1 The Black-Scholes Formula
In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper, coupled with closely related work by Robert Merton, revolutionized both the theory and practice of finance.
1.1 Call Options
The Black-Scholes formula for a European call option on a stock that pays dividends at the con- tinuous rate δ is
c(S, K, σ, r, T, δ) = Se−δTN (d1) − Ke−rTN (d2) (1) where
d1= ln(S/K) + (r − δ +12σ2)T σ√
T d2= d1− σ√
T
There are six inputs to the Black-Scholes formula: S, the current price of the stock; K, the strike price of the option; σ, the volatility of the stock; r, the continuously compounded risk-free interest rate; T , the time to expiration; and δ, the dividend yield on the stock.
N (x)in the Black-Scholes formula is the cumulative normal distribution function, which is the probability that a number randomly drawn from a standard normal distribution (i.e., a normal distribution with mean 0 and variance 1) will be less than x.
Example 1 (Practice Problem 8) Let S = $41, K = $40, σ = 0.3, r = 8%, T = 0.25 (3 months), and δ = 0. Compute the Black-Scholes call price.
1.2 Put Options
The Black-Scholes formula for a European put option is
p(S, K, σ, r, T, δ) = Ke−rTN (−d2) − Se−δTN (−d1) (2) where d1and d2are given in the Black-Scholes formulation for a European call option.
Since the Black-Scholes call and put prices, equations (1) and (2), are for European options, put-call parity must hold:
p(S, K, σ, r, T, δ) = c(S, K, σ, r, T, δ) + Ke−rT − Se−δT
This version of the formula, together with the fact that for any x, 1 − N (x) = N (−x), satisfies equations (1) and (2).
Example 2 (Practice Problem 9) Let S = $41, K = $40, σ = 0.3, r = 8%, T = 0.25 (3 months), and δ = 0. Compute the Black-Scholes put price.
1.3 When is the Black-Scholes formula valid?
Derivations of the Black-Scholes formula make a number of assumptions that can be sorted into two groups: Assumptions about how the stock price is distributed, and assumptions about the economic environment. For the version of the formula we have presented, assumptions about the distribution of the stock price include the following:
• Continuously compounded increase in the stock price are normally distributed and inde- pendent over time.
• The volatility of continuously compounded increase in the stock price is known and con- stant.
• Future dividends are known, either as a dollar amount or as a fixed dividend yield.
Assumptions about the economic environment include the following:
• The risk-free rate is known and constant.
• There is no transaction costs or taxes.
• It is possible to short-sell costlessly and to borrow at the risk-free rate.
Many of these assumptions can easily be relaxed. For example, with a small change in the formula, we can permit the volatility and interest rate to vary over time in a know way.
1.4 Options on Stocks with Discrete Dividends
When a stock makes discrete dividend payments, we have to find an appropriate dividend yield for the known discrete steam of dividends.
From the original Black-Scholes formula given by equation (1), we can rewrite d1 as follows:
d1 = ln(Se−δT /Ke−rT)+12σ2T σ√
T
When d1 is rewritten in this way, it is apparent that the dividend yield enters the formula only to discount the stock price, as Se−rT, and the interest rate enters the formula only to discount the strike price, as Ke−rT. Notice also that volatility enters only as σ2T.
We can price a European option with discrete dividends by subtracting the present value of dividends from the stock price, and entering the result into the formula in place of the stock price.
1.5 Perpetual Calls
Suppose we have a perpetual American call with strike K. If we decide to exercise the option whenever S hits a certain level H, then at exercise we receive H − K. The present value of $1 payable when the stock price reaches H is
Value of $1 received when S first reaches H from below = S H
h1
where
h1 = 1
2− r − δ σ2 +
s
r − δ σ2 − 1
2
2
+ 2r σ2 Therefore, the value of receiving H − K when S reaches H is
(H − K) S H
h1
In order to finish computing the value of the call we need to specify H, the price at which the call should be exercised. We simply need to pick a value for H that makes the value of the call as great as possible. If we make H too small, when we prematurely throw away option value (i.e., protec- tion against a subsequent price decline). If we make H too large, then we forgo dividends for too long while waiting to exercise. It is possible to show that the exercise level H∗that maximizes the value of the call is
H∗ = K
h1
h1− 1
Since h1 > 1, we have H∗ > K. Making this substitution, the value of the perpetual call is
Price of perpetual call = K h1− 1
h1− 1 h1
S K
h1
(3)
If δ = 0, then H∗ = ∞; i.e., it is never optimal to exercise a call option on a nondividend-paying stock.
1.6 Perpetual Puts
For a perpetual put, we need to know the present value of $1 payable when the stock price reaches Hfrom above, which is
Value of $1 received when S reaches H from above = S H
h2
where
h2 = 1
2− r − δ σ2 −
s
r − δ σ2 − 1
2
2
+ 2r σ2
Therefore, the value of a perpetual call if we exercise when S = H is given by
(K − H) S H
h2
Again selecting the exercise level H∗to maximize the value of the put, we get
H∗ = K h2 h2− 1 which implies the price of the perpetual put is
Price of perpetual put = K 1 − h2
h2− 1 h2
S K
h2
(4)
2 Back to Real Options
The decision to invest in a project involves a comparison of net present value; in what sense is this an option? We can view the investment cost as an exercise price and the value of the project as the underlying asset.
The decision to exercise an option prior to expiration involves an implicit comparison of three factors: The dividends forgone by not acquiring the asset today; the interest saved by deferring the payment of the strike price; and the value of the insurance that is lost by exercising the option.
It turns out that the same three considerations govern the decision to invest in a project.
Example: Widget ProjectSuppose we can invest in a machine, costing $10, that will produce one widget a year forever. In addition, each widget costs $0.90 to produce. The price of widgets will be $0.55 next year and will increase at 4% per year. The effective annual risk-free rate is 5% per year. We can invest, at any time, in one such machine. There is no uncertainty.
Static NPV
A natural first step is to compute the NPV if we invested in the project today. We obtain
NPV(invest today) = $0.55 ×
1
1.05 + 1.04
1.052 +1.042 1.053 + · · ·
− $0.9 ×
1
1.05 + 1
1.052 + 1
1.053 + · · ·
− $10
= $0.55
1.04 × 1
1.05 1.04− 1
!
− $0.9 0.05− $10
= $0.55 0.01 − $28
= $27
This calculation tells us that if widget production were to start next year, we would pay $27 for the project.
If we delay investment, the project is worth more than $27. In the early years, the project has an operating loss. If we activate the project today, then next year we will have negative operating cash flows, spending $0.90 to produce a $0.55 widget. In addition, at a 5% rate of interest, the opportunity cost of the $10 investment is $0.50/year.
Although the initial cash flows are negative, the widget price is growing. The project will be- come profitable in the future. Suppose we wait 5 years to invest instead of investing immediately.
NPV is then
NPV(wait 5 years) = 1 1.055
(1.04)5$0.55 0.01 − $28
= $30.49 Thus, it is better to wait 5 years than to invest today.
In this example it would be correct to invest in the project today if not activating the project today meant that we would lose it forever. Under this assumption, the mutually exclusive alter- native (never taking the project) has a value of 0, so taking it today would be correct.
To decide whether and when to invest in an arbitrary project, we need to be able to compute the value of delaying that investment, if doing so is allowed (flexibility).
The Project as an Option
The decision to invest in the project involves a comparison of net present values; in what sense is this an option? As suggested earlier, we can view the investment cost as an exercise price and the value of the project as the underlying asset.
The decision to exercise an option on a stock prior to expiration involves an implicit compar- ison of three factors: The dividends forgone by not acquiring the asset today; the interest saved by deferring the payment of the strike price; and the value of the insurance that is lost by exercis- ing the option. It turns out that the same three considerations govern the decision to invest in a project.
In the widget project, there is no uncertainty and, hence, non insurance value. However, there are interest and forgone dividends.
Once we begin widget production, we are committed to spending the present value of the marginal widget cost, $18, along with the $10 initial investment. The value of delaying investment is interest on the total investment cost, or 0.05 × $28 = $1.40 per year.
In delaying investment, we lose the cash flow from selling widgets. This forgone cash flow is analogous to a stock dividend not received. The lost cash flow is initially $0.55. The present value of future cash flows is $0.55/0.01=$55. Thus, the dividend yield is approximately 1%. (We can also think of the dividend yield as the difference between the discount rate (5%) and the growth
We can compute the value of the widget project option using the perpetual call calculation . The formula assumes continuously compounded rates, so for the interest rate we use ln(1.05) = 4.879%, and for the dividend yield we use the difference between the continuously compounded interest rate and growth rate, or ln(1.05) − ln(1.04) = 0.9569%.
With S = $55(the present value of revenue), K = $28(The present value of costs), r = 0.04879, σ = 0, and δ = 0.009569, the value of the perpetual call by equation (3) is $35.03 and investment when the widget price equals $1.4276. We will call this price the investment trigger price. We reach this price after about 24.32 years.
