C = $26.30 Since the option grant is for 30,000 options, the value of the grant is: Grant value = 30,000($26.30) Grant value = $789,123.34 465 b. Because he is risk-neutral, you should recommend the alternative with the highest net present value. Since the expected value of the stock option package is worth more than $750,000, he would prefer to be compensated with the options rather than with the immediate bonus. If he is risk-averse, he may or may not prefer the stock option package to the immediate bonus. Even though the stock option package has a higher net present value, he may not prefer it because it is undiversified. The fact that he cannot sell his options prematurely makes it much more risky than the immediate bonus. Therefore, we cannot say which alternative he would prefer. c. 2. The total compensation package consists of an annual salary in addition to 15,000 at-the-money stock options. First, we will find the present value of the salary payments. Since the payments occur at the end of the year, the payments can be valued as a three-year annuity, which will be: PV(Salary) = $375,000(PVIFA9%,3) PV(Salary) = $949,235.50 Next, we can use the Black-Scholes model to determine the value of the stock options. Doing so, we find: d1 = [ln(S/K) + (R + σ 2/2)(t) ] / (σ 2t)1/2 d1 = [ln($34/$34) + (.05 + .742/2) × (3)] / (.74 × d2 = .7579 – (.74 × 3 ) = .7579 3 ) = –.5238 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(0.7579) = 0.7757 N(d2) = N(–0.5238) = 0.3002 Now we can find the value of each option, which will be: C = SN(d1) – Ke-–RtN(d2) C = $34(0.7757) – ($34e–.05(3))(0.3002) C = $17.59 Since the option grant is for 15,000 options, the value of the grant is: Grant value = 15,000($17.59) Grant value = $263,852.43 The total value of the contract is the sum of the present value of the salary, plus the option value, or: Contract value = $949,235.50 + 263,852.43 Contract value = $1,213,087.93 466 3. Since the contract is to sell up to 5 million gallons, it is a call option, so we need to value the contract accordingly. Using the binomial mode, we will find the value of u and d, which are: u=e
σ/ n u = e.46/ 12/3 u = 1.2586 d=1/u d = 1 / 1.2586 d = 0.7945 This implies the percentage increase if gasoline increases will be 26 percent, and the percentage decrease if prices fall will be 21 percent. So, the price in three months with an up or down move will be: PUp = $1.74(1.2586) PUp = $2.19 PDown = $1.74(0.7945) PDown = $1.38...
View Full Document