BS formula in exam formula sheet

Question 1

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:

d1 = [ln($27,500/$25,000) + (.05 + .552/2) 1] / (.55 ) = .5392

d2 = .5392 – (.55 ) = –.0108

N(d1) = .7051

N(d2) = .4957

Putting these values into the Black-Scholes model, we find the equity value is:

Value of firms equity = $27,500(.7051) – ($25,000e–.05(1))(.4957) = $7,603.04

The value of the debt is the firm value minus the value of the equity, so:

DA = $27,500 – 7,603.04 = $19,896.96

We need to redo the calculations for project B.

And the value of the firm if it accepts Project B is:

d1 = [ln($27,900/$25,000) + (.05 + .342/2) 1] / (.34 ) = .6399

d2 = .6399 – (.34 ) = .2999

N(d1) = .7389

N(d2) = .6179

Putting these values into the Black-Scholes model, we find the equity value is:

EB = $27,900(.7389) – ($25,000e–.05(1))(.6179) = $5,921.30

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:

ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – .5339 ProbabilityFall = .4661 or 46.61%

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