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23. We can use the Black–Scholes model to value the equity of a firm. Using the asset value of $27,200
as the stock price, and the face value of debt of $25,000 as the exercise price, the value of the firm’s
equity is:
1
Putting these values into the Black–Scholes model, we find the equity value is:
The value of the debt is the firm value minus the value of the equity, so:
The return on the company’s debt is:
24. a. The combined value of equity and debt of the two firms is:
b. For the new firm, the combined market value of assets is $48,900, and the combined face value of
debt is $45,000. Using Black–Scholes to find the value of equity for the new firm, we find:
Putting these values into the Black–Scholes model, we find the equity value is:
Equity = $48,900(.7271) – ($45,000e–.05(1))(.6232)
c. The change in the value of the firm’s equity is:
The change in the value of the firm’s debt is:
d. In a purely financial merger, when the standard deviation of the assets declines, the value of the
25. a. Using the Black–Scholes model to value the equity, we get:
10
Putting these values into Black–Scholes:
b. The value of the debt is the firm value minus the value of the equity, so:
c. Using the equation for the PV of a continuously compounded lump sum, we get:
d. The new value of assets is the current asset value plus the project NPV. Using the Black–Scholes
model to value the equity, we get:
Putting these values into Black–Scholes:
e. The value of the debt is the firm value minus the value of the equity, so:
Using the equation for the PV of a continuously compounded lump sum, we get:
When the firm accepts the new project, part of the NPV accrues to bondholders. This increases
the present value of the bond, thus reducing the return on the bond. Additionally, the new project
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.
The stock price today is $62. It will either increase to $76 or decrease to $54 in one year. If the
stock price rises to $76, the call will be exercised for $65 and a payoff of $11 will be received
If the stock price rises, its return over the period is 22.58 percent [= ($76/$62) – 1]. If the stock
price falls, its return over the period is –12.90 percent [= ($54/$62) – 1]. We can use the
following expression to determine the risk-neutral probability of a rise in the price of the stock:
This means the risk neutral probability of a stock price decrease is:
Using these risk-neutral probabilities, we can now determine the expected payoff of the call
option at expiration. The expected payoff at expiration is:
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:
b. Yes, there is a way to create a synthetic call option with identical payoffs to the call option
described above. In order to do this, we will need to buy shares of stock and borrow at the risk-
free rate. The number of shares to buy is based on the delta of the option, where delta is defined
as:
Therefore, the first step in creating a synthetic call option is to buy .50 of a share of the stock.
Since the stock is currently trading at $62 per share, this will cost $31.00 [= (.50)($62)]. In
Call Option
The payoff of this synthetic call position should be identical to the payoff of an actual call
option. However, owning .50 of a share leaves us exactly $27.00 above the payoff at expiration,
regardless of whether the stock price rises or falls. In order to reduce the payoff at expiration by
c. Since the cost of the stock purchase is $31.00 to purchase .50 of a share and $26.34 is
borrowed, the total cost of the synthetic call option is:
27. a. In order to solve a problem using the two-state option model, we first 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.
The stock price today is $45. It will either decrease to $37 or increase to $68 in six months. If
If the stock price rises, its return over the period is 51.11% [= ($68/$45) – 1]. If the stock price
The risk-free rate over the next six months must be used in the order to match the timing of the
Which means the risk-neutral probability of a decrease in the stock price is:
Using these risk-neutral probabilities, we can determine the expected payoff of the put option at
expiration as:
Since this payoff occurs 6 months from now, we must discount it at the risk-free rate in order to
find its present value, which is:
b. Yes, there is a way to create a synthetic put option with identical payoffs to the put option
described above. In order to do this, we need to short shares of the stock and lend at the risk-
free rate. The number of shares that should be shorted is based on the delta of the option, where
delta is defined as:
Since the put option will be worth $0 if the stock price rises and $13 if it falls, the swing of the
call option is –$13 (= $0 – 13). Since the stock price will either be $68 or $37 at the time of the
Therefore, the first step in creating a synthetic put option is to short .42 of a share of stock.
Since the stock is currently trading at $45 per share, the amount received will be $18.87 (= .42
Delta shares
The payoff of the synthetic put position should be identical to the payoff of an actual put
option. However, shorting .42 of a share leaves us exactly $28.52 below the payoff at
expiration, whether the stock price rises or falls. In order to increase the payoff at expiration by
c. Since the short sale results in a positive cash flow of $18.87 and we will lend $27.83, the total
cost of the synthetic put option is:
28. a. The company would be interested in purchasing a call option on the price of gold with a strike
price of $1,380 per ounce and 3 months until expiration. This option will compensate the
b. In order to solve a problem using the two-state option model, first draw a price tree containing
both the current price of the underlying asset and the underlying asset’s possible values at the
time of the option’s expiration. Next, draw a similar tree for the option, designating what its
value will be at expiration given either of the 2 possible stock price movements.
The price of gold is $1,270 per ounce today. If the price rises to $1,465, the company will
exercise its call option for $1,380 and receive a payoff of $85 at expiration. If the price of gold
falls to $1,120, the company will not exercise its call option, and the firm will receive no payoff
at expiration. If the price of gold rises, its return over the period is 15.35 percent [= ($1,465 /
The risk-free rate over the next three months must be used in order to match the timing of the
expected price change. Since the risk-free rate per annum is 6.50 percent, the risk-free rate over
the next three months is 1.59 percent [= (1.0650)1/4 – 1], so:
And the risk-neutral probability of a price decline is:
Using these risk-neutral probabilities, we can determine the expected payoff of the call option
at expiration, which will be.
Since this payoff occurs 3 months from now, it must be discounted at the risk-free rate in order
to find its present value. Doing so, we find:
Therefore, given the information about gold’s price movements over the next three months, a
c. Yes, there is a way to create a synthetic call option with identical payoffs to the call option
described above. In order to do this, the company will need to buy gold and borrow at the risk-
free rate. The amount of gold to buy is based on the delta of the option, where delta is defined
as:
Since the call option will be worth $85 if the price of gold rises and $0 if it falls, the swing of
the call option is $85 (= $85 – 0). Since the price of gold will either be $1,465 or $1,120 at the
Therefore, the first step in creating a synthetic call option is to buy .25 of an ounce of gold.
Since gold currently sells for $1,270 per ounce, the company will pay $312.90 (= .25 × $1,270)
Call Option
Delta Shares
The payoff of this synthetic call position should be identical to the payoff of an actual call
option. However, buying .25 of a share leaves us exactly $275.94 above the payoff at
expiration, whether the price of gold rises or falls. In order to decrease the company’s payoff at
expiration by $275.94, it should borrow the present value of $275.94 now. In three months, the
d. Since the company pays $312.90 in order to purchase gold and borrows $271.63, the total cost
of the synthetic call option is $41.27 (= $312.90 – 271.63). This is exactly the same price for an
