9°° = 28.2a tag" F288 oe = HF (28.2)Equation (28.2) is structurally very similar to the Black-Scholes differential equation (12.15). Aswe would expect, it is exactly the same as the Black-Scholes differential equation when 6 is S, theprice of a non-dividend-paying stock. This is because @ is then the price of an investment asset andfrom equation (28.1) must satisfy m — r= As, so that the second term in equation (28.2) becomesafro—a0Comparing equation (28.2) with equation (13A.4), we see that the differential equation for theprice of an asset dependent on @ is the same as that for a derivative dependent on an assetproviding a dividend yield equal to g, where g=r—m-+As. This observation leads to a way ofextending the traditional risk-neutral valuation result.
Extension of Traditional Risk-Neutral Valuation
Any solution to equation (134.4) for S is a solution to equation (28.2) for @, and vice versa, whenthe substitutiongG-—r—mtisis made. As explained in Section 13.2, we know how to solve (13A.4) using risk-neutral valuation.This involves setting the expected growth rate of S equal to r — gq and discounting expected payoffsat the risk-free interest rate. It follows that we can solve equation (28.2) by setting the expectedgrowth of @ equal tor—(r—~m+aAs)=m—hs
and discounting expected payoffs at the risk-free interest rate.Real Options 663
In general, any asset dependent on @ can be valued by reducing the expected growth rate of 6by As, from m to m — As, and then behaving as though the world is risk neutral. The procedure is tocalculate expected cash flows based on the assumption that 6’s expected growth rate is n: — As anddiscount these cash flows at the risk-free rate.
Example 28.1 The cost of renting commercial real estate in a certain city is quoted as the amountthat would be paid per square foot per year in a new five-year rental agreement. The current cost is$30 per square foot. The expected growth rate in the cost is 12% per annum, its volatility is 20% perannum, and its market price of risk is 0.3. A company has the opportunity to pay $1 million now forthe option to rent 100,000 square feet at $35 per square foot for a five-year period starting in twoyears. The risk-free rate is 5% per annum (assumed constant). Define V as the quoted cost per squarefoot of office space in two years. We make the simplifying assumption that rent is paid annually inadvance. The payoff from the option is
100,000A max(V — 35, 0)where A is an annuity factor:
ARTEL x COE 1 ye e052 4 1 ye pA OSF 4 Tye gu OOSH4 _ 4 5355The expected payoff in a risk-neutral world is therefore
100,000 x 4.5355 x £[max(V — 35, 0)] = 453,550 E[max(V — 35, 0)}where & denotes the expectation in a risk-neutral world. Using the result in equation (12A.1), this is
453,550[E(V)(d,) — 35M(d2)]
whereai Pry (2st ne 44d= MLL J/ JI] PUL K Lh' 0.22InfE(V)/35] — 0.2? x 2/2dy =0.2./2
The expected growth rate in the cost of commercial real estate in a risk-neutral world is m — As,where m is the real-world growth rate, » is the volatility, and 4 is the market price of risk. In this case,m = 0.12, s = 0.2, and 4 = 0.3, so that the expected risk-neutral growth rate is 0.06 or 6% per year. Itfollows that £(V) = 30e°°%? = 33.82 Substituting this in the expression above gives the expectedpayoff in a risk-neutral world as $1.5015 million. Discounting at the risk-free rate the value of theoption is 1.5015e °°*? = $1,3586 million. This shows that it is worth paying $1 million for theoption.
The risk-neutral valuation result in Chapter 12 is a particular case of the more generai result wehave derived here. To see this, suppose that @ is the price of a non-dividend-paying stock. Fromequation (28.1),
m—r=Asorm—-As=r
which shows that changing the expected growth rate of @ to m — As involves setting the return fromthe stock equal to the risk-free rate of interest.
It is interesting to note that our extension of the traditional risk-neutral valuation arguments inChapters 10 and 12 is more subtle than it first appears. When @ is not the price of an investment664 CHAPTER 28
asset, the risk-neutral valuation argument does not necessarily tell us anything about what wouldhappen in a risk-neutral world. It simply states that changing the expected growth rate of 6 from mto m—As and then behaving as though the world is risk neutral gives the correct values forderivatives. For convenience, however, we will refer to a world where expected growth rates arechanged from m to m—As as a risk-neutral world.
Several Underlying Variables
We now extend the analysis by supposing that there are » variables, 6,, @2,..., 9, followingstochastic processes
i
dO,a =m; dt + s; dz; (28.3)
for i=1,2,...,”, where the dz; (1 <i <n) are Wiener processes. The parameters m; and s; areexpected growth rates and volatilities and may be functions of the 6, and time t. The instantaneouscorrelation between 6; and 6; will be denoted by p;;. The process for the price, f, of an asset that isdependent on the 0; has the form
a 28.4)
i=l
In this equation, yz is the expected return from the asset and o; dz, is the component of the risk ofthis return attributable to 6,.Equation (21.13) shows that
hors Sais; (28.5)i=l
where A; is the market price of risk for 6;. 1t6’s lemma in Appendix 21A can be used to express uand o; in terms of the m; and s;. Equation (28.5) then leads (see Problem 28.4) to
ale
awag, 7! (28.6)
of of yalat dea, — Aisi) +5 » iS: 9;9,
Our risk-neutral valuation arguments presented earlier concerning differential equation (28.2) canbe extended to cover the more general differential equation (28.6). As a result, we find that an assetcan always be valued as if the world were risk neutral, provided that the expected growth rate ofeach underlying variable is assumed to be m; — d,s; rather than m;. The volatility of the variablesand the coefficient of correlation between variables are not changed. This result was first developedby Cox, Ingersoll, and Ross and represents an important extension to the basic risk-neutralvaluation argument.? The valuation methodology is valid when the asset’s cash flow at a time T
> To illustrate this point, suppose that 6 is the temperature in the center of New Orleans. The process followed by 8clearly does not depend on risk preferences (at least, not the risk preferences of human beings), but it is possible thatthere is a nonzero market price of risk associated with this vartable because of the relationship between temperaturesand agricultural production. We might, therefore, have to adjust the drift of the process when valuing derivativesdependent on 6.
3 See Lemma 4 in J. C. Cox, J. E. Ingersoll, and $. A. Ross, “An Intertemporal General Equilibrium Modet of AssetPrices,” Keonometrica, 53 (1985), 363-84. 
Real Options 665
is some function of the paths followed by the 9; up to time T, as well as when it is dependent onlyon the values of the 6; at time T.
28.3 ESTIMATING THE MARKET PRICE OF RISK
 
The risk-neutral valuation approach to evaluating an investment avoids the need to estimate risk-adjusted discount rates in the way described in Section 28.1, but it does require the market price ofrisk parameters for all stochastic variables. When historical data are available on a particularvariable, its market price of risk can be estimated using the capital asset pricing model. To showhow this is done, we consider an investment asset dependent solely on the variable and define:
w: Expected return of the investment asseto: Volatility of the return of an investment assetA: Market price of risk of the variablep: Instantaneous correlation between the percentage changes in the variable and returns on abroad index of stock market pricesMm: Expected return on broad index of stock market pricesOm: Volatility of return on the broad index of stock market pricesr: Short-term risk free rateBecause the investment asset is dependent solely on the market variable, the instantaucouscorrelation between its return and the broad index of stock market prices is also p. From thecontinuous-time version of the capital asset pricing model, we have
por= —(Ly, — 7)On
From equation (28.1), another expression for yz — r is
w-r=hoIt follows that
h= 5 (Hm -r) (28.7)
This equation can be used to estimate A.
Example 28.2 A historical analysis of company’s sales, quarter by quarter, show that percentagechanges in sales have a correlation of 0.3 with returns on the S&P 500 index. The volatility of theS&P 500 is 20% per annum and, based on historical data, the expected excess return of the S&P 500over the risk-free rate is 5%. Equation (28.7) estimates the market price of risk for the company’ssales as
0.32 x 0.05 = 0.075
When no historical data are available for the particular variable under consideration other similarvariables can sometimes be used as proxies. For example, if a plant is being constructed tomanufacture a new product, data can be collected on the sales of other similar products. The666 CHAPTER 28
 
correlation of the new product with the market index can then be assumed to be the same as that ofthese other products. In some cases the estimate of p in equation (28.7) must be based on subjectivejudgment. If an analyst is convinced that a particular variable is unrelated to the performance of amarket index, its market price of risk should be set to zero.
For some variables it is not necessary to estimate the market price of risk because the processfollowed by a variable in a risk-neutral world can be estimated directly. For example, if the variableis the price of an investment asset, its total return in a risk-neutral world is the risk-free rate. If thevariable is the short-term interest rate, r, Chapter 23 shows how a risk-neutral process can beestimated from the initial term structure of interest rates. Later in this chapter we will show how therisk-neutral process for a commodity can be estimated from futures prices.
28.4 APPLICATION TO THE VALUATION OF A NEW BUSINESS
 
Traditional methods of business valuation, such as applying a price/earnings multiplier to currentearnings, do not work well for new businesses, Typically a company’s earnings are negative duringits early years as it attempts to gain market share and establish relationships with customers. Thecompany must be valued by estimating future earnings and cash flows under different scenarios.
The company’s future cash flows typically depend on a number of variables such as sales,variable costs as a percent of sales, fixed costs, and so on. Single estimates should be sufficient forsome of the variables. For key variables a risk-neutral stochastic process should be estimated asoutlined in the previous two sections. A Monte Carlo simulation can then be carried out togenerate alternative scenarios for the net cash flows per year in a risk-neutral world. It is likely thatunder some of these scenarios the company does very well and under others it becomes bankruptand ceases operations. (The simulation must have a built-in rule for determining when bankruptcyhappens.) The value of the company is the present value of the expected cash flow in each yearusing the risk-free rate for discounting.
Schwartz and Moon (2000) applied this approach to the internet start up, Amazon.com, at theend of 1999.4 They assumed that the company’s sales revenue, R, and its revenue growth rate, p,are stochastic.* Their model is
é = pdt+a(t)dz,withdp = K(f — p)dt + n(t) dz
These equations show that the revenue has an expected growth rate y, which itself follows astochastic process. The stochastic process for jz is mean reverting to a long-run average growthrate ji at rate K.
The volatility of revenues, o(f), was assumed to be deterministic and to decrease exponentiallyfrom an initial level of 10% per quarter to a long-run average level of 5% per quarter. Thestandard deviation of the sales growth rate, 7(t), was also assumed to be deterministic. It decreasedexponentially from an initial level of 3% per quarter to zero. The initial sales level was $356 million
4 B.S. Schwartz and M. Moon, “Rational Pricing of Internet Companies,” Financial Analysts Journal, May/June2000, 62-75.
> Strictly speaking, R is the instantaneous rate at which revenues are generated. When discretized, the stochasticprocesses can be used to calculate the sample changes in the revenue and its growth rate from one quarter to the next.Real Options 667
 
per quarter and the initial sales growth rate was 11% per quarter. The values of « and ji were 7%per quarter and 1.5% per quarter, respectively. The two Wiener processes dz, and dz, wereassumed to be uncorrelated.
Schwartz and Moon assumed that cost of goods sold would be 75% of sales, other variableexpenses would be 19% of sales, and fixed expenses would be $75 million per quarter.° The initialtax loss carry forward was $559 million and the tax rate was assumed to be 35%.
The market price of risk, 2g, for R was estimated from historical data using the approachdescribed in the previous section. The market price of risk for 2 was assumed to be zero. The risk-neutral stochastic process for R was therefore
a = (uu — Apo)dt + odz,
32.332>Db5323>2B
while the risk-neutral process for uw was the same as the real ¢Whe ine msx-neutrar process ror was the Same as tne rea: Wor
The time horizon for the analysis was 25 years and the terminal value of the company wasassumed to be ten times pretax operating profit. The initial cash position was $906 million and thecompany was assumed to go bankrupt if the cash balance became negative.
Different future scenarios were generated using Monte Carlo simulation. The evaluation of thescenarios involved taking account of the possible exercise of convertible bonds and the possibleexercise of employee stock options. The value of the company to the shareholders was calculated asthe present value of the net cash flows discounted at the risk-free rate.
The market price of Amazon.com’s shares at the end of 1999 was $76.125. The price per sharegiven by the Monte Carlo simulation based on the assumptions we have outlined was $12.42.However, Schwartz and Moon point out that a small change in the assumption about the initialvalue of 7 leads to a big change in the price per share. For example, when (0) was increasedfrom 3% to 6%, the share price given by the Monte Carlo simulation increased to about $100.
28.5 COMMODITY PRICES
 
 
Many investments involve uncertainties related to future commodity prices. Often, futures pricescan be used to estimate the risk-neutral stochastic process for a commodity price directly.Commodity prices are therefore similar to investment assets in that the need to estimate themarket price of risk directly can be avoided.
From equation (13.16) the expected future price of the commodity in a risk-neutral world is itsfutures price. If we assume that the growth rate in the commodity price is dependent solely on timeand that the volatility of the commodity price is constant, the risk-neutral process for thecommodity price, S, has the form
dsi Lt) dt + odz (28.8)
thenF(t) = EI S(O] = Selo"
© This means that, in the simulation, variable costs were 94% of sales. Schwartz and Moon point out thatAmazon.com’s variable costs had historically been more than 100% of sales. Amazon.com’s future profitabilitydepended critically on its ability to reduce the cost of goods sold as a percent of sales, and Schwartz and Moonsuggest that it could be appropriate to develop a more elaborate model where cost of goods sold is modeledstochastically.668 CHAPTER 28
where F(t) is the futures price for a contract with maturity t and £ denotes the expected value ina risk-neutral world. It follows that
In F(t) = In S(O) + i p(t) dr0
Oileentiating hath <idee with «VUICICNLAUE DULL SIUCS ILE
Gas
GaSotCG5
=&=a
w(t) = fin FCO]
Example 28.3 Suppose that the futures prices of live cattle at the end of July 2002 are (in cents perpound) as follows:
August 2002 62.20October 2002 60.60December 2002 62.70February 2003 63.37April 2003 64.42June 2003 64.40
These can be used to estimate the expected growth rate in live cattle prices in a risk-neutral world.For example, when the model in equation (28.8) is used, the expected growth rate in live cattle pricesbetween October and December 2002 in a risk-neutral world is
62.70in(F) = 0.034
or 3.4% with continuous compounding. On an annualized basis this is 20.4% per annum.Ip ig p
Example 28.4 Suppose that the futures prices of live cattle are as in Example 28.3. A certainbreeding decision would involve an investment of $100,000 now and expenditures of $20,000 in threemonths, six months, and nine months. The result is expected to be that extra cattle will be availablefor sale ai the end of ihe year. There are two major unceriaintiies: ihe number of pounds of exiracattle that will be available for sale and the price per pound. The expected number of poundsis 300,000. The expected price of cattle in one year in a risk-neutral world is, from Example 28.3,64.40 cents per pound. Assuming that the risk-free rate of interest is 10% per annum, the value of theinvestment (in thousands of dollars) is
= 100 ~ 2067719925 — a9 e999 _. 2 @°O1*9-75 4.300 x 0.64428 *! = 17.729
This assumes that any uncertainty about the extra amount of cattle that will be available for sale haszero systematic risk and that there is no correlation between the amount of cattle that will be
available for sale and the price.
A Mean-Reverting Process
It can be argued that the process in equation (28.8) for commodity prices is too simplistic. Inpractice, most commodity prices follow mean-reverting processes. They tend to get pulled back toacentral value. A more realistic process than equation (28.8) for the risk-neutral process followed bythe commodity price, S, is
din S = [@(t) — aln S]dt + odz (28.9)Real Options 669
This incorporates mean reversion and is analogous to the lognormal process assumed for theshort-term interest rate in Chapter 23, The trinomial tree methodology in Section 23.12 can beadapted to construct a tree for S and determine the value of O(t), so that F(t) = E[S(d)].
We will illustrate this process by building a three-step tree for oil. Suppose that the spot price ofoil is $20 per barrel and the one-year, two-year, and three-year futures prices are $22, $23, and $24,respectively. Suppose that a = 0.1 and o = 0.2 in equation (28.9). We first define a variable X thatis initially zero follows the process
dX = —adt +odz (28.10)
Using the procedure in Section 23.12, a trinomial tree can be constructed for X. This is shown inFigure 28.1.
The variable In S follows the same process as X except for a time-dependent drift. Analogouslyto Section 23.12, we can convert the tree for X to a tree for InS by displacing the positions ofnodes. This tree is shown in Figure 28.2. The initial node corresponds to an oil price of 20, so thedisplacement for that node is In 20. Suppose that the displacements of the nodes at one year is a).The values of the X at the three nodes at the one-year point are +3.464, 0, and —3.464, Thecorresponding values of In S are 3.464+ a), a;, and —3.464+a,. The values of S are thereforee361 61 and e3364+%1 | respectively. We require the average value of § to equal the futuresprice. This means that
0.16672 3441 + 0.6666e"! + 0.1667e 34rt — 22
The solution to this is a, = 3.071. The values of S at the one-year point are therefore 30.49, 21.56,and 15.25.
           
B0.34640,0000
i iNrene Zsa]
Node: A B Cc D E F G H I
Pu 0.1667 0.1217 0.1667 0.2217, «0.8867 0.1217 0.1667 = 0.2217 0.0867Pm 0.6666 0.6566 = 0.6666 = (0.6566 = 0.0266 »=—- 0.6566 = 0.6666 = 0.6566 = 0.0266Pa 0.1667 0.2217 0.1667 0.1217, 0.0867, 0.2217 0.1667 0.1217 0.8867
Figure28.1_ Tree for X. Constructing this tree is the first stage in constructing a tree for the spot price ofoil, S. Here, py, Pm, and py are the probabilities of “up”, “middle”, and “down” movements from a node670 CHAPTER 28
     
‘Lit Lie |
 
Node: A B Cc D E F G H I
 
Pu 0.1667 0.1217 0.1667) 0.2217 0.8867 0.1217 «0.1667 = 0.2217 (0.0867Pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266Pa 0.1667 0.2217, 0.1667) 0.1217 0.0867) 0.2217, 0.1667 = 0.1217 0.8867
Figure28.2 Tree for spot price of oil. Here, p,, Py», and py are the probabilities of “up”, “middle”, and“down” movements from a node
At the two-year point, we first calculate the probabilities of nodes E, F, G, H, and I beingreached from the probabilities of nodes B, C, and D being reached. The probability of reachingnode F is the probability of reaching node B times the probability of moving from B to F plus theprobability of reaching node C times the probability of moving from C to F. This is
0.1667 x 0.6566 + 0.6666 x 0.1667 = 0.2206
The probabilities of reaching nodes E, F, G, H, and I are 0.0203, 0.2206, 0.5183, 0.2206, and0.0203, respectively. The amount, a2, by which the nodes at time two years are displaced mustsatisfy
0.020398 ta2 +. 9.22060" M+" 4. 0.5183e% + 0.2206e 9 454442 + 0.9203e7 078 te2 = 23
The solution to this is a = 3.099. This means that the valucs of S at the two-year point are 44.35,31.37, 22.18, 15.69, and 11.10, respectively. A similar calculation can be carried out at time threeyears. Figure 28.2 shows the resulting tree for S. We will illustrate how the tree can be used in thevaluation of a real option in the next section.
28.6 EVALUATING OPTIONS IN AN INVESTMENT OPPORTUNITY
 
 
Most investment projects involve options. These options can add considerable value to theproject and are often either ignored or valued incorrectly. Examples of the options embeddedReal Options 671
 
in projects are:
1. Abandonment option. This is an option to sell or close down a project. It is an American putoption on the project’s value. The strike price of the option is the liquidation (or resale) valueof the project less any closing-down costs. When the liquidation value is low, the strike pricecan be negative. Abandonment options mitigate the impact of very poor investmentoutcomes and increase the initia] valuation of a project.
2. Expansion option. This is the option to make further investments and increase the output ifconditions are favorable. It is an American call option on the value of additional capacity.The strike price of the call option is the cost of creating this additional capacity discounted tothe time of option exercise. The strike price often depends on the initial investment. Ifmanagement initially chooses to build capacity in excess of the expected level of output, thestrike price can be relatively small.
3. Contraction option. This is the option to reduce the scale of a project's operation. It is anAmerican put option on the value of the last capacity. The strike price is the present value of
the future expenditures saved as seen at the time of exercise of the option.
as
Option to ae (One of the most im rt nt
wa
. Option to extend. Somciimes ii is possibie to extend the life of an asset by paying a fixedamount. This is a European call option on the asset’s future value.
As a simple example of the evaluation of an investment with an embedded option, consider acompany that has to decide whether to invest $15 million to obtain 6 million barrels of oil from acertain source at the rate of 2 million barrels per year for three years. The fixed costs of operating theequipment are $6 million per year and the variable costs are $17 per barrel. We assume that the risk-free interest rate is 10% per annum for all maturities, that the spot price of oil is $20 per barrel, andthat the one-, two-, and three-year futures prices are $22, $23, and $24 per barrel, respectively. Weassume that the stochastic process for oil prices has been estimated as equation (28.9) with a = 0.1and o = 0.2. This means that the tree in Figure 28.2 describes the behavior of oil prices in a risk-neutral world.
First assume that the project has no embedded options. The expected prices of oil in one, two,and three years’ time in a risk-neutral world are $22, $23, and $24, respectively. The expectedpayoff from the project (in millions of dollars) in a risk-neutral world can be calculated from thecost data as 4.0, 6.0, and 8.0 in years 1, 2, and 3, respectively. The value of the project istherefore
15.0 + 4.0e°'*! 4 6.067? 4 8.0693 = 0,54
This analysis indicates that the project should not be undertaken because it would reduce share-holder wealth by 0.54 million.
Figure 28.3 shows the value of the project at each node of Figure 28.2. This is calculated fromFigure 28.2. Consider, for example, node H. The is a 0.2217 probability that the price of oil at theend of the third year is 22.85, so that the third-year profit is 2 x 22.85-2x17-6=5.70.Similarly there is a 0.6566 probability that the price of oil at the end of the third year is 16.16,so that the profit is —7.68 and there is a 0.1217 probability that the price of oil at the end of the672 CHAPTER 28
 
Node: A B Cc D E F G H J
 
Pu 0.1667 0.1217, 0.1667) 0.2217 0.8867) 0.1217 0.1667 = 0.2217 (0.0867Pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266Pa 0.1667 0.2217 0.1667) 0.1217, 0.0867, 0.2217 0.1667 0.1217 (0.8867
Figure 28.3 Valuation of base project with no embedded options. Here, p,, Pm, and py are theprobabilities of “up”, ‘‘middle”, and “down” movements from a node
third year is 11.43, so that the profit is —17.14. The value of the project at node H in Figure 28.3 istherefore
[0.2217 x 5.70 + 0.6566 x (—7.68) + 0.1217 x (-17.14)JeO'™! = -5.31
As another example, consider node C. There is a 0.