19. If the standard deviation is zero, d1 and d2 go to +8 because standard deviation is in the denominator of the d1 formula.
As d1 is +8, so there is d2
Therefore N(d1) and N(d2) equal to 1.
So: C = SN(d1) – EN(d2)e–rt C = $84(1) – $80(1)e–.05(6/12) = $5.98
22a.
Note: The value of the firms asset and face value of outstanding debt is found in question 21.
We can use the Black-Scholes model to value the equity of a firm. Chapter 22 has a very thorough explanation how equity can be considered as a call or put.
Using the asset value of $27,500 (the $26,300 current value of the assets plus the $1,200 project NPV) as the stock price, and the face value of debt of $25,000 as the exercise price, the value of the firm if it accepts project A is:
The value of the debt is the firm value minus the value of the equity, so:
DB = $27,900 – 5,921.30 = $21,978.70
b. Although the NPV of project B is higher, the equity value with project A is higher. While NPV represents the increase in the value of the assets of the firm, in this case, the increase in the value of the firm’s assets resulting from project B is mostly allocated to the debtholders, resulting in a smaller increase in the value of the equity. Stockholders would, therefore, prefer project A even though it has a lower NPV.
26. a. In order to solve a problem using the two-state option model, we first need to draw a stock price tree containing both the current stock price and the stock’s possible values at the time of the option’s expiration. Next, we can draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements.
Price of stock
Call option price with a strike of $75
Today
1 year
Today
1 year
$93
$18
=Max($93 – 75, 0)
$78
?
$65
$0
=Max($65 – 75, 0)
The stock price today is $78. It will either increase to $93 or decrease to $65 in one year. If the stock price rises to $93, the call will be exercised for $75 and a payoff of $18 will be received at expiration. If the stock price falls to $65, the option will not be exercised, and the payoff at expiration will be zero. We use the following forumla to determine the risk-neutral probability of a rise in the price of the stock:
Using these risk-neutral probabilities, we can now determine the expected payoff of the call option at expiration. The expected payoff at expiration is:
Expected payoff at expiration = (.5339)($18) + (.4661)($0) Expected payoff at expiration = $9.61
Since this payoff occurs 1 year from now, we must discount it back to the value today. Since we are using risk-neutral probabilities, we can use the risk-free rate, so:
PV(Expected payoff at expiration) = $9.61 / 1.025 PV(Expected payoff at expiration) = $9.38
b. Yes, there is a way to create a synthetic call option with identical payoffs to the call option described above. In order
