4.Risk-free rate. As the risk-free rate increases from 12 to 16 percent, the value of
the option increases slightly, from $1.88 to $1.98. Equations 17-1, 17-2, and 17-3
suggest that the principal effect of an increase in rRFis to reduce the present value
of the exercise price, XerRFt, hence to increase the current value of the option.^9
The risk-free rate also plays a role in determining the values of the normal distri-
bution functions N(d 1 ) and N(d 2 ), but this effect is of secondary importance. In-
deed, option prices in general are not very sensitive to interest rate changes, at least
not to changes within the ranges normally encountered.
5.Variance. As the variance increases from the base case 0.16 to 0.25, the value of the
option increases from $1.88 to $2.27. Therefore, the riskier the underlying secu-
rity, the more valuable the option. This result is logical. First, if you bought an op-
tion to buy a stock that sells at its exercise price, and if 2 0, then there would be
a zero probability of the stock going up, hence a zero probability of making money
on the option. On the other hand, if you bought an option on a high-variance
stock, there would be a fairly high probability that the stock would go way up,
hence that you would make a large profit on the option. Of course, a high-variance
stock could go way down, but as an option holder, your losses would be limited to
the price paid for the option—only the right-hand side of the stock’s probability
distribution counts. Put another way, an increase in the price of the stock helps op-
tion holders more than a decrease hurts them, so the greater the variance, the
greater is the value of the option. This makes options on risky stocks more valuable
than those on safer, low-variance stocks.Myron Scholes and Robert Merton were awarded the 1997 Nobel Prize in Eco-
nomics, and Fischer Black would have been a co-recipient had he still been living.
Their work provided analytical tools and methodologies that are widely used to
solve many types of financial problems, not just option pricing. Indeed, the entire
field of modern risk management is based primarily on their contributions. The next
section discusses the application of option pricing to real options.What is the purpose of the Black-Scholes Option Pricing Model?
Explain what a “riskless hedge” is and how the riskless hedge concept is used in
the Black-Scholes OPM.
Describe the effect of a change in each of the following factors on the value of a
call option:
(1) Stock price.
(2) Exercise price.
(3) Option life.
(4) Risk-free rate.
(5) Stock price variance, that is, riskiness of stock.Introduction to Real Options
According to traditional capital budgeting theory, a project’s NPV is the present value of
its expected future cash flows, discounted at a rate that reflects the riskiness of the expected634 CHAPTER 17 Option Pricing with Applications to Real Options
(^9) At this point, you may be wondering why the first term in Equation 17-1, P[N(d 1 )], is not discounted. In
fact, it has been, because the current stock price, P, already represents the present value of the expected
stock price at expiration. In other words, P is a discounted value, and the discount rate used in the market to
determine today’s stock price includes the risk-free rate. Thus, Equation 17-1 can be thought of as the pres-
ent value of the end-of-option-period spread between the stock price and the strike price, adjusted for the
probability that the stock price will be higher than the strike price.