Chapter Six
The Black-Scholes Option Pricing Model
Answers to Problems and Questions
1.
Interest rates may have changed, the expected amount of the next dividend
may have changed, or the market may anticipate a different future level of
volatility in the stock. Also, another day has passed, so the option has less
time value.
2.
Call premiums rise as interest rates go up. This can be explained by
noticing that the “second half” of the BSOPM is discounted by the riskless
interest rate. If you discount by a larger number, you get a smaller value.
Because the second half of the model is subtracted from the first half, a
larger discount rate means that the model will predict a higher option
premium.
3.
The option trader is more concerned about what will happen in the future
than what has already occurred. Historical volatility deals with the past,
while implied volatility gives an indication about the future.
Implied volatility is probably more important in this respect, but the best
answer to this question recognizes that implied volatility should be
compared with historical volatility. If implied volatility is very high
relative to the historical figure, this may mean that option premiums are
“too high,” or that there is excessive speculation in the underlying security.
4.
The Options Clearing Corporation will direct that striking prices, the
number of options you hold, and the number of shares covered by each
option be adjusted to account for the stock split. No one benefits or loses
in the options market simply because a firm split its stock.
5.
Option premiums are affected by interest rates because of the time value of
money. Higher interest rates reduce the value of having to pay the striking
price, which is to the advantage of the call holder. They also reduce the
value of cash received in the future, which is to the detriment of the put
holder.
6.
We know from financial research that investors care primarily about the
risk and expected return of their investments. For this reason, securities
that offer the same expected return and have the same risk should sell for
the same price.
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Chapter Six. The Black-Scholes Option Pricing Model
7.
Share prices decline on the ex-dividend dates. A decline in share price
reduces the value of a call option. The higher the dividend yield, the
greater the negative impact on call values.
8.
A dividend cut normally is because of adverse developments at the firm.
If this was unanticipated news, it would most likely cause the share price
and the call premium to decline despite the reduced “dividend pressure”
on the call premium.
9.
A perpetual European put option can never be exercised. This means you
would never be able to collect any cash from exercising the option.
10. The standard deviation of a series of raw data is not the same as the
standard deviation of the logarithms of returns on the same data. The
standard deviation using logarithms is usually less.
11. If a company announced its intent to go out of business and pay a large
liquidating dividend it would make sense to exercise the option before the
ex-dividend date. After payment of the dividend the stock would be
worthless, and so would the option.
12. If the options were European style, they could never be exercised. The
options would have to be American style.
13. The first step would be to estimate the volatility of Outback Steakhouse
common stock. You can find the current stock price (which equals the
striking price). We have no information on the life of the option, so you
have to assume something here. The interest rate should match the term
of the option.
14. The Black-Scholes inputs are as follows.
S = value of the assets (analogous to stock price) = $10 million
K = face value of the bonds (striking price) = $6 million
R = risk free rate = 5%
T = 730 days or two years
σ = 25%
According to the BSOPM, the value of the call is $4,602,623,
corresponding to the value of the equity. The market value of the bonds,
then, is $10,000,000 - $4,602,623 = $5,397,377. Answers will vary a bit
depending on whether you assume the debt is callable (i.e. an American
style option) or not (European style).
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Chapter Six. The Black-Scholes Option Pricing Model
If volatility increases to 40%, the new value of the call is $4,850,612 and
the market value of the bonds falls to $5,149,388.
15. According to an official at Nasdaq, they valued the warrants using a
volatility of 30%.
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