Chapter 21 Real options

Chapter 21 Real options

R. Jarrow et al., Eds., Handbooks"in OR & MS, Vol. 9 © 1995 Elsevier Science B.V. All rights reserved Chapter 21 Real Options Gordon Sick Faculty of...

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R. Jarrow et al., Eds., Handbooks"in OR & MS, Vol. 9 © 1995 Elsevier Science B.V. All rights reserved

Chapter 21

Real Options Gordon Sick Faculty of Management, University of Calgary, Calgary, AB T2N IN4, Canada

1. Introduction A real option is the flexibility a manager has for making decisions about real assets. These decisions can involve adoption, abandonment, exchange of one asset for another or modification of the operating characteristics of an existing asset. In many cases, real options amount to American options on an underlying asset, so the real option literature has been viewed by some as a subset of the much larger financial option literature. On the other hand, the analysis of real options is closely related to dynamic programming in the operations research literature and to optimal control theory in the mathematics literature. Below, I argue that real options have some special characteristics that give them a special growing niche in the finance and capital budgeting literature. 1.1. Real vs. financial options A call option is the right to acquire an underlying stock or asset (of price P) on a specific exercise or maturity date T at a pre-determined exercise price K. Similarly, a real option to develop a project at some future date provides a call option to acquire the underlying asset, which is the value of a going-concern operating project. The acquisition is done by paying an exercise price, which is the capital cost of developing the project. For example, the underlying asset could be developed urban property, while the option is agricultural land that can be converted to urban use. 1.2. Term

Financial options typically have a finite term to maturity, whereas many real options have a perpetual term to maturity. Many financial options and essentially all real options are American options. This means that they can be exercised early or at maturity. The determination of optimal exercise policy is central to the real options analysis and somewhat tangential to financial options analysis. Much of the financial options analysis involves modifying the Black-Scholes option pricing 631

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formula to suit special situations. The beauty of the Black-Scholes formula is that it provides an easy-to-compute analytic formula that is based on observable financial data. However, it provides the value of European options, which cannot be exercised prior to maturity. Its value in real options analysis is to provide a lower bound to the value of an American real option that has comparable payoff characteristics plus the value of flexibility of choice of exercise date.

1.3. Who will use real options first? Real options analysis is not nearly as common in either academia or practice as is the analysis of financial options. The main reason for this is the availability of data. Academics have a wealth of data with which to build and test contingent claim models. Practitioners have many markets on which to trade, and the same data with which to parameterize their trading models. The deep and broad contingent claims markets of the 1980s and 1990s are based on the sophisticated models developed by academics. Real options analysis is similar to financial options analysis insofar as practice will follow academic theory and empirical study. However, there is less data available for real options analysis. Thus, the first areas to adopt real options analysis will be those involving commodity production, such as resource and agricultural industries because of the wealth of data on commodity prices. Many leading-edge petroleum and mining companies are analyzing their developments with the help of real options analysis. The next areas to adopt real options analysis will be those with a great deal of data, but data that is unfortunately of lower quality. The real estate industry has many databases but nonuniform products. Thus, data on average resale prices are widely available but not useful. Data on a standardized three bedroom bungalow or a square foot of class-A downtown office space are not direct data but data synthesized from a hedonic model of supply and demand. The quality of the data is reduced by the statistical error. Nevertheless, some useful theoretical and empirical models have been built to analyze real estate from a real options perspective. The area to adopt real options last will probably be the broad area of research and development and strategic planning that is important to almost all firms. Past experience on the risky payoffs to research and development is difficult to codify into an empirical model. However, real options analysis does provide some very helpful comparative statics that will help guide the decision maker. This is a refreshing improvement over the cavalier strategic planning models that are so popular in business and business schools. For example, the pharmaceutical company Merck uses Monte Carlo simulation to assess research and development strategy, and Black-Scholes option analysis to assess the merits of a joint venture, 1 It also uses game theory to analyze competitive responses to introduction of new products. While these may not See Nichols [1994].

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be the perfect analytic tools for the problems at hand, they will give useful quantitative measures of value and optimal decisions. More importantly, they yield useful comparative statics, such as the notion that the ability to limit downside risk makes a real option more valuable than NPV analysis would suggest. Also, structuring an option to allow for delay enhances value. Many more firms will come to see their strategic planning in a more sophisticated light, using financial analysis, including real options analysis. Merck is just an early adopter of these techniques.

1.4. The relative importance of real asset decisions A great deal of research and analysis goes into making financing decisions, such as decisions involving capital structure, dividend policy and the management of exotic securities such as warrants, convertibles, caps, collars and futures or forward contracts. Under Modigliani-Miller theory, these decisions, as a first approximation, are all irrelevant because financial markets can be used to readily undo or replicate any financial decisions the firm may make. Thus, a firm cannot create significant value by its financing policies. However, a firm can create significant value by its operating or capital budgeting policy, because it can have proprietary access to a portfolio of projects that it may undertake. This arises because of market power, patents, special corporate expertise and operating synergy. Real options analysis is directed toward managing this value-creating activity. Clearly, it makes sense to devote as many resources (academic and applied) to analyzing real options as it does to analyzing financial options. The field of real options has a great deal of catching up to do.

1.5. Real options and the traditional NPVrule The traditional net-present-value (NPV) rule says 'develop as soon as NPV exceeds zero'. This is also true for real options, as long as we understand that the NPV of an option is the NPV of development minus the opportunity cost that consists of the loss in value of killing the option. To think of this in a different way, the traditional NPV rule for mutually exclusive projects advocates adopting the most valuable project (in order to maximize firm value). In a real options setting, what initially appears to be one project is an infinite number of projects. There is one project for each starting date of the basic project - - the Year-0 project, the Year-l-start project, and so forth. The traditional NPV rule then says that we should select the start date that maximizes the value of the firm. This is what real options analysis is all about. The problem is in calculating the value gained or lost by delaying the project.

1.5.L Random beta and cost of capital The development cost of a project tends to be fixed, while the benefits of development tend to be random. The fixed cost tends to provide operating leverage for the option, which affects the beta and hence the cost of capital

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at which one would discount the option payoffs. The degree of this leverage is random, since it increases as the underlying project value falls toward the cost of development. Offsetting this is the risk-reducing effect from the option flexibility. The option passes through upside potential while protecting its owner from significant downside risk. Overall, there is no guarantee that the beta or cost of capital of an option is constant, so we will see that a certainty-equivalent approach is more appropriate. Once an option is analyzed, a cost of capital can be calculated, but this is of no benefit, since the value of the option will have been determined before its cost of capital is determined.

1.6. Real options vs. dynamic programming in operations research Real options analysis bears a significant resemblance to the dynamic programming literature in operations research. Both deal with the flexibility of future decision making by a backward induction process. The critical difference is that real options analysis takes advantage of and requires an assessment of market risk. Markets are an important source of data for the real options analyst, but the presence of economic markets also imposes some discipline upon the analyst. Financial economists distinguish between systematic risk, which commands a risk premium, and unsystematic risk, which does not. The real options approach is to model these risk premia and adjust the probability distribution for them by calculating a 'risk-neutralized' probability distribution. This distribution can be determined by arbitrage analysis or by fundamental economic analysis. Financial markets provide other valuable data about stochastic processes and their parameters, which are essential to real options analysis.

2. Primary principles in valuation

2.1. General asset-pricing models There are two major approaches that can be taken to option pricing. Fortunately, the approaches are compatible,, but their intuition is slightly different. One approach involves replicating the option payoffs with the payoffs of a portfolio which has a dynamically updated composition. In the absence of an arbitrage opportunity, the option value equals the value of the portfolio. This approach is popular in the financial option pricing literature and was made popular by Black & Scholes [1973]. Consider the valuation of a call option by the following Black-Scholes replication approach. The only random factor in the payoff to the call option is the random value of the underlying asset at the maturity date. The random portfolio that replicates the call option will thus take a long investment position in the underlying asset. It will also take a short position in the riskless asset (borrowing), since the fixed exercise price of the option provides leverage. The portfolio weights must be continuously updated to ensure that the portfolio replicates the

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option. The updates are done on a self-financing basis - - the value of asset purchases equals the value of asset sales in the update procedure. If someone offers to pay more for the option than the value of the replicating portfolio, then an arbitrageur could sell the option short and use the proceeds to purchase the replicating portfolio, for an immediate net cash inflow or arbitrage profit. Since the dynamic portfolio strategy would not require any intermediate cash flows and would replicate the option cash flows at maturity, this would result in a riskless profit to the arbitrageur. Similarly, if someone were willing to sell the option for less than the value of the replicating portfolio, an arbitrageur could earn a riskless profit by going long on the option and short on the replicating portfolio. Arbitrage opportunities should be exhausted by trading in financial markets, so we conclude that the value of the option must equal the value of the replicating portfolio. While this line of analysis is very useful for financial options because there are many arbitrageurs on the floors of options exchanges trading with strategies of this type, many people find this approach to be less compelling with real options, because it is so difficult to sell a real asset short. One can construct a dynamic portfolio strategy of riskless borrowing and ownership of the underlying asset that replicates the payoff of the option, as in the Black-Scholes analysis. However, the underlying asset may not exist as a tradeable asset if such assets only exist as a result of the exercise of real options. Thus, it may not be feasible to form the replicating portfolios needed to validate the arbitrage analysis. Even if the underlying asset exists, it may not be very liquid and could be hard to sell short. For example, when the underlying asset is developed urban property, agricultural land owners have a real option to convert their land to urban use. Neither urban nor agricultural land are so liquid that the replicating dynamic portfolio strategies are very realistic. Unfortunately, these difficulties lead some people to discard all of the systematic analysis and resort to some very crude rules of thumb when dealing with real options. For example, they may ignore risk, they may accept a hurdle NPV of zero, or they may introduce arbitrary 'fudge factors' into the analysis. Clearly these people need something to grasp in order to make consistent value-maximizing decisions. To serve these people, I advocate pricing real options with the capital assetpricing model (CAPM) and all of its variations and extensions. The variations include arbitrage pricing theory, the consumption capital asset-pricing model, the intertemporal capital asset-pricing model, and martingale pricing theory. These linear pricing operators are the most general pricing operators that are consistent with the absence of arbitrage opportunities. This class of pricing operators will stand the test of time in financial economics. The only things we can expect to learn are details about the characterization and parameters of the pricing operators. The popularity of the CAPM and its variants make them easy to accept for many financial decision makers, even though their proofs often rely on assumptions about perfect markets, absence of arbitrage or optimizing behavior on the part of

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economic agents. Indeed, most financial decision makers are prepared to accept the present-value approach to making decisions about riskless cash-flow streams, even if" these streams are illiquid. If they are offered an illiquid stream at a price less than its present value, they will still acquire it even if they do not intend to earn an immediate arbitrage profit by offsetting it with a riskless stream of outflows. Similarly, if someone offers to buy from them an illiquid stream at a price greater than its present value, they will sell, even if the stream offered a suitable time pattern of future cash inflows. Buying and selling these illiquid streams merely amount to portfolio adjustments that can ultimately be offset by future trades in other assets. More generally, these same decision makers are willing to use the CAPM to value the risk coming in or out of their portfolio by buying and selling risky assets. If they are offered the opportunity to buy or sell assets at favorable prices that are inconsistent with the CAPM, they will do so. If they want to offset the change in risk exposure, they can do so by trading broad and liquid stock portfolios. They are not forced to conduct their risk offsets with the illiquid or nontraded underlying asset. Indeed, decision makers may manage their own nontraded human wealth by trading financial assets to hedge their own personal risk. The only limitation would arise if there are incomplete markets (for the risk of human wealth), or compelling agency reasons for an owner-manager of a firm to not become well-diversified, Even in these situations, the CAPM provides a suitable starting point for risk analysis.

2.1.1. Martingale and state-space pricing Although martingale valuation is also discussed by Carr & Jarrow [1995] elsewhere in this volume, I will review these ideas below to help keep this chapter self-contained. Readers who are only interested in applications can skim this material for the notation and refer back only as needed to interpret the later material. The applied material starts with the Section 2.2 on Hotelling valuation. More rigorous discussions of martingale pricing appear in Harrison & Kreps [1979] and Harrison & Pliska [1981]. The ideas are also discussed succinctly in Ross [1989]. The broad extensions to the CAPM are discussed in Cox, Ingersoll & Ross [1985a]. An overall discussion of these issues appears in Ingersoll [1987] and Jarrow [1988]. First, I will discuss the martingale pricing operators, and interpret them in the context of state-preference theory, the consumption CAPM, and the CAPM. I will discuss the restrictions necessary to ensure that the operator is positive in the sense that it will always assign a nonnegative value to a nonnegative cash flow. Suppose the riskless rate of interest r is known and constant. Consider the valuation today of risky cash-flow payoffs T periods hence. That is, we wish to determine the price 2 Vt[[email protected]] to be paid at time t for the right to receive the risky cash flow ) ( r at a future time T > t. The risky cash flow f;r is a real-valued 2 Tile future cash flow Xr could be positive or negative. Thus, the value at time t could be positive, negative or even zero.

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Ch. 21. Real Options

function X~,: ~ - -÷ !)t on the state space ~2~ that describes what amount of cash in )(T(S) is paid in each state s c ~2v. Suppose the probability distribution of this state space ~2~ is a- = 7v0. Over time, information arrives, and this distribution is updated by all agents in the economy to the conditional distribution 7ct at time t. Under martingale valuation, there exists a (martingale) probability measure ~ on the payoff space ~ v such that the time-t value or price Vt[~'T] of any time-T cash flow XT is given by the present value of the expected payoff under this martingale measure:

v,[2r] = e -~(T ') f 2r(s) dJt,(s) ,]

(1)

a,

= e -m

')*},[2r]

(0 < t < :r)

where ~i and /}t['] are, respectively, the distribution and expectation conditional on information available at time t with respect to the martingale measure. The sequence of valuation measures over time forms a martingale in the sense that the stochastic process of discounted asset values { e - " V , [ 2 r ] } ~ 0 forms a martingale under the distribution zt: < t < T ~ e-r'v~[2r]

~[e-r'v,[2r]]

=

This can be verified by an application of the law of iterated expectations:

e-rrvr[f(r]

=

e - " r e - r ( r r)/~r[J~r] =

e-~r~[~t[2r]] =

= e r t / ~ [e-r(r-t)~t[J~r]] = e-rt Er[Vt[f(T]].

This is just a consistency condition on the pricing operator. For example, setting r = 0, we can interject an intermediate time period and compute the same value: vo[2r] = Vo[V,[2r]]. The martingale measure or expectation is sometimes called risk-neutral or riskneutralized since values are computed by taking the present value of expected payoffs as though no risk-premium is required. In fact, we shall see that any risk-premium is embodied in the transition from the true probability distributions rrt to the martingale distributions ~'t. To see that ~t must be a probability distribution, first note that a riskless security paying $1 for certain at time T must have a value e - r ( r - 0 , so e -r(r-t)

=

Vt[1]

=

e -r(T-t) f

1 dz~t(s)

=

e-r(T-t)frt(~T)

, 1

Q'I"

and hence the probability measure integrates to unity over the probability space. Moreover, the measure should be positive in the sense that the probability of any event must be nonnegative. If we have a security or contingent claim that pays $1 if an event A occurs and nothing otherwise, this security has a nonnegative payoff, and should have a nonnegative price to prevent arbitrage. The risk-neutral expected payoff of the security is the risk-neutral probability, ~:t(A), and the price

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of this security is the present value of this amount, so the measure ~t itself must be positive. Thus, the risk-neutral measure 77t is nonnegative and assigns a value of i to the whole state space, so it must be a probability measure. To make things more concrete, suppose we have a discrete state space, and replace the continuous discounting factor e - r ( r - 0 with the discrete discounting factor (1 + r) - ( r - t ) , recycling the discount rate r with this new meaning. T h e n the present value of the martingale probability of a state s at date T is simply the date-t Arrow-Debreu state-space price for a claim to $1 contingent on occurrence of that state,

Ps,t = (1 -[- r) -(r t)frt(s).

(2)

If markets are not complete, the A r r o w - D e b r e u prices are not unique, but we do know that in the absence of arbitrage, there exists a set of positive A r r o w D e b r e u prices such that the value of a risky cash flow is the sum of its payoffs in each state times the price of an A r r o w - D e b r e u claim to $1 in that state. This is consistent with (1):

vt[2rl

= (1 + r) -(T-t) ~

frt(s)Xr(s) = ~

[email protected]~T

ps,tf(r(s).

(3)

SE~T

Note that, if we are given a set of A r r o w - D e b r e u prices of basic state-contingent claims, we can solve (2) for the martingale probabilities ~t, so the martingale approach can be regarded as an outcome of A r r o w - D e b r e u state-space theory with complete markets. We can also relate the martingale measure to the true probability measure. First, observe that, if a state has a probability of 0, a claim to a payoff of $1 in that state should also have a price of zero. The rationale is that nobody would pay anything for a security that pays nothing with probability 1. This means that :rt (s) = 0 ~ Ps,t = 0. For pricing purposes, we can restrict ourselves to discrete states that have positive probability. We can define the ratio of time-t price to the present value of the probability as the r a n d o m variable:

Ps,t g,t,~'(s) = (1 + r ) - ( r - t ) r r t ( s ) '

(4)

The o u t c o m e of the r a n d o m variable gt,T is learned at time T > t, when state s is revealed. T h e date-t value of a risky payoff X r is the present value of the expected product of the risky payoff times gt,T:

Vt[f(r] = (1 + r)-('r-t)Et[~t,r2T].

(5)

Note from (2) and the fact that ~t has the properties of a probability distribution implies that the r a n d o m variable gt,r >_ 0 and Et[g,t,r] = 1.

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2.1.2. Capital asset-pricing model and its relatives In a utility-optimization model such as the consumption C A P M of Breeden [1979] the r a n d o m variable gt,r(s) = Ut(s) is the marginal utility 3 of date-T consumption for a representative investor in a market equilibrium, conditional on information available at time t about the probability of state s. Assuming that the investor is risk averse, the utility function is concave, so that its derivative (marginal utility) is positive. From (2) and (4),

;r~(s) = #~,r(s)zrt(s) = U'(s)Jr~(s).

(6)

This means that the risk-neutral probability distribution 72t differs from the true distribution zrt by an adjustment factor equal to the marginal utility of consumption at the final payoff date. Since Et[gt,r] = 1 , the average adjustment factor is 1. If the marginal or representative investor is risk-neutral, this adjustment factor is a constant, and must be 1. Thus, the risk-neutral probability distribution ~t is the one that is consistent with market prices being set by risk-neutral investors. If the marginal investor is risk averse, then marginal utility is less than 1 for high-consumpti0n states and more than 1 for low-consumption states. Thus, a security is relatively m o r e valuable if it pays off when investors need the consumption the most (in the low-consumption states). There is another important condition on the payoffs to a security that allow it to be priced as if the marginal investor were risk-neutral. For example, if the r a n d o m payoff J)r is distributed independently of marginal utility gt,r, then the expectation of their product in (5) can be written as a product of expectations. Since Et[g,t,r] = 1, this means that the value or price of the r a n d o m payoff is the present value of its expected payoff under the true conditional probability distribution rrt. This result can be generalized slightly by characterizing the pricing operator in terms of returns to resemble the more traditional versions of the C A P M or arbitrage pricing theory (APT). Using the fact that a covariance between two r a n d o m variables is the mean of their product minus the product of their means, we can replace (5) with

Vt[2r]

= (1 +

r)-(r-OEt[~,t,r2r]

= (1 +

r)-(r-t)(E~[~,t,r]Et[2r] +

= (1 +

r)-(r-O(Ed2r] + covd2r~t,rl).

cov,[2~.~,,r]) (7)

U p to this point, we have allowed the cash flow ))r to be positive or negative. Thus, the valuation equation (7) is useful for valuing general contracts, such as 3 We have scaled prices relative to current consumption as a numeraire. Tile investor's Von Neumann-Morgenstern utility function U is unique up to a linear transformation, so the marginal utility U~ is unique only up to a positive scale factor. We also choose, for simplicity, a scaling of the utility function so that, for investment decisions made at time t about the date-T consumption in state s, g'(s) = ~t(s).

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futures and forward contracts for which future net cash flows can be positive or negative. Thus, it could impose future liabilities on the owner. Now, suppose the cash flow imposes no future liability on the individual who is to receive it. T h e r e is limited liability and ~71- > 0. The claim to such a cash flow is a fully paid-up capital asset, and has a nonnegative value Vt[Jfr] >_ 0. A n investment strategy of paying Vt [J)r] at time t to acquire a claim to the future cash flow X r , is self-financing in the sense that no other cash injections are required to maintain ownership of the asset. Thus, Vt[J(r] > 0 is the value of a capital asset. Now, take T - t = 1 to be one year. We can rewrite (7) to resemble the C A P M in its rate of return form on the capital asset by defining ZV-

J)T

Vr-l[2V]

1.

to be the one-period rate of return realized at time T on the investment in cash flow 5:r. T h e n from (7), the expected rate of return is ET'-l[Zr] = r + c o v r - l [ S r ,

fr-l,r]

(8)

where we define the systematic risk factor

fT-l,r = --gT-1,T.

(9)

M o r e generally, we have the following certainty-equivalent valuation equation 4, which applies for a contingent contract that is not fully paid up and may incur future outflows:

VV-I[2T]

= E T - I [ J ) T ] --

covv-l[2rfT'-l,rl

(10)

l+r This is the consumption CAPM, since f7"-1,'1" is minus the marginal utility of consumption at date T. The measure of systematic risk of the security paying a rate of return ZT is the 'consumption beta' COVT-I[ZT,fr-l,T]. This measure is normalized so that the return premium per unit risk is 1. Since marginal utility is decreasing in consumption, the marginal utility g T - l , r tends to be low in high.wealth states. Thus, fT-1,T is positively correlated with the marginal investor's consumption or aggregate wealth. As in the C A P M , the expected rate of return of a security is higher if the security payoff varies positively with aggregate wealth. If the expected rate of return is uncorrelated with marginal utility gT-1,T, then the expected rate of return is the riskless return, and hence the risk-neutralized probability is the true probability. The difference between the true probability distribution of the rate of return on a security and the risk-neutralized probability distribution is the adjustment for systematic risk. T h e standard C A P M also has the form (8) or (10), but the systematic risk factor f r - I , T is replaced by a factor that is proportional to the rate of return on the 4 The numerator is the certainty equivalent of the risky cash flow )('r because discounting such a certain payment at the riskless rate of interest gives the value of the cash flow.

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market index security JZm,T, with the constant of proportionality chosen to make the relation correctly fit the expected return on the market index:

(OT_I, T

(ET-I[rm,T]=F'~

-

(11)

=-- \ varT-l[fm,T] jrm'T"

Let us relate ~ r 1,7" to f r - l , r , to see what restrictions this places on qST-l,r. Applying (9) we must have Et[fi,r] = --1 and )~,T _< 0. Since covariance is not affected by adding a constant to a r a n d o m variable, define

.[T<,T =- ?Pr-l,r + A, and choose the constant A so

that

Et[ft,T ] =

- 1 . This means that

A = -- II + ( ET-I[Ym'T] ~-r'~ ET_I[~-m,T]1 \ varr-l[~m,T] / Ensuring that j~, T < 0 is equivalent to the following u p p e r - b o u n d condition 5: ( var T-l[rm,r] "]

rm,r <--ET-I[#'m,T] -[- \ ET-I[rm,T] -- r ) "

(12)

In general, we do not m o d e l market rates of return to be b o u n d e d from above in discrete time, so the traditional C A P M does not give a positive operator. The C A P M can still correctly price the universe of (existing) assets that lead to the m e a n - v a r i a n c e optimization, such as joint normally distributed payoffs. However, once we start requiring that our pricing operator also correctly price assets with m o r e general payoffs, then we must take care to ensure that we are using a positive pricing operator. For example, if (12) fails, then the value of a d e e p - o u t - o f the m o n e y call option on the m a r k e t index that only pays off when (12) is violated will have a negative price under the CAPM, even though it has nonnegative payoffs. To ensure that we have a consistent valuation operator for call options on such joint normally distributed payoffs, we could assume that the marginal investor has negative exponential utility, where g T - l , r is proportional to exp(--al~T), where WT is the aggregate payoff to all risky assets. This will still give the C A P M for the joint normally distributed assets (because the expected utility functions are m o n o t o n e transformations of a linear combination of m e a n and variance of payoff), and it will give meaningful nonnegative prices to call options on these assets since the pricing factor gT-1,T is nonnegative. These techniques can be generalized to the continuous state space, where ~ is the R a d o n - N i k o d y m derivative of the martingale measure r? with respect to the true probability m e a s u r e Jr. T h e martingale measure :? and the pricing factor are uniquely determined if the markets are completefi In general, markets will 5 Assuming that Er 1 [rm,T] ~>r, which is to say that the risk of the market index commands a positive risk premium.

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not be complete, so the pricing operator is not unique. Moreover, it is possible that introducing a new unspanned security (one that does not pay off like a portfolio of existing securities), such as an option, will change the prices of existing securities. This could invalidate earlier choices of the martingale measure or the pricing factor. This can make it difficult to use some of the option-pricing techniques of this chapter and other papers with absolute impunity. Indeed, there may fail to be unanimity in such a situation, so that different investors may rank different investment strategies of the firm differently. In these situations, all of our existing capital budgeting techniques can fail, and choosing an optimal investment program can come down to game theory, which is not the focus of this p a p e r ] Thus, we will assume, as do other authors in this field, that markets are either complete, or that the payoffs of any project under consideration can be spanned by the payoffs of existing securities.

2.1.3. Steps to finding a martingale valuation operator Harrison & Pliska [1981] show that, even if markets are incomplete, all the martingale measures that correctly price traded assets give identical prices for claims to risky payoffs that are attainable by trading strategies for existing assets, s Call options on traded assets are attainable claims, using the standard Black & Scholes [1973] replication technique whereby the option payoff is synthesized by a dynamic portfolio strategy of investments in the underlying risky and riskless assets. This creates a useful technique for valuing options and other claims that are attainable by dynamic trading strategies in existing assets: 1. Find a 'risk-neutral' or 'martingale' probability measure ~ on the same probability space 9 as the true probability measure Jr such that the stochastic 6 The market is complete if, given a random scalar variable defined on the probability space, there exists a priced security that pays off according to the values of the random variable. If the state space is finite, then the market is complete if, given state s, it is possible to find a security or portfolio of securities that pays $1 if state s occurs, and nothing otherwise. The prices p.~ of such securities define the martingale measure and pricing factor uniquely, as above. 7 Here is an example of how incomplete markets can break down unanimity. Suppose a pharmaceutical firm is trying to decide whether to develop a treatment for a very rare disease, with no existing treatment. Given the rarity of the disease, analysts in the firm conclude that the research and development program will be more costly than the potential sales revenue from the treatment. On a net present value basis, the R&D program has a negative value and would be rejected. However, if some of the shareholders of the firm are afflicted by this disease, they may find personal benefits to the research that exceed their share of the negative NPV. This is the classic agency problem. If these shareholders can control the firm's decisions, they will undertake the project, even though the other shareholders disagree. The market failure here is the absence of a market for treatment in the rare disease. If treatments were available, then all shareholders would agree to make decisions based on value maximization, since even the ill shareholders would prefer to be wealthier, in order to be able to buy more of the existing treatment. 8 For discrete-time processes, this is Harrison and Pliska's Proposition 2.9~ In continuous time, see their Proposition 3.31. 9 By 'same space' we mean that both the true probability distribution 7r and the martingale distribution 77 assign a zero probability to the same events. In this sense, these distributions are 'equivalent'.

Ch. 21. Real Options

643

process of discounted values of the underlying traded assets {e-rtPt}f__o form a martingale under 72. Thus, if the probability space is a continuous process over time that achieves the values on the interval (0, co), the martingale process shall achieve values on the same interval. Or, if the true process is a sequence of Bernoulli trials over time, with increments drawn from a two-point set {zl, z2} the martingale process must be also be a sequence of Bernoulli trials drawn from {zl, z2}. The probability of selecting the two points will be different for the martingale process and the true process, but the state space is the same for both. Note that we only need the marginal probability distribution for events defined by the payoff levels of the asset that we wish to value. 2. Value any payoff that is attainable by a trading strategy involving the underlying assets by the present value of the expected payoff under the selected martingale probability measure. The choice of the martingale measure may not be unique, but for any attainable payoff (i.e., one that can be replicated by a trading strategy in underlying assets), all such martingale processes will agree on the value of the new claim to this payoff as the present value of the expected payoff under the martingale probability. To state this more mathematically, suppose XT is a risky payoff at date T that is a function of the date-T values of some existing, traded assets (i = 0, 1, 2 . . . . . n), which have price processes {/5i}~t = 0 " Also, suppose that there is a dynamic trading strategy in these traded assets that generates the payoff XT. This is the case, for example, if X~ is a call or put option on the traded assets. Then, we only need to find a way to adjust the parameters of the stochastic processes for the traded assets to get a risk-neutral martingale process for the values of these assets. That is, if the assets i = 0, 1, 2 . . . . . n pay no dividends 10, then letting ^ denote expectations with respect to this martingale distribution, we have

pi (t) = e - r ( ~ - t ~ , [ ~ i (T)] whenever 0 < t < T. Even if the joint probability distribution giving these marginal martingale distributions is not unique, the value of the claim to XT- is the present value of the expected payoff under such a joint distribution, in the sense that (1) holds. Often it is relatively simple to determine an eligible martingale probability, since under the martingale probability, the expected rate of return on each of the traded assets if the riskless rate of return. For example, in discrete time,

~i Z t --

P]

P/-1

l

is the one-period rate of return realized at time t on asset i. From the consumption CAPM (8), the true expected rate of return is

Et--l[z[] = r + covt-t[~/, j~].

(13)

10 We will see how to adjust for dividends when they arise. The technique is straightforward, but merely makes the notation m o r e clumsy at this point.

G. Sick

644

A suitable martingale distribution is one in which the expected rate of return on this asset is the riskless rate of return: Et-l[Z/]

= r.

To get such a risk-neutralized martingale process, we only need to subtract from the true m e a n rate of return the consumption CAPM risk premium covt-1 [~[, J~]. This technique will also work if the underlying asset pays a dividend. In that case, we merely need to choose a martingale process for which the underlying asset pays an expected capital gain plus dividend at the rate r. Given this risk-neutral probability distribution, we can determine the value of a claim to the new cash flow as:

Vt[Xr] = (1 +

r)-(r-°P~t[2r].

(14)

We will take the liberty of extending this technique to situations in which the risky payoff X r cannot be replicated or attained by a trading strategy in existing assets. As long as this new payoff occurs in sufficiently small quantities compared to the payoffs of existing assets, it is unlikely to change the parameters of the consumption CAPM, including the riskless rate of return r or the parameters of the marginal utility factor g't,r. In this environment, the risk neutral probability distribution can be obtained by adjusting the drift to ensure that the value process has an expected rate of return equal to the riskless rate under the risk-neutral probability 77. This concludes our review of martingale pricing theory.

{v~[2r]}~0

2.2. HoteUing valuation of operating resource properties In many real options situations, we can relate the value of the underlying asset to the spot price of some commodity. For example, under the Hotelling valuation principle 11, the value of an optimally managed operating mineral property equals the current contribution margin per unit volume times the volume of reserves. Since the rate of resource extraction is optimally managed, the value today of extracting one unit of reserves must be the same as the value of reserves extracted at any later date. If not, value can be increased by shifting production ahead or back in time. Developing the property corresponds to exercising a call option. At the time of exercise, the net proceeds from exercising equal the net value of commencing production of a resource property, which is NPV=

Q(P-VC)-K.

(15)

where P Q

= spot price of commodity; - quantity of reserves;

Jl See Hotelling [1931], Miller & Upton [1985a, b] or Sick [1989b, pp. 10-11] for more discussion.

Ch. 21. Real Options

645

VC --- unit cost of production; K = capital cost of development. The net proceeds of exercise per unit of reserves is P - (VC + K / Q ) . This is the payoff to a call option that has an exercise price of VC ÷ K / Q , and the underlying asset is a unit of the commodity, with spot price P. In more general problems, the spot price of the commodity is still a sufficient statistic for the random value arising from development of a property. In these situations, the net proceeds of exercise can be calculated by a discounted-cashflow analysis of the NPV of development at a variety of spot prices. Bjerksund & Ekern [1990] develop models of this sort. Laughton & Jacoby [1991] suggest that the nonlinearity of tax and royalty structure, coupled with the uncertainty of future spot prices can best be modelled by Monte Carlo simulation of spot prices subsequent to the point at which the project is developed. The net proceeds of exercise is the expected present value of the after-tax net cash flows from the operating project, minus the cost of development.

2.3. Nature of financing (tax-adjusted discount rate) Financial options are often priced by consideration of arbitrage opportunities amongst bonds, an underlying asset and the option. This arbitrage is typically conducted by a floor trader who trades in and out of arbitrage positions in just a few days. Thus, capital gains are not deferred, and become taxed at the same rate TI, as dividend income and debt income, in the hands of the arbitrageur. The after-tax net payoffs to arbitrage positions are simply the pre-tax net payoffs multiplied by the retention factor (1 - Tp). If taxation of gains and losses is linear and symmetric, this leads to the same arbitrage relations as in a pre-tax analysis. Thus, it is customary to ignore taxes in the analysis of financial options. The arbitrage analysis generates certainty-equivalents, which are simply discounted at the riskless rate of return r t on debt. On the other hand, real options are long-term in nature, so they fit naturally into the analysis of the capital budget of the firm. In capital budgeting analysis, it is customary to adjust for the differential taxation of long-term returns on debt and equity. This is accomplished by either adjusting the discount rate to get a weighted average cost of capital, or by adjusting a base-case value, corresponding to the value if all-equity financed, by the present value of the interest tax shields. This is discussed in most textbooks, such as Brealey and Myers [1991, chapter 19] or Ross, Westerfield & Jaffe [1993, chapter 17]. However, Sick [1990] and Taggart [1991] argue that the traditional treatment of interest tax shields is inconsistent with the modern literature on taxation and capital structure equilibrium stemming from Miller [1977]. Since real options are priced as certainty-equivalents of long-term cash-flow streams that will be taxed at a personal level when passed to investors, we do need to adjust for the differential taxation of debt and equity by using a tax-adjusted discount rate. In contrast, financial options are priced as certainty. equivalents of claims to underlying securities that are already priced in the market

646

G. Sick

in a m a n n e r that already incorporates adjustments for personal taxation. In particular, Paddock, Siegel & Smith [1988] and Sick [1989b] advocate the use of a tax-adjusted b o n d return as the discount rate for the certainty-equivalents in real options. First, let us review the arguments advanced by Miller and others regarding the integration of corporate and personal taxes into a capital structure equilibrium. Miller assumes there are different personal tax rates for various individuals but the same tax rate for all corporations Zc. In equilibrium, the marginal investor faces a marginal tax rate of rpb on interest income and ~;pe on equity income. The after-all-tax cash flow on $1 of riskless pre-tax debt income for the marginal investor is (1 - ~:pb) and the after-all-tax cash flow on $1 of riskless pre-tax equity income for the same investor is (1 - r~.) x (1 - Zpe). In equilibrium, the investor is indifferent between debt and equity, so (1 -- Tc) x (1 -- rpe) = (1 -- rpb).

(16)

The claim to this cash flow would have the same value as debt or equity, so suppose it is $V. The market rates of return on riskless debt, denoted rf, and equity, d e n o t e d rz, are after corporate tax and prior to personal tax. Thus, r/=

1

~

and

rz -

1-r~ V

(17)

Substituting for V, from (17), we have: r z = (1 - rc)r[.

(18)

T h e r e is a tax wedge (1 - rc) between the market return on riskless debt and the return on riskless equity. The wedge arises because debt income is untaxed at the corporate level, but is taxed m o r e heavily than equity income at the personal level. U n d e r the Miller theory, the after-all-tax return on debt equals the after-all-tax return on equity for the marginal investor (1 - rpb)r t = (1 rpe)r z. Now, we can generalize the Miller theory to assume that not only do various individuals pay tax at different rates, but also assume that various corporations pay tax at different rates. Different firms or organizations can face different statutory tax rates, and a firms' effective tax rate will be lower than its statutory rate if it carries forward tax losses. In this case the effective tax rate rc is the present value of the statutory rate, discounted from the time the taxes are actually to be paid in the future. Now equilibrium in the debt and equity markets requires that the marginal firm and the marginal investor be indifferent between debt and equity flows. Let the marginal firm have tax rate rm. T h e n this generalized t h e m y requires that -

(1 - Cm) x (1 - rpe) = (1 -- rp~).

(19)

r"z = (1 - 7:m)rf.

(20)

and that

Ch. 21. Real Options

647

2.3.1. D e t e r m i n a t i o n o f the e x p e c t e d rate o f return o n riskless e q u i t y

In theory, the expected rate of return on riskless equity should be a composite rate of return on the dividends and capital gains on a riskless equity security. Unfortunately, such securities are hard to find. Thus we must search for a proxy. A practical solution is to study the yield on high-grade preferred shares. Unfortunately such shares are not perfectly riskless, so they embody a small risk premium, and they provide dividend income, which may be taxed somewhat differently from capital gain income. One must also be careful to study a preferred share that has no conversion features, since these conversion features are valuable and substitute for yield. That is, convertible preferreds have lower yields than straight preferreds, even though their cost of capital is approximately the same, because they also provide some income in the form of capital gains from the conversion feature. If straight high-grade preferred shares have a yield that is 70% to 80% of straight corporate bond yields, one can argue that the marginal firm's tax rate is approximately rm = 20% to 30%. Another approach to estimating rz is to note that market-line asset-pricing theories, like the Capital Asset Pricing Model or Arbitrage Pricing Theory are typically estimated on equity securities, so the intercept or zero-beta rate of return in these models should be r e . Unfortunately, most tests of these theories yield estimates of rz that exceed the yield on riskless bonds, rt, which would imply a negative tax rate for the marginal firm. This could be a result of misspecification of the market portfolio or errors in the measurement of beta. This debate is beyond the scope of this paper. However, a study by Breeden, Gibbons & Litzenberger [1989] does find rz < r t , and is consistent with marginal tax rates rm in the range of 20% to 30%. If we assume that personal equity is untaxed, rpe = 0, so that "cm = "gpb. This suggests estimating rm from the relationship between the yields on tax-exempt bonds (such as US municipal bonds) and taxable bonds. In this situation, rz would be approximated by the yield on riskless tax-exempt bonds. More generally, let rmb represent the yield on a riskless municipal bond. If the investor who is indifferent between municipal bonds and taxable bonds is also the investor who is indifferent between riskless debt and riskless equity, we have r.~b = (1 - r p b ) r f .

(21)

Using (19) and (21), this implies that rmb re -- 1 - rpe"

(22)

Thus, if the marginal investor is taxable 12 and rpe > 0, then r z > rmb, 12Paddock, Siegel & Smith [1988] advocate using a real after-tax return based on the real municipal bond yield.

648

G. Sick

To summarize the argument to this point, a firm with a tax rate (rc) equal to that of the marginal firm (rm), is indifferent between debt and equity on a tax basis, and should discount certainty-equivalents (for example, the risk-neutral expected payoffs to a real option) at the after-tax cost of riskless equity: r = rz. On the other hand, if the firm has a tax rate different from that of the marginal firm, (re ~ rm), it should adjust for this in its valuation of a real option. If the firm maintains a constant or predictable debt level $D, then the adjusted present value approach can be used, in which the base case value of the option consists of discounting the certainty-equivalents at the riskless rate for all-equity financing, which is rz. To this should be added the present value of the interest tax shields. Sick [1990] shows that the certainty-equivalent of the net (risky) interest tax shield is the net tax shield (rc - Vm) on the rate of return to a riskless investment (e~I - 1). This certainty equivalent should be discounted at the all-equity riskless rate. To keep the notation simple, temporarily assume that the option will be financed with SD of debt in year t, the present value of the interest tax shield for year n is PV of ITS = e - ( t r z ) ( r c - rm)(e'~t - 1)D.

(23)

An alternative case is the one in which the firm continually refinances as option or firm value changes, and keeps the debt ratio L constant. In this case, Sick shows that the appropriate discount rate for certainty-equivalents is (24)

r = r z - (re - r m ) r f L .

This is equivalent to a weighted-average cost of riskless capital: r =L(1-Tc)rf+(1-L)r

z.

(25)

3. Derivative asset-pricing techniques 3.1. T h e s t o c h a s t i c p r o c e s s f o r t h e u n d e r l y i n g a s s e t

With real assets, there are so many uncertainties about precise asset values, and parameter values of stochastic processes that the analyst cannot hope to get precise values of real options and the critical values for decision making. Thus, the choice between continuous-time or discrete-time should be made to help the analyst understand the economics of the problem in order to be able to choose a meaningful model for the analysis. Real options analysis has not yet been adopted as widely in the practical world as has financial option analysis, so an important desideratum for the analyst is to m a k e the model understandable by those who will have to make a decision with it. We find it convenient to develop intuition with both continuous-time and discrete-time representations for stochastic processes. We will prove some results, but merely reference a source of proof for more technical results, if the intuition can be made clear here.

Ch. 21. Real Options

649

3. 2. Continuous-time processes' for price In continuous time, we will typically assume that the underlying asset price P follows the diffusion process 13

d P = or(P, t)dt + ~r(P, t)dco

(26)

where o~

cr do)

= expected growth in dollars per unit time of the underlying asset price P; = annual standard deviation of underlying asset returns; = Wiener process with zero drift and unit variance per unit time, i.e., co(t) - co(t - 1 ) is normally distributed with m e a n zero and variance one, and is i n d e p e n d e n t of co(T) - co(r - 1) for any time interval [r - 1, v] that does not intersect the time interval (t-l,t).

Suppose also that 8(P, t) = T h e rate at which cash value is conveyed to the owner of the underlying asset (a 'dividend'). T h e expected rate of dollar return to the owner of the underlying asset is the sum of the capital gain and dividend:

E[dP + 8(P, t)dt) = el(P, t)dt ÷ 8(P, t)dt. By the consumption C A P M (10)14, this expected return is the risk-free rate of return on the investment P plus a p r e m i u m dependent on the covariance between 13 We will follow the custom in finance and denote risk in discrete time by a tilde, but drop the tilde in continuous time. Thus, the asset price could also be read as /5t. 14 We have from (7) and (9) the discrete-time valuation model: V,[J(t+l](1 + r) = Et[fft+l] - covt[Xt+lft,~+l]. Let t+l

=t+At,

Xt+At

= fir+At + 6(P, O A t

V,[2~+A,] = P, ~kff)t+At

z

e t + A t -- Pt*

Also, replace r by r a t , and f ' r - l , T by A f to allow for At to vary and approach 0. (These models were based on the assumptions that r was the interest rate for one period and .)ct,t+l is the accumulated systematic risk factor over one period.) Then, we have Pt(1 + r a t ) = Et[ Dt+At + 8( P, t)At] - covt[(/St+Ar + 8( P, t )At, A f ] . Noting that 8(P, t ) A t and Pt are known at time t and can be removed or added without affecting the covariance, we can arrange and allow At to approach 0 getting E[d P + 8(P, t)dt] : r Pdt + c o v [dP, d r ] , which is (26). Note that this derivation does not assume that the value of the claim is nonzero, so it can be used to describe the change in value of assets that do not have limited liability.

G. Sick

650

the price P a n d the pricing factor is f ( t ) :

oz(P, t)dt + 5(P, t)dt = r P d t + coy [ ~ ( P , t)do), d f ] ,

(27)

for s o m e p r i c i n g factor dr. W e c a n write (27) as

a ( P, t) + 6( P, t) = r P + cr( P, t ) ¢ t . ( f , t)p[o),

~t]

(28)

where

p[o), o~t ] w ( f , t) v / ~

= t h e u n i t l e s s c o r r e l a t i o n b e t w e e n t h e W i e n e r processes do) a n d d f ; = s t a n d a r d d e v i a t i o n of d f .

S i n c e W i e n e r processes have u n i t v a r i a n c e p e r u n i t time, the c o v a r i a n c e is cov[do), d f ] = p[c0, o)t]dt.

3.2.1. Pricing derivative assets: options Now, we i n t r o d u c e the definitions for the o p t i o n asset:

W ( P , t) = v a l u e of t h e real o p t i o n or c o n t i n g e n t claim. T h i s real o p t i o n c o u l d b e

D(t)

t h e right to acquire, e x c h a n g e or a b a n d o n the u n d e r l y i n g asset u n d e r specified t e r m s ; = t h e rate at which cash v a l u e is c o n v e y e d to t h e o w n e r of t h e o p t i o n (e.g., n e t r e v e n u e o n a p r o p e r t y t h a t m u s t b e e x c h a n g e d for t h e u n d e r l y i n g asset w h e n the o p t i o n is exercised).

W e t a k e t h e c u s t o m a r y a p p r o a c h a n d a s s u m e t h a t the o p t i o n is a fully p a i d

capital asset t h a t g e n e r a t e s n o f u t u r e liabilities a n d has a positive value. W e will alter this r e s t r i c t i o n w h e n we discuss f u t u r e s a n d f o r w a r d contracts. A s s u m i n g the f u n c t i o n W ( P , t) is twice differentiable, by It6's l e m m a t h e stochastic p r o c e s s for the o p t i o n is 16

d W = Wp d P + W t d t + ½~r2(P, t ) W p p dt = Wpot(P, t)dt + W e c r ( P , t)do) + Wt dt + ½~r2(P, t) Wpp dt.

(29)

J5 From the martingale theory, we know that f can be represented as a scalar random variable. However, to write a diffusion equation for f, it may be necessary to use a vector of state variables. All we need here is that the risk premium depends on the product ~rf(f, t)p[co, (of]. To see the full multivariate diffusion analysis, see Cox, Ingersoll & Ross [1985b]. 16 ItS's lemma is the chain rule for stochastic calculus. In ordinary calculus, the chain rule for a nonstochastic function W(P, t) of two variables P and t is dW(P, t) -- Wp dP + Wt dt. When P is a random variable, we get an extra term (1/2)cr2(P, t)Wpe, which is the drift in the option value induced by the combination of risk (o2(P, t)) and nonlinearity of the option price as a function of the underlying asset price. For example, if the option price is a convex function of the underlying asset price, then Wpp > 0 and by Jensen's inequality, the expectation of the option price (at the next instant of time) is higher than the option value at the expected underlying price: E[W(P + dP, t + dr)] > W(P + E[dP], t + dt). This drift is represented by (1/2)cr2(p, t)Wpp. It6's lemma comes from a Taylor series expansion of E[W] to order dt around the point (P, t). For a further discussion, see Ingersoll [1987, chapter 16], for example.

Ch. 21. Real Options

651

The expected return (capital appreciation plus dividend) to the option owner, per unit time, is then

(WEo~(P, t) + Wt + ½cr2(p, t)WEE + D(t)) dt.

(30)

But, by the consumption CAPM and It6's lemma, this expected return must also equal

r W d t +cov[dW, df] = (rW + WEcr(P,t)af(f,t)p[co, o~r])dt.

(31)

Equating (30) and (31), noting the CAPM relation for the underlying asset (28), and simplifying, yields the fundamental partial differential equation for valuation of options: lcr2(P, t) Wpp -t- Wt + WE[r P -- 8( P, t)] + D(t) = rW.

(32)

This pricing equation was derived by equating the return required by investors for an investment in the option (bearing in mind its risk) to the expected capital gain plus dividend payable to the owner of the option. These returns are dollar returns per unit time. The option inherits the systematic risk of the underlying asset through the factor WE, and this generates a return premium for the option in the consumption CAPM. However, the option also inherits the same extra return from the underlying asset, through It6's lemma. Equal terms in the consumption beta of the option appear on both sides of the equation and drop out. Thus, the option valuation equation (32) does not reference the consumption beta of the option or the underlying asset. The equation would be the same whatever the risk preferences of investors. Indeed, equation (32) readily admits the following 'risk-neutral' interpretation: The right side represents the required return (per unit time) of a risk-neutral investor for an investment in the option. The left side represents the expected return to ownership of the option (per unit time), assuming the underlying asset is priced by a risk-neutral investor. The first term on the left side is the drift in option price induced by nonlinearity of the function W(P, t) and the random variation of P. The second term on the left side is the direct dependence of the option value on time. The third term is the drift in the option price induced by drift in the underlying asset price, assuming that the underlying asset is priced by a risk-neutral investor who demands a total return of r P per unit time, but against this gets an amount S(P, t) in the form of a dividend. Finally, the option holder also is compensated with a direct payment of a dividend in the amount D(t). While (32) does admit this risk-neutral interpretation, it does not necessarily require that investors be neutral towards risk. Adjustments for systematic risk are still embodied in the equation through the price of the underlying asset P. What is important from the risk-neutral analysis is the fact that, so long as P is the price of an asset, the value of P contains all the information about systematic risk that is necessary to determine the value of any asset whose value is contingent on future values of P. Equation (32) still directly depends on the total risk of the underlying

G. Sick

652

asset o-(P, t), because the underlying asset price does not have to reflect the level of total risk under the consumption CAPM. We have derived this equation from the consumption CAPM, but it is the same equation that is commonly derived by no-arbitrage techniques. The no-arbitrage analyses of Black & Scholes [1973] and Cox & Ross [1976] were originally presented with risk-neutral interpretations. The appealing aspect of the approach we use here is that it does not require an assumption of the ability to perform short sales or continuous trading of the underlying asset, which may be very hard to justify if the underlying asset is a real asset that is not traded. For example, the outcome of research and development would be a call option to acquire an underlying operating project of uncertain value P, in exchange for a fixed construction cost of K. If we would like to use (32) to calculate the value of the real option, it is useful to know that the validity of the equation does not require any assumptions about liquidity of the underlying asset. Such an asset is very illiquid if it does not exist until the option is exercised! The option inherits its systematic risk from the systematic risk of the underlying asset through the hedge ratio Wp. This allows us to use the consumption CAPM to adjust for the systematic risk of the option. The martingale pricing operators adjust for this systematic risk in pricing assets, so all we need to know is how much systematic risk is conveyed to the option at various points in its life. If any hedging or arbitrage is done to price the option (or the underlying asset), the hedging can be done by transactions in liquid assets, matching the level of systematic risk in a portfolio of liquid assets to the level of systematic risk of the option. No trading in illiquid assets is required. Indeed, this pricing approach can be readily extended to price options on assets that don't exist or never would be held. For example, suppose we have an option with a payoff that is a function of a random variable P. This function could be the standard payoff to a call option: max{0, P - K}. Alternatively, it could be any nonlinear function representing the net payoff of some production process as a function of some basic economic variable 17 p. In the latter case, there is no asset that always has the value P, and we cannot directly infer the value of the dividend accruing to ownership of an asset with this value. However, given the process (26) for the variable P, we can use the consumption CAPM (27) to impute a dividend:

3(P, t) = r P + cr(P, t)crf(f, t)p[w, wf] - o~(P, t).

(33)

Substituting into (32): ½a2(p, t ) W p p + Wt +

+ Wp(c~(P, t) - a ( P , t ) ~ t ( f , t)p[co, mr]) + D(t) = rW.

(34)

This equation can be used instead of (32) to derive the option value. It does require knowledge of the drift a ( P , t) and a risk premium or(P, t)c[t() , t)p[co, co/], 17Later, we will see that options analysis can be driven by uncertain interest rates, so we can replace P by r.

Ch. 21. Real Options

653

in order to c o m p u t e value. In particular, it does require an explicit modelling of the pricing factor f , unlike equation (32). In a risk-neutral context, the factor c~(P, t) - or(P, t)~rf(f, t)p[o), o)f] of (34) is the risk-neutralized drift for the r a n d o m variable P. This is the true drift minus the risk p r e m i u m that would be c o m m a n d e d by an asset whose price is perfectly correlated with P.

3.2.2. Pricing derivative assets: Jbrward and futures contracts It is useful to have an understanding of the pricing of forward and futures contracts because price data on these contracts provide useful information on the imputed convenience dividends on commodities as well as on their spot prices] 8 A forward contract is a contract for the future delivery of a commodity. The contract is f o r m e d between two agents - - the long agent agrees to take delivery against receipt of the forward price, and the short agent agrees to make delivery at time T in exchange for payment of the forward price. Since both agents could engage in spot market transactions at time T, the net swap p a y m e n t is a gain to one and a loss to the other. The contract is a zero-sum game. N o m o n e y changes hands at the time of negotiation, nor prior to delivery. Unlike options, forward contracts are not fully paid up and hence are not capital assets. M o r e formally, the forward price Ft,r negotiated at time t for delivery of the commodity at time T > t is the certain payment to be made at time T that has the same time-t value as the claim to the risky future spot price/St. That is, 0 = Vt[Ft,r - Pr].

(35)

H e r e it is understood that the net swap payment Ft,r - / S t takes place at time T. Thus, the forward price Ft,r is the certainty-equivalent set at time t of the risky future spot price Pr- Defining the certainty-equivalent operator set at time t by CEt [.], w e have

Ft,r = C E t [ P r ] .

(36)

The certainty-equivalent operator is the risk-neutral expectations operator. That is, the certainty-equivalent formed at time t of a claim to a risky payoff X r to be received at a future date T is the time-t risk-neutral expectation of that claim: C E t [ 2 r ] = Et[2r].

(37)

To see this, note that (35) and (36) implies (for 2 r = P r ) Vt[/ST] = e-r(r-t) Ft,r = e-r(r-t)CEt[ PT ], while (1) establishes that Vt[/Sr] = e-r(T-t)l~t[pr]. 18Carr & Jarrow [1995] also take the view that futures markets are a practical source of information for contingent-claim valuation.

654

G. Sick

A f o r w a r d c o n t r a c t does n o t c o n v e y a capital asset, b e c a u s e t h e c o n t r a c t is n o t fully p r e p a i d . M o n e y is e x c h a n g e d b e t w e e n t h e c o n t r a c t i n g a g e n t s only at the m a t u r i t y d a t e T, a n d n o t at the e a r l i e r n e g o t i a t i o n d a t e t, so t h e f o r w a r d price is n o t t h e p r e p a i d price p a i d to a c q u i r e a capital asset. A f u t u r e s contract, like a f o r w a r d contract, is a c o n t r a c t b e t w e e n two a g e n t s to p u r c h a s e a c o m m o d i t y at a f u t u r e date. H o w e v e r , t h e futures c o n t r a c t is r e s e t t l e d or m a r k e d to m a r k e t by h a v i n g p a y m e n t s f r o m t h e losing a g e n t to the g a i n i n g a g e n t o n a daily basis t h a t k e e p s t h e v a l u e of t h e swap c o n t r a c t at $0 every day 19 T h a t is, t h e f u t u r e s price F t - l , r set at t i m e t - 1 for delivery at t i m e T > t is t h e "2

price t h a t is t h e c e r t a i n t y e q u i v a l e n t of t h e n e x t - p e r i o d futures price, F t , r : ^

"X

F,-1,T = CG[F,,r].

(38)

A n a l o g o u s to (35), we c a n write: 0 = V t - I [ F t - I , T -- F t , r ] .

(39)

A t m a t u r i t y , t h e f u t u r e s c o n t r a c t settles at a v a l u e e q u a l to the p r e v a i l i n g spot price, so t h e t i m e - T - 1 f u t u r e s price is set to reflect this: o = Vr-l[Fr-l,r

-

(40)

['r].

I t e r a t i n g a p p l i c a t i o n s of (38), a n d u s i n g (40), we see that t h e f u t u r e s price e q u a l s a n e s t o f c e r t a i n t y - e q u i v a l e n t o p e r a t o r s a p p l i e d to t h e final spot price: %

Ft,r = CEt[CE¢+I ... CEr-l[fir]

"-'].

(41)

If i n t e r e s t r a t e s a r e c e r t a i n , f u t u r e s a n d f o r w a r d prices are i d e n t i c a l 2o. If i n t e r e s t rates a r e stochastic a n d c h a n g e s i n i n t e r e s t rates a r e c o r r e l a t e d w i t h c h a n g e s in 19This mark-to-market feature, combined with interest-earning margin requirements, removes the incentive of a losing agent to default on a futures contract. This facilitates exchange trading of futures contracts. In contrast, forward contracts are traded over the counter by direct negotiation between buyer and seller. 2o From (35) and (40), it is clear that forward and futures prices are the same at time T - 1. More generally, suppose for simplicity that the continuously compounded interest rate every period is the constant r. The value at time t of a claim to receive the spot price at time T > t is the present value of the forward price: Vt[/Sr] = e r(T-t)CEt[/Sr] = e-'(r-t)F, r. But an agent could also acquire the same payoff by purchasing a contract that will pay at time t + 1 an amount equal to the time-t + 1 value of a claim to the spot price at time T. Thus, Vt[/ST] = Vt[Vt+l[/fiT]] = e-rCEt[Vt+l[Pr]]. iterating this process, we have that Vt[/Sr] = e-rCEt[e-"CEt+~ ... e-rCEr_l[/Sr] ...] = e '(r-t)CEt[CEt+l -.. CEr_I[Pr] ...] = e r(T-t)ft,

T.

Ch. 21. Real Options

655

t h e f u t u r e s price, t h e n f o r w a r d prices c a n b e d i f f e r e n t f r o m f u t u r e s prices b e c a u s e i n t e r m e d i a t e cash inflows a n d outflows o n t h e f u t u r e s c o n t r a c t a r e c o r r e l a t e d w i t h t h e i n t e r e s t r a t e at w h i c h they c a n b e i n v e s t e d (or b o r r o w e d ) . E m p i r i c a l l y , t h e d i f f e r e n c e b e t w e e n t h e s e does n o t s e e m to b e large, h o w e v e r . F o r m o r e i n f o r m a t i o n o n t h e s e d i s t i n c t i o n s b e t w e e n f u t u r e s a n d f o r w a r d prices, see Cox, I n g e r s o l l & Ross [1981], J a r r o w & Oldfield [1981] a n d R i c h a r d & S u n d a r e s a n [1981]. W e will use f o r w a r d a n d f u t u r e s prices i n t e r c h a n g e a b l y in t h e r e m a i n d e r of this article. I n c o n t i n u o u s time, f o r w a r d a n d f u t u r e s contracts will still satisfy t h e f u n d a m e n tal p r i c i n g e q u a t i o n s (32) or (34) t h a t g o v e r n o p t i o n assets, b u t with o n e i m p o r t a n t v a r i a t i o n to a c c o u n t for t h e fact that t h e f u t u r e s price is always m a r k e d to m a r k e t in o r d e r to k e e p t h e c o n t r a c t v a l u e at $0. D e f i n e t h e f u t u r e s price as a f u n c t i o n of the u n d e r l y i n g c o m m o d i t y spot price P as F ( P , t) = Ft,v. T h e n t h e f u t u r e s price f u n c t i o n satisfies t h e a n a l o g u e of (32): 1 2 ( p , t ) F p p + Ft + F p ( r P - 3(P, t))

O.

(42)

or, e q u i v a l e n t l y , a n a l o g o u s to (34): ½~r2(P, t ) F p p + Ft + Fp(c~(P, t) - (~(P, t ) ~ f ( f , t)p[co, ~ / ] ) = 0.

(43)

T h e s e e q u a t i o n s differ f r o m their o p t i o n a n a l o g u e s b e c a u s e they a r e m i s s i n g t h e t e r m s D(t) a n d r W . I n t u i t i v e l y 21, o n e c a n ascribe this to two facts. First, a f o r w a r d c o n t r a c t pays n o i n t e r m e d i a t e d i v i d e n d s D(t) = 0. S e c o n d , t h e v a l u e of the f u t u r e s c o n t r a c t is z e r o at t h e start of the p e r i o d 22, so t h a t the r i s k - n e u t r a l r e q u i r e d d o l l a r r e t u r n r F = 0. T h a t is, t h e s e e q u a t i o n s describe t h e r i s k - n e u t r a l i z e d e x p e c t e d payoff to a n a g e n t w h o goes l o n g o n e f u t u r e s c o n t r a c t (for a cost of $0) a n d m a r k s it to m a r k e t at t h e n e x t i n s t a n t of t i m e (for a n e t cash flow of d F ) . Since t h e r e is n o i n v e s t m e n t , t h e r i s k - n e u t r a l e x p e c t e d dollar p a y o f f is $0, w h i c h is the r i g h t - h a n d side of (42) or (43). T h e f u t u r e s c o n t r a c t only g e n e r a t e s capital gains, Thus, the futures price must equal the forward price. Note that we needed interest rates to be nonstochastic to be able to float the present value factors into and out of the certainty-equivalent operators. 21 More rigorously, We can develop (42) and (43) from (32) and (34), respectively, by considering the pricing of a portfolio strategy that consists of holding a fully-margined futures contract. At time t, such a portfolio consists of a long position in a futures contract (costing $0), plus an investment in a bond worth $Ft. The risk-neutral expected return on such a contract is r(0 + Ft)dt. The mark-to-market generates a capital gain or loss of dFt. The portfolio strategy is self-financing because, at the end of the period, the agent has exactly the level of wealth required to acquire the portfolio prescribed by the strategy for the next period, plus a dividend equal to the interest earned on the margin. Thus, the portfolio cost is the value of a fully paid-up capital asset for which equations (32) and (34) apply. In these equations, the dividend rF~ dt paid to the portfolio equals the risk-neutral required return on the portfolio, so these terms cancel on either side of the equations, resulting in (42) and (43). The remaining terms in these equations are a result of the futures contract rather than the bond contract, so these equations apply equally well to the portfolio strategy as to the futures contract. 22 Agents must also post margin to engage in futures markets to guarantee against default. Since they earn market rates of interest on the margin, the margin has no effect on the pricing of the contracts.

656

G. Sick

and no dividend payments, so the risk-neutral expected returns are the capital gains as identified in the left sides of (42) and (43). 3.3. Boundary conditions

Equations (32), (34), (42) and (43) describe the motion of the price W of any option asset or the futures contract, so long as its risk is derived from an underlying random variable P. Typically these differential equations have a general solution, with several parameters that must be determined to get a specific option value function. To determine a value for a specific option, boundary conditions must be specified to describe how the option price behaves in certain limiting circumstances. In practice, there are 3 types of boundary conditions: 1. Payoff boundaries describe the value of the option at the time the option is converted into another asset, which is often the underlying asset. Sometimes the option is a compound option, and exercising the option gives rise to another option or operating state of an asset. In these cases, the boundaries describe the transition between such states. 2. Free boundary conditions describe the freedom the option owner has to decide when to exercise the option. The owner acts to optimize the value of the option, so this condition often becomes a first-order condition of optimality. This typically is the high-contact or smooth-pasting condition, and says that at exercise or transition into another state, the option function is tangent to the payoff boundary described above. 3. Technical conditions are often necessary to preclude meaningless solutions to the differential equations. For example, the option may be expected to increase or decrease in value with the value of the underlying asset. Or the option may reasonably be expected to converge to a specific value as the underlying asset value becomes arbitrarily large or small. 3.4. Example: perpetual American call option on a log-normal asset

Consider a perpetual option to convert farm land earning income at the fixed rate $D per year to urban land with uncertain value $P by paying a fixed development cost K. This is a call option with an exercise price of K and dividend paid to the option owner of D. As with most real options, a key issue is to determine when to exercise the option, so the option is an American option. Suppose the standard deviation of the price of the underlying asset (urban land) is o-(P, t) = o'0P for a one-year period. The dividend paid to the underlying asset is the net annual rental rate earned by the owner of developed urban land, which we assume to be 6(P, t) = 6oP. Thus, the underlying asset price follows a log-normal diffusion with a constant dividend yield 30. Equation (32) becomes: 1o-2p2 o Wpp + Wt + Wp(r - ao)P + D = rW.

(44)

To exercise the call option, the owner pays a conversion cost K, so the net payoff if the option is exercised when the value of the underlying asset P = P* is

Ch. 21. Real Options

657

W(P*'t) = max l D' p* - K]

(45)

This says that the owner has the right to walk away from the option and receive the value of pure nonconvertible agricultural land, D/r, or exercise the option and receive the net present value of conversion, P* - K. The owner chooses the larger of the two values. Condition (45) is a payoff boundary, as discussed above. The owner of the option must decide on an optimal exercise policy. There are two state variables in this problem - - the underlying asset value P and the time t. Since the option is perpetual and the dividends and exercise prices do not change over time, the exercise decision should not depend on the state variable t. That is, the exercise decision is time-invariant - - it can depend on P but not t. The optimal exercise policy is characterized by a region of values for P, such that exercise occurs the first time the underlying asset value enters the region. Let P* be a boundary point of the region. Then (45) must hold whenever the underlying asset value passes into the region for the first time, at price P = P*. After the underlying asset price passes into the exercise region the first time, there is no further exercise decision in this problem, since the option is not reversible. Later, we shall see that there is only one boundary point for the exercise region. So far, we have seen that all the characteristics of the model are invariant to time, since the dividends, drift, boundaries, and exercise price do not vary with time. Thus, the option value should not be dependent on time and we can write W(P, t) = W(P) and set Wt = 0. We can rewrite (44) as 1 2~2,,, gcr o r wpp + Wp(r-•o)P + D = r W .

(46)

This second-order ordinary differential equation is the Euler equation. A particular solution to (46) is W = D/r. The general solution is obtained by adding the general solution to the homogeneous equation, to obtain: D r

W(P) = A + P v+ + A _ P g- + - -

(47)

where 1

60-_____~r i ( 1

+

~0-r~ 2

+ 4

)

2r

(48)

+72o2

The constants A+, A_ are to be determined by the boundary conditions. Note that y+._ > 0 if and only if ~ (1

±

~o--r~2

+ 41+702

2r

(1

a0--r~

>-

Thus, y+ > 0 and y_ < 0. To determine signs for the constants A+, A and B, we need to apply technical boundary conditions that consider the limiting behavior of (47) for large and small P. The option can be exercised at any time

G. Sick

658

SO (45) places a floor under the value of the option. Also, the option cannot provide any more value than the value of its perpetual dividend stream and the value of the underlying asset into which it can be converted. Thus, we must have D / r < W(P) <_ P + D/r. Since PY- becomes unbounded as P -+ 0, this means that A_ = 0. We would expect W(P) to be increasing in P, so A+ > 0. We would expect the owner to exercise into the underlying asset only if the proceeds of exercise (P* - K ) exceed the value obtained by discarding the option and only maintaining the pure agricultural land value ( D / r ) , so (P* - K) > D/r. In (45), then:

W(P*) = P* - K.

(49)

Using this with (47) and the fact that A_ = 0, we have that

A+ = p,(_y+) ( p , _

K

_D)r "

(50)

We can determine the values of A+ and P* equivalent ways. One way is to note that, with A_ globally maximized if we maximize the value of the of the option should choose P* to maximize A+ as the unique value

p. _

?/+

jointly in either one of two = 0 in (47), the value of W is constant A+. Thus, the owner defined by (50). This occurs at

( K + D~

k

?/+-1

(51)

r}

If the dividend yield on the underlying asset, So, is positive, then },+ > 1 and the boundary value P* is positive.23Substituting into (50) and (47), and simplifying, we have

W(P)-

1

(

- -

×+ -

K+

D)(P~V+D

+--.

1

(52)

r

There are several observations to make about this solution and our derivation. First, note that if ownership of the option conveys no dividends (D = 0) then (52) gives the well-documented solution for a perpetual call option on a log-normal asset. This allows an alternative derivation of this model by regarding convertible farm land to be a portfolio of two assets. One asset is pure nonconvertible land of value D/r. The other asset is an option to exchange the farmland for urban land by paying an exercise price of K + D/r, which consists of a conversion cost K plus 23 Note that y+>l

<-+ l (

1 + S0-r~2ff2 ] + 7>%2r 2 ~1 0 - - r~2 2

But this latter inequality holds if 60 > 0, for then

as desired.

659

Ch. 21. Real Options

Option Value, W(P) Payoff boundary max {D/r, P - K}

Infeasible / Optimal

D/r

~

/

I /

~

f

I Subo Ximal

/ 0

/ K

I P1 P*

P2

Underlying Asset Value, P Exhibit A. At the optimal exercise price P*, the option value graph is tangent to the payoff boundary. the value of surrendering the pure nonconvertible farm land of value D / r . The pure option value is

?/+-1 in this interpretation. This solution is illustrated in Exhibit A, which shows various solutions to the pricing differential equation (46), along with the payoff boundary W ( P ) > m a x { D / r , P - K}. The lowest solution shown cuts the exercise boundary twice. It corresponds to a policy of exercising the option when the underlying asset price reaches one of the two intersection points P1 or P2, since at those points W ( P i ) = m a x { D / r , Pi - K } = Pi - K (for i = 1, 2). Setting the value of A+ to satisfy either of these boundary conditions results in the lower graph. But this solution is suboptimal, since we could clearly increase the function values everywhere by selecting an exercise price between P1 and P2. The highest solution to (46) shown is infeasible because it never intersects the exercise boundary. Its 'option' values are only hypothetical. The optimal solution is the graph that is tangent to the exercise boundary. The point of tangency is the optimal exercise price P*. This tangency condition is the smooth-pasting or high-contact condition. For a heuristic derivation of it, see Merton [1973]. In general, the condition is that the option value function (regarded as a function of underlying asset price) is tangent to the boundary at the point of exercise: - K ) P=p* W p ( P * , t) -- 3 ( P-O-P

(53)

G. Sick

660

Since the slope of the exercise boundary is 1 in this call option, we have

Wp(P*, t) = 1.

(54)

O n e can verify that this is the same as the first order condition that led to (53) in this particular example. So far, we have not been particularly careful to describe the exercise region. Instead, we have merely studied the option behavior at an exercise point P*. In the perpetual call option we have just seen, the exercise region only has one b o u n d a r y point, and it is the interval [P*, co). This can be seen, for example, by noting that the global m a x i m u m of the coefficient A+ is attained at one point only. 24 In other real option models, to be discussed below, there can be multiple critical or exercise points. Finally, we can set the dividend yield on the underlying asset 30 = 0, to note that V+ = 1 in (48). Thus in (51), P* approaches + o c as 30 approaches 0. T h e r e is no incentive to exercise an A m e r i c a n call option early, if the underlying asset does not pay a dividend. T h e option is exercised in order to capture the dividend on the underlying asset. In other situations, we may have other reasons to exercise an A m e r i c a n call option early. These could be changing exercise prices, the arrival of a maturity date in finite time, or stochastic interest rates. We can calculate the cost of capital for the option by dividing the right side of equation (31) by W and dropping the differential dt:

r+

Wpo-( P, t ) ~ f ( f , t)p[co, O)f] W WpaoP = r + ~af(f, t)p[co, o)t.] = r + (Y+) ( W T / r ) a o ~ f ( f , t ) p [ o J ,

wf].

W e are assuming that the risk p r e m i u m on the underlying asset (namely, a0o).(f, t)p[o~, o~f]) is constant, so the cost of capital for the option is n o n r a n d o m if and only if the ratio (W - D / r ) / W is n o n r a n d o m , which is to say that D = O. T h a t is, D/r is the value of the perpetual deterministic dividend to the owner of the unexercised call option. If there is no such dividend, the cost of capital of the option is constant. W h e n D/r ~ O, the option value consists partly of the riskless dividend value and partly of the risky pure option to exchange this riskless asset and the exercise price for the risky underlying asset. T h e fact that the cost of capital for the option is constant when D = O, is the result of a knife-edge condition arising from the log-normal distribution of the underlying asset and the perpetual nature of the option. As the underlying asset price falls toward the fixed exercise price, the option cost of capital tends 24 The only zero of the derivative of the function A+(P*), is the solution (51), where it is positive. Noting that A+(P*) is continuously differentiable on (0, co) and that limp*~oo A+(P*) = 0 and limp*_,0 A+(P*) = -oo, we can conclude that the maximum of A+(P*) is unique.

Ch. 21. Real Options

661

to increase because of the higher operating leverage. However, this is mitigated by the way the option limits downside risk. As the underlying asset price falls, the right to allow the option to die worthless becomes more valuable, and this absorbs risk. As long as 3o > 0, one can prove that y+ > 1, so that the cost of capital of the option is larger than that of the underlying asset, when D = 0. In this situation, the call option does lever up the risk of the underlying asset. For other stochastic processes on the underlying asset price, the cost of capital on the underlying asset is random, so that of the option is random as well. This example illustrates many of the important features of real options.

3.5. Modelling the dividend on the underlying asset In the previous example, we saw that the dividend payout to the underlying asset is an important determinant of optimal exercise policy and call option value. In a financial option analysis, it is easy to determine the dividend on the underlying asset, since it is a cash payout that goes to the owner of the underlying asset but not to the owner of the option. In a real option setting, the dividend is a more elusive concept, since the underlying asset may not even exist until the option is exercised. Alternatively, the underlying asset may not be a capital asset that is held for capital appreciation and cash dividend income. The underlying asset is simply the value of an operating project after the initial construction cost has been sunk. In some situations, it may be convenient to think of the underlying asset as a unit of a commodity, such as petroleum, lumber or copper. In other situations, the value of the underlying asset may be a function of a risky economic variable, such as an interest rate. It is still useful to think of the option value as being a function of this risky variable. At first it is not obvious how to impute a dividend to a risky economic variable that does not represent an asset price. We now turn to the problem of modelling this dividend, which can be called a convenience value or convenience yield.

3.5.1. Commodity convenience dividend The convenience dividend may arise because there is value to being able to quickly use the spot commodity to meet an emergency demand. The convenience dividend we define here is net of the storage costs of the spot commodity. If these costs are larger than the benefit of being able to use the spot commodity in an emergency, then the convenience dividend could have a negative value. If P is a risky future payoff, we define the commodity convenience dividend by the imputed dividend in (33). This imputed dividend is the difference between the expected dollar return under the CAPM (if the payoff were a capital asset) and the actual expected rate of dollar return. This is just a reexpression of the identity: Convenience dividend + Capital appreciation = Return on investment. For commodities, it is possible to determine the convenience dividend from market data on forward prices. For example, suppose that the spot price of a commodity P follows a log-normal diffusion so that o-(P, t) = c~0P for some

662

G. Sick

volatility or0, and the convenience dividend is proportional to the underlying asset value for some yield 80. Thus, 6(P, t) = 8oP. T h e n (42) provides a differential equation in the futures price F: ~o o r r p p + Ft + F p ( r P - 8oP) = 0.

(55)

T h e futures contract converges to the underlying spot price at maturity, providing the b o u n d a r y condition: F ( P , T) = P.

(56)

This equation is solved by 25 F ( P, t) = e(r-5°)(T-t) P.

(57)

This assumes a constant convenience yield and a constant interest rate. If convenience yields and interest rates vary deterministically over time, so that rr and 80,~ are the forward interest rates and forward convenience yields that will prevail at time T, then we can generalize (58) and examine contracts with adjacent maturities T = r and T = T + AT to get the approximation:

\ Ft,~+Ar ) "

(58)

This model assumes that the (forward) interest rates and (forward) convenience yields are deterministic, and a test of this assumption is to examine the variation in the forward yields and rates implied by (58) at various points in time t. If they do vary, then we must consider one of the stochastic convenience yield models below. However, if the variation is not significant, real option models can be safely built by assuming them to be constant. If we assume that convenience yields are deterministic and that the cost of capital for discounting the spot commodity is constant, then we can also calculate the convenience yields from a vector of forecasts of future spot prices. This is important, because in many practical situations, the value of the underlying project will be c o m p u t e d by a present-value analysis in which expected future cash flows are discounted at a cost of capital or risk-adjusted discount rate. By imposing a consistency between the convenience yields implied by the forecasts (typically m a d e within the corporation) and the convenience yields implied by the market in forward prices, the analyst can get more economically relevant valuation models. In order to get a constant discount rate for a future spot price, one has to assume a constant risk p r e m i u m for the expected rate of return to an investment 25 This is equivalent to e - ' ( T t ) F = e -~o(T t)p. This common value is the price at time t of a claim to the delivery of the spot commodity at time T. The left side is the present value of the certainty-equivalent of this claim, The right side is the value of the spot commodity today, discounted for the convenience dividends that will not be received.

Ch. 21. Real Options

663

in the project. 26 In the notation of (28) or (33), the expected rate of growth in the spot price of the c o m m o d i t y is

c~(P, t) P The

discount

r P + (x(P, t)cg.(L t)p[co, coil - ~(P, t) P rate

for the

spot

price

at

time

v >

0 is k~ =

r~ +

cr(P, r)o).(f, r)p[co, cof]/P. In (33), this means that the 'beta' of the spot price cash flow c~(P, v)c~t.(f, r)p[co, o~f] must be proportional to the spot price P. Thus, we assume that a ( P , r ) a f ( f , v) = ao,vPc[t and oe(P, v) = al),r P for some constants c~0,~, o'0,~ and o)., so that oe0,~ = r~ + o-0,~c~fp[co, cof] - 30,~. The discount rate then becomes k~ = r~ + cro,~c[tp[co, co/]. The time-0 forecasts of future spot prices are c o m p o u n d e d at this growth rate net of the convenience dividend:

E0[/St] = exp

(kr - 50,T) d r

P0.

(59)

Analogous to (58), we can then approximate the forward convenience yield for time r by comparing price forecasts for adjacent future points in time, r and r + At: 3o,~ = k~ + ~

In

\e0[P +ad/

.

(60)

3.5.2. Stochastic convenience models The previous section modelled the convenience dividend as a constant proportion of the spot price. If the convenience dividend and yield are both stochastic, we need some useful models. Gibson & Schwartz [1990] and G a r b a d e [1993] suggest modelling the convenience dividend by a mean-reverting process. Others, such as Pindyck [1991] and L a u g h t o n & Jacoby [1993] suggest modelling the commodity price itself as a mean-reverting process. A commodity price can be mean-reverting to the extent that there are imperfect substitutes for it in various product markets and various sources of the commodity that can be brought on stream (with a capital cost or time lag) at different production costs. A mean-reverting commodity price could be a good representation if a cartel, such as O P E C , controls a significant fraction of the supply. If prices fall below some target, the cartel will cut back production to increase revenue. If prices rise above the target, the cartel produces m o r e to preclude the development of alternative supplies. T h e strength of the cartel's response increases as the prices deviate m o r e strongly f r o m the mean. T h e intuition for a mean-reverting convenience dividend or yield is m o r e indirect, and perhaps less compelling. With many commodities, there is a seasonal 2o See Myers & Turnbull [1977], Fama [1977] and Sick [1986, 1989a] for a discussion of environments in which expected cash flows can be discounted at a deterministic risk-adjusted discount rate. Little useful generalization can be made beyond the assumptions made here.

664

G. Sick

convenience yield induced by costly storage and either seasonal supply (particularly agricultural commodities) or seasonal d e m a n d (particularly energy sources). Since the period of the seasonality is one year or less, there should be little impact upon real option decisions that have a longer term. Alternatively, the short-run elasticity of supply could be much smaller than the long-run elasticity of supply. 27 If using a mean-reverting convenience model amounts to using the level of the convenience to proxy for the time of the season, a more efficient explanatory variable for the time of the season is the actual date! Models of mean-reverting commodity prices a m o u n t to making choices of the functional form o~(P, t) and or(P, t) in (26), which is reproduced here for convenience: d P = ol(P, t)dt + or(P, t)dw

(61)

Mean-reverting processes are just continuous-time analogues of basic autoregressive and integrated auto-regressive processes. The simplest such process is the O r n s t e i n - U h l e n b e e k process, in which the change in price level has a m e a n reverting drift that elastically depends on a strength of mean-reversion )~, and the distance b e t w e e n price level and the long-run m e a n / 5 : d P = Z([' - P ) d t + ~0dco.

(62)

This process is the analogue of the discrete-time AR(1) process: Pt = X/5 + (1 - X)Pt-, + Acot.

(63)

To keep the process stationary, one would normally require that )~ > 0, and to keep the process from overshooting the target, one would normally require )~ < 1. Iterative substitution in (63) for Pt-1, Pt-2, etc. yields the infinite moving average process in the disturbances Acot_n: p~ = 2,/5 + (1 - )~)Pt-1 + Acot =

-----)v/5 q- (1 -- X)(),./5 q- (1 -- )OPt-2 q- Acot-1) q- A(.ot . . . . . oo

= E

(Z(1 - z)n/5 + (1 - Z)nAcot_n)

(64)

rt=0

Thus, the effect of a disturbance Acot on the future price P~ is reduced at an exponentially declining rate by the factor (1 - X)~-t. The half-life of the process is the time required for this weight to decline to 1/2, which is n = - In 2/[ln(1 - )~)]. Clearly, (63) can be easily be estimated by a multiple regression model, where Acot could either be pure white noise or some function of the systematic risk factor f . This process could b e c o m e negative, but the probability of this is small if or0 is small c o m p a r e d to )~ a n d / 5 . 27 It is also possible to model production decisions by the real option to start or stop production as a short-term planning measure after the capital investment has been made. This is discussed by Brennan & Schwartz [1985], Pindyck [1991] and Cortazar & Schwartz [1993].

Ch. 21. Real Options

665

To ensure that P never becomes negative, one could take the exponential of a r a n d o m variable that follows an O r n s t e i n - U h l e n b e c k process 28: d P = )~(L - in P ) P dt + croP do).

(65)

It is convenient to assign a new constant/5, such that L = In/5 + crc~/2;~" This is the analogue of the time-series technique of taking the logarithm of the price and modelling the transformed variable as following an auto-regressive process. These processes imply a convenience dividend or yield by equation (33). D e n o t e p = p[o), o)f] and a t. = o 7 ( f , t). Thus, the convenience dividend consistent with (62) is ~ ( P , t) = r P + croatP - )~(/5 - P).

(66)

T h e basic pricing equation (32) or (34) for a real option with value W ( P , t) becomes: ~cr612W p p + Wt + Wp(Z(/5 - P ) - crocrfP) + D ( t ) = r W .

(67)

Similarly, when the commodity follows (65), the convenience yield is ~- ( P- , t )

-- r + crooT p -

( cln/5+~--~ r02_lnP)

(68)

and the basic pricing equation is 1 2 ~crO P W p p Jr" Wt +

In both of these models, the size of the convenience dividend increases with P. This is consistent with the notion that the value of ready access to the commodity rises as the commodity becomes m o r e valuable. Moreover, in (65) and (68), we see that the convenience yield rises with P. This is consistent with the notion of a temporary shortage (low inventories) driving up price and a sufficiently inelastic d e m a n d in the short run that the convenience dividend of a spot supply rises at an even faster rate. This m o d e l also implies that the convenience yield follows an O r n s t e i n - U h l e n b e c k process itself, since there is 28 That is, assume that P = e p where dp = ),(ln/5 p)dt + aodw. By It6's lemrna, dP 1 d2p 2 dP = __-~-Pd p + ~ ~-p2 a0 dt = x(L - In P)P dt + croP dw. _

This is the process proposed by Laughton & Jacoby [1993]. There are other variations on this, such as that which Pindyck [1991] proposes: dP = ~(/5 - P)P dt + aoP do)

or the process Trigeorgis [1995] proposes: dP = X(P - P)dt + a0P do.

G. Sick

666

an invertible linear relationship between In P and g0 by (68), and In P follows an Ornstein-Uhlenbeck process. As we did earlier for the log-normal price process, we can infer the meanreverting parameters from the futures markets by determining the futures prices that are consistent with the spot process. If the spot price follows (62), we can modify (67) as in (43) to get the partial differential equation for the futures price

Ft,T : 1 2 Fp p + Ft q . Fp(k(/5 ~0.d . . .

P)

~roc{f

(f,t)p[co, cof]) = 0

(70)

subject to the boundary condition (56). The solution to this is the futures price term structure (assuming that the systematic risk measure p[w, cof]0.f (f, t) is constant): Ft T : /5 + e-~(r-t)(p, _ /5)

0.00.fP ( 1 - e-~(g-t)) .

'

(71)

X

The futures price consists of the long-run mean price /5, plus a revision in expectations e-X(T-0(P -- /5) arising from the difference between current spot price P and /5, minus a systematic risk term [(cr0afp)/)~](1- e -x(T 0). The expectations revision term vanishes as the term of the futures contract increases to oo. The systematic risk term increases to a maximum asymptotic value of (0.ocr/p)/)~ as the futures maturity approaches oc. Setting this latter term to zero (for example, by setting crtp = 0) gives the term structure of expected future spot prices. If the spot price follows (65), the fundamental pricing equation is the modification of ( 6 9 ) : 1 2 PFpp + Ft + FpP I ()~l n / 5 + ~ - £°"2° -lnP)-0.o0.fpl=O. 7o-0

(72)

The solution to this is the futures price term structure (assuming that p[w, o)t.] at(f, t) is constant):

Ft,T= /5 (~)exp[-)~(T-t)] exp I ~o.g - ( 1 - - e -2)'(T-t))

crOC~tP(l--e-X(T-t))] )~ .

(73)

Again, by setting crt p = 0, we obtain the term structure of expected future spot prices. Gibson & Schwartz [1990] assume that the spot price and the convenience yield follow joint stochastic processes. The spot price follows a log-normal diffusion and the convenience yield follows an Ornstein-Uhlenbeck process. This does not generalize our mean-reversion models, because their spot price is assumed to follow a log-normal diffusion rather than a mean-reverting process. Their convenience yields are estimated from futures market data. Since futures markets

Ch. 21. Real Options

667

have maturities measured in months, while real options have maturities measured in years, care must be taken in extrapolating their convenience yield models to real option situations. They also find that the market price of convenience yield risk seems to vary over time. Their model is d P = o~oP dt +

CrlP dcol (74)

d3 ---=Z(~ -- ao)dt + ff2do)2 P12 = corr [de01,

do)2].

Here, the convenience yield 3o = 3 ( P , t ) / P is measured f r o m futures prices by the continuous-time analogue of (58), and is assumed to be independent of term to maturity of the contract. Any contingent claim on future delivery of oil prices will have a value W ( P , 3o, t) that depends on oil price, convenience yield and time. Since both oil price and convenience yield are stochastic, the fundamental pricing equations (32) and (34) are expanded to incorporate a multivariate It6's lemma: l_2rJ2~iz

5Ol r

l

2

vvp p + P 3orYlrTZplz Wpa + ~y~ W~a + Wt + Wp(r - 3o) +

+ W~(X(6-3°)-c°v[r~2dc°2'df])

(75)

where cov[o-2doo2, d f ] / d t is the market price of convenience risk per unit time. The right side of this equation represents the risk-neutral return required for an investment of W in the contingent claim. The term D(t) represents any payout to the owner of the contingent claim at time t. If we instead want to determine the time-t price F,,~ of a futures or forward contract for delivery at time r > t, there is no initial payment, so the risk-neutral expected return is zero. The fundamental pricing equation for a futures contract becomes 1 ~2~2~1 2 2~1 r vpp @ P3oCrl~2pFpa + g¢r~ Faa + Ft +

+ Fp (r - 3o) + Fa (Z(8 - 30) - c o v [crzdco2, d f ] / d t ) = 0.

(76)

Gibson and Schwartz estimate the convenience yield (once per week), using (58) for the two New York Mercantile Exchange ( N Y M E X ) crude oil price contracts nearest to maturity. They then estimate a discretized version of the stochastic_ processes (74) for the spot oil price and the near-term convenience yield. They find, for example, that the long-run m e a n convenience yield 6 is approximately 18% per year. They also test for m e a n reversion in oil prices, but find none. 29 T h e resulting parameter estimates are then used to determine the value of the market price of convenience-yield risk by fitting the whole term structure of 29 The failure to find mean reversion in oil prices is probably an artifact of the weekly sampling and short 5-year time interval they use. Monthly sampling over longer time periods does seem to suggest mean reversion in oil prices, with a half life of disturbances of less than 1 year.

668

G. Sick

actual futures prices to theoretical prices derived from (76) with the terminal boundary condition F~,T = P~. They find that the market price of convenience yield risk is quite variable, which could be interpreted as a misspecification of their model. They also determine risk-adjusted present value factors for long-term future delivery of oil, that are consistent with their estimates of the short-term processes. Their results should be regarded as preliminary, since they use an inefficient two-step procedure for estimating the parameters of (76) and the market price of convenience-yield risk. Moreover, if they cannot find the riskless rate of interest used by commodity traders, they will make corresponding errors in their estimate of the convenience yield. Nevertheless, their model does share the following stylized fact with models (71) and (73): spot oil prices are more volatile than forward oil prices. This is because the short-term convenience yield is quite random. Laughton & Jacoby [1993] find three general effects of mean reverting spot prices on the valuation of real options. First, because the mean reversion tends to reduce long-term uncertainty it can increase underlying asset value. Second, the reduced risk can reduce option value for a given underlying asset value. Finally, the future reversion of cash flow can have direct valuation effects. Thus, ignoring mean reversion can lead to early or to late project adoption, as well as under- or over-estimates of the value of the real option. The nature of the error depends on the particular circumstances of the problem, which can only be assessed by a full analysis of the real option.

3.6. Discrete-time economic analysis of real options It is often the case that the continuous-time partial differential equations like (44) do not admit analytic solutions. In such a case, numerical methods [such as in Schwartz, 1977, or Brennan & Schwartz, 1978] can be employed to solve the partial differential equation, subject to the boundary conditions. These techniques convert the continuous-time economic problem into a discrete mathematical problem, and then allow the analysis to be done with mathematics. An alternative approach is to regard the economic problem as a discrete-time problem in the first place. This allows the analysis to be done with economic reasoning backing up all the mathematical steps. Little, if anything, is given up by taking a discrete-time approach to analyzing real options, because accurate determination of value and optimal exercise hurdles is not as important as development of a consistent, economically meaningful model. The primary problem for the practitioner is to have analytic tools that are reliable and unlikely to result in bizarre normative prescriptions. By keeping the economic intuition in the problem as long as possible, there is less likelihood of a serious modelling error. 3° 30The situation is similar to a traditional capital budgeting setting in which determination of NPV consists of discounting expected cash flows at a cost of capital. It is often possible to calculate a cost of capital to a reasonable degree of accuracy using accepted financial techniques that make

Ch. 21. Real Options

669

Consider the discrete points in time {0, h, 2h . . . . . nh . . . . } for some increment of time h > 0. A representative time from this set is t. The discrete-time analogue of the process (26) is: APt+h =-- Pt+h -- Pt = oe( Pt, t)h + cr( Pth, t)"~/hzt+h

(77)

where Zt+h = independent, identically distributed error term, with mean zero and unit variance 31 For each point in time, t~ one can calculate the net value of an immediately adopted project as a function of the underlying asset price as discussed, for example, in Section 2.2 (Hotel ling valuation of resource properties). Denote this by NPV(Pt, t). Denote by W(Pt+h, t + h) the value of the option at the next point in time, t + h. Viewed from time t, the price ['t+h is random, and the value at time t -t- h of the policy of keeping the option alive at least for one more period is the present value of the risk-neutral expected value of the option next period, or

e-rhE, t[W(Pt+h, t q- h)].

(78)

Since we have a Markov process in two state variables (underlying asset price and time), the risk-neutral conditional expectation operator Et [.] and the value in (78) are functions of Pt and t. By the Bellman equation (or principle of optimality in dynamic programming), the value of the option at time t is the larger of these two values: W(Pt, t) = max {NPV(Pt, t), e rh/~t[W(/St+h, t + h)]}.

(79)

If the option has a finite maturity date T = N h by which the option must be exercised, then we have W(t'r+h, T + h) = 0.

(80)

This allows us to analyze the option value and optimal exercise decisions by recursively working backwards from date T, using the principle of optimality (79). That is, from (80) in (79) for t = T, we can evaluate the function W(-, T), as a function of P. Given W[., T], we use (79) to evaluate W(., T - h). This process is continued until the function W(-, 0) is evaluated. Along the way, all the option price functions for various intermediate times have been evaluated, and all the optimal exercise decisions have been made, by remembering which element of reference to various market relationships. The job of forecasting cash flows is often done outside a market context, and it is easy for an analyst to glibly extend historic start-up growth rates into a forecast of unrealistically high growth rates for a mature project. By recognizing that N P V is an economic rent, and noting that economic rents are scarce, the analyst with economic insight can provide m o r e useful estimates of project NPV. 31 For such increments, the variance and m e a n of increments to the process increase proportionally to the time increment h. Thus, the standard deviation, ¢Y(Pt+nh,t-[-nh)~,/h increases proportionally to ~/%

670

G. Sick

the m a x i m a n d (79) has b e e n selected. 32 This is the standard b a c k w a r d - i n d u c t i o n analysis of a decision tree. I f t h e r e is n o maturity date and the real o p t i o n is perpetual, t h e n there is n o starting b o u n d a r y point for the induction, and d e p e n d e n c e o n time will vanish (unless the terms of exercise are still t i m e - d e p e n d e n t ) . To solve the p r o b l e m o n e must find a n o p t i o n function W that solves the B e l l m a n e q u a t i o n (79), with the dependence o n t dropped. T h e solution is a fixed p o i n t in some function space, a n d can be a n intractable p r o b l e m to solve. I n practice, it is often simplest to approximate the p r o b l e m by one with the maturity date T t a k e n to be in the distant future. By letting the time i n c r e m e n t h shrink towards zero, the discrete process usually converges towards the c o n t i n u o u s process. We have to be careful to say 'usually' since convergence d e p e n d s o n the n a t u r e of the distribution a n d the behavior of the drift a n d the variance as functions of the underlying asset price. A t the level of real options, convergence might n o t be a critical issue, if the analyst feels that the e c o n o m i c r e p r e s e n t a t i o n of the p r o b l e m in discrete terms is itself a good a p p r o x i m a t i o n to reality. That is, economic events do occur in discrete time, so a discrete m o d e l may have direct validity, rather t h a n n e e d i n g to inherit validity as an a p p r o x i m a t i o n to a c o n t i n u o u s - t i m e model. T h e convergence of various discrete-time models to c o n t i n u o u s - t i m e models (typically w h e n the discrete-time m o d e l also has a discrete state space) is discussed in greater detail by Boyle, E v n i n e & G i b b s [1989], H e [1990], N e l s o n & R a m a s w a m y [1990] a n d A m i n [1991]. T h e general methodology is to assume that the true u n d e r l y i n g asset price follows a diffusion process, such as (26) with dist u r b a n c e s that are locally normally distributed: do) ~ N (0, dt). By approximating the diffusion process with a discrete process (77) that locally has the same m e a n and variance as the diffusion, it is often possible to show that the distribution of the discrete process (even when the distribution of price i n c r e m e n t s is a discrete B e r n o u l l i process) converges to the distribution of the diffusion process. 33 T h e n , the probability distribution for the discrete price process is replaced by 32 This feature makes the recursion very efficient, because no analysis is done of suboptimal exercise policies. In general, the optimal exercise policy is a function of time - - for example, a call option at maturity is optimally exercised when the underlying asset price exceeds the exercise price, but when the call option has an infinite time to maturity, the optimal exercise price is much higher. If we were to use Monte Carlo simulation to analyze a real American option, we would have to simulate the whole model for all possible exercise functions, and then choose the function that provides the highest option value. This is computationally inefficient, because the analyst (or her computer) must consider the myriad of wrong answers before finding the right answers. 33 If the volatility and drift are reasonably well-behaved functions of the underlying asset price, convergence can often be proven. Problems occur if these functions grow too quickly as the asset price increases or drop to zero at prices that are attainable. In this respect, the normal, log-normal and mean-reverting processes discussed here are well-behaved. One of the most straightforward approaches, as discussed by Boyle, Evnine & Gibbs [1989] is to borrow from the proofs of the central limit theorem, and analyze the convergence of the characteristic function of the discrete distribution to the characteristic function of the normal distribution, after some appropriate transformation of variables. The characteristic functions can be approximated by a second order Taylor series, which depends on the first and second moments of the distribution. The question is whether the error term of the Taylor series vanishes as h --+ 0.

Ch. 21. Real Options

671

the martingale distribution, as discussed in the Section 2.1.3 (Steps to finding a martingale valuation operator). Since this typically only amounts to subtracting a risk premium from the drift of the true probability distribution, the convergence properties carry over to the discrete martingale distribution. That is, the discrete martingale distribution will converge to the continuous martingale distribution. Since an asset price (such as an option price) is the present value of the risk-neutral expectation of the (continuous) payoff functions, the asset prices calculated for the discrete approximations will converge to the asset price for the continuous diffusion process. Of course, this discussion is rather informal and heuristic. The real options analyst desiring more rigor could go to the literature cited above and go through the convergence proofs for the particular model of interest. Not only does this tend to dampen the enthusiasm of many who would want to analyze a real option, but it does not even address the important question of speed of convergence. How many steps are needed to get a good approximation? To get a practical grip on all these problems at once, the following checklist is useful: 1. Does the discrete approximation seem to be economically meaningful in the sense that it fits the important stylized facts known to the analyst? 2. In some problems, analytic solutions can be obtained for problems that bound the real option. Are these bounds are satisfied by the approximation? If not, shorten h and add more steps to the problem. If this doesn't address the violation of the bound, it may be that the process doesn't converge, or a modelling error has been made by the analyst. Useful bounds include: a. European options are less valuable than otherwise-similar real American options. b. Finite-lived American options are less valuable than perpetual American options. Often it is possible to model a contract payoff so that the discrete solution should approximate that of a known analytic solution. Thus, if the Bellman equation (79) for an American option is replaced by W ( Pt, t) = e-rh E t [ W (fit+h, t -}- h)],

(81)

and the terminal condition (80) is replaced by W(/Sr, T) = N P V ( P r , T),

(82)

we get a European option, for which there may be a known analytic solution for the continuous diffusion. Similarly, if we replace the Bellman equation by the certainty-equivalent characterization of the futures contract (37), and replace the terminal condition by futures-spot convergence (56), we get the futures price, which can be compared to the analytic formulas for futures prices given earlier for various processes. To obtain analytic formulas, one can determine a risk-neutral probability distribution of the terminal payoff (option payoff or spot price in the case of futures). The European option value is the present value of the risk-neutral expectation of the option payoff. The futures price is the risk-neutral expectation of the future spot price.

G. Sick

672

3.6.1. Choosing an approximating process Consistent with the diffusion process, one could model Zt+h as a normally distributed random variable, with mean 0 and variance 1. This could make the backward induction analysis unwieldy, since it might not be possible to parametrize the functional form of intermediate option values W(., t). In general, this is a nonlinear function of Pt for underlying asset prices below the critical value at which the option is optimally exercised. Alternatively, one can model at as a discrete random variable taking on only two or three different values in its domain. Instead of needing a parametric representation of W(Pt, t), as a function of Pt, one only needs the actual values of the function at a finite number of discrete points in time and at discrete underlying asset prices. The option values are only calculated for a tree-like lattice structure of discrete points (P, t). If the incremental process takes only two different values at each point in time, it is a Bernoulli process. Since Bernoulli processes can be made to converge to many useful continuous processes, such as normal, lognormal, and Poisson processes 34, little is lost by using a Bernoulli process as the main analytic tool. In a real options context, very useful analysis can be conducted with as few as 50 or 100 time steps, which can often be conveniently represented on a spreadsheet, as discussed in Sick [1989b].

3. 6. 2. Discrete Bernoulli approximations of continuous processes For simplicity, we will restrict ourselves to discrete Bernoulli processes. At each time t, the random variable Z,t+h can assume only two values: { ~+(P, t), zt =

e _ ( P , t),

with probability :r(P, t), with probability 1 - a-(P, t)

(83)

These processes can be represented by a simple tree structure, and they can be analyzed with familiar decision-tree methods. By allowing Jr to depend on P, the increments of the zt process may not be independent, as were the increments in the Wiener process co in (26). However, the process is still a Markov process, and this extra freedom is needed to approximate some processes, such as a mean-reverting process. We will discuss various approaches selecting functional representations of the coefficients for the discrete process to approximate the continuous process. We are only interested in the true probability distribution ~r in so far as knowing that it will converge to the true continuous distribution. Our analysis only involves the risk-neutral probability distribution ~. We find it convenient to rewrite the risk-neutral versions of (77) and (83) as: A bt+h

- - P t + h -- Pt =

= [ u(Pt, t, h), ! d(Pt, t, h),

with probability ~ ( P - t, h), with probability 1 - z?(P - t, h),

(84)

where u( Pt, t, h) = a( Pt, t)h + e+(P, t) and d(Pt, t, h) = o~(Pt, t)h + ~_( P, t). 34 See, for example Madan, Milne & Shefrin [1989].

Ch. 21. Real Options

673

We have 3 g e n e r a l objectives in this a p p r o x i m a t i o n : 1. T h e drift of t h e price should equal the riskless return, a d j u s t e d for the c o n v e n i e n c e dividend 35:

(85)

/~t[A/St+h] = erh pt -- 6( Pt, t)h In (84), this implies that 5-(P,, t, h) =

[erh p, - a( Pt, t)hl - d(Pt, t, h) u(Pt, t , h ) - d ( P t , t , h )

(86)

Equivalently, we c o u l d use the C A P M , as in the transition f r o m (32) to (34) to write:

erh pt -- ~(Pt, t )h = (ot( Pt, t) - ~ ( Pt, t)crf(f , t)p[co, cof ])h.

(87)

2. T h e v a r i a n c e of the price changes should equal the variance of the c o n t i n u o u s distribution. In practice, we will see that this is often h a r d e r to achieve, and s o m e t i m e s it is only necessary to have this condition hold as a limit w h e n h --> 0. Simplifying the n o t a t i o n by suppressing the a r g u m e n t s of the functions, the strictest f o r m of this b e c o m e s :

~2h

vart[APt+h] =

~(U

2 --

Et [(A/St+h) 2]

^

-

d 2) ned 2 -- (Or -- crcrfp)2h 2.

2 (88)

Using (86) and (87), this simplifies to

cr2h + (~ - cr~.f p)2h 2 = (c~ - cr~r¢p ) ( u + d) - ud.

(89)

G i v e n u, e q u a t i o n (89) can be solved for d and (86) can be solved for 7?. T h e flexibility in choosing u gives one ' d e g r e e of f r e e d o m ' , which can be useful in m e e t i n g the final objective. 3. T h e tree of price levels over t i m e m u s t recombine, so that the state is fully described by the total n u m b e r of up moves and down moves, a n d is i n d e p e n d e n t of the o r d e r in which the moves occur. Thus, a down move followed by and up m o v e must l e a d to the s a m e u n d e r l y i n g asset price as an up m o v e followed by a down move. M a t h e m a t i c a l l y , this m e a n s that the up and down o p e r a t i o n s m u s t b e c o m m u t a t i v e :

d ( u ( e t , t, h), t q- h, h) = u(d(Pt, t, h), t -t- h, h).

(90)

This basically forces us to use additive or multiplicative functional forms, or to use a t r a n s f o r m a t i o n of price and time in which the up a n d d o w n moves are 35 In (85), we are taking a dividend that is analogous to simple interest but accruing the riskless interest at a compound rate. If h is sufficiently small, this distinction is not important. If we know more about the functional form of the dividend, a more accurate expression can be derived. For example, if 8(P, t) - ~0P for a constant yield 60, then it is best to replace (85) with Et[APt+h] = e(r-8°)h Pr,

G. Sick

674

additive. 36 It also limits our flexibility in allowing the parameters of the u and d functions to vary with price, because a sequence of N / 2 ups followed by N / 2 downs must lead to the same price as N / 2 downs followed by N / 2 ups. The former sequence resides in the high-priced portion of the tree, while the latter resides in the low-priced portion of the tree. Similarly, the up and down functions cannot directly depend on time. However, if the variance function ¢r2( ., .) depends on price or time, it is often useful to transform the stochastic process to have constant variance over price and/or time. If we represent the underlying asset price and time as transformations of a stochastic variable and rescaled time, then there is an open question as to whether the first two criteria should involve the means and variances of the increments of the price process or of the transformed process. T h e two can be related by It6's l e m m a and any resulting biases will generally vanish as h ---> 0. The important question is the accuracy of the approximation for larger h.

3. 6.3. Examples of discrete-time approximations In the following examples, we will suppose that the underlying diffusion process has a constant risk p r e m i u m o-0~fp. First, consider an additive diffusion in which c~(P, t) = or0 and or(P, t) = or0, where o~0 and o-0 are constants. T h e n the assignment

u( P, t) = - d ( P, t) = ~cr~h + (or - cro¢tp)2he l

7? = ~ +

~

-

~oc~tp

2u

(91) (92)

satisfies our three criteria. It is also possible to verify that this gives 0 < 7? < 1. The resulting lattice structure has equal up and down jumps, from a starting point of P0. Next, consider a log-normal diffusion in which ol ( P , t) = oe0P and cr ( P, t) = or0P. By (33), this m o d e l implies a constant dividend yield 80 = r + a 0 o ) p - ~0- For this popular model, Cox, Ross & Rubinstein [1979] (hereafter C R R ) proposed using

u ( P , t) = P ( e ~°'/h - 1), d ( P , t) = P ( e -~ro'/h - 1)

(93)

7? = ~ +

(94)

and ~/h.

They assumed that dividends were paid on discrete dates and adjusted the prices in the tree by the factor (1 - 80) ~, where v = the n u m b e r of dividends paid in the a m o u n t 30P per payment. This assignment satisfies our third criterion, but only satisfies the first two criteria in the limit as h ---> 0. This assignment of values is 36 If we take a logarithmic transformation, a multiplicative function becomes additive, so we generally only need to investigate additive functional forms.

675

Ch. 21. Real Options

based on a logarithmic transformation of prices, so that the diffusion is an additive diffusion, as in our first example. That is, the CRR assignment sets r h - ln(1 + d / p ) ~" = ln(1 + u / p ) - ln(1 + d / p )

(95)

Also, this assignment sets the variance of the diffusion process equal to the second moment of the discrete process. That is instead of subtracting the mean square to get a central moment, in (88) CRR use the noncentral moment by setting: ~r2h =- Et [(A ln(/St+h))2] .

(96)

While the CRR approximation is simple and converges to the continuous model solution, it sometimes produces violation of some bounds for 'large' values of h. For example, it can result in discrete values of American options that are less than the Black-Scholes values of otherwise similar European options. Hull & White [1988] and Sick [1989b] suggest setting the mean and variance of the process for prices (rather than logarithms of prices) to be the same in the discrete and continuous models. This behaves better for large values of h, and one might wonder which of the two major changes gave most of the improvement - - dealing with prices rather than logs, or measuring varifince accurately, rather than as a second moment. Certainly a simple approach is to improve the accuracy of the mean and dividend approximation by setting e ( r - 3 0 ) h _ e-rr~,/h

7~ =

(97)

e~4~ _ e-~4~

while maintaining the volatility model of (93). This yields approximately the same results as the Hull-White model, but has a simpler formula. 37,38 37 If we desire to also have the logarithm of the price process have the same variance in the continuous and discrete cases, we m u s t set cr2h

=

t,t 2 _

(Et[~x

ln(/St+h)]) 2.

Using It6's l e m m a or the properties of the log-normal distribution, we can see that this requires setting

u=~/h crg + ( r - 8 o - ~ ' ) h . T h e second term under the square root converges to 0 as h -* 0, so the expression is asymptotic to (93). 38 A m i n [1991] also uses the degrees of freedom in selecting u and d to model the mean price accurately, without addressing the volatility approximation: In(1 +

u( P, t )') = (r - 8o)h - l n ( P

/

ln(l+d(P't))=(r-8o)h-lnIp1 and

1 2

e'o'/h - - e - ~ ° v ~ ~ + ¢0x/h 2 / / e c~0~/~- e c~0~/h\

2

) -~r°'~

G. Sick

676

Now, consider the mean-reverting diffusion (62). Using the information available at time t, the risk-neutral expected value of the spot price at time t 4- h is the certainty equivalent or futures price Ft,t+h. Thus, by (71), the risk-neutral expected drift over the interval h is

Et[A/5t+h] = Ft t+h -- Pt = (1 -- e_Xh ) r1(/5 _ Pt) '

t.

crocrfp n| . Z

(98)

J

Thus, solving (84) for 77, we have 39 =

t?,[A/5,+h]

u -d

-- Cl

=

(1 -- e

[(/5

-- P,)

--

-

u -d

(99)

where we must select u and d to satisfy criteria 2 and 3. Consistent with the C R R analysis, we set u = -d = ~0v~.

(100)

If the time to maturity is large and there are few steps so that h is large, (99) may give values of 7? that are outside the interval [0, 1]. In such cases, one could simply set the risk-neutral probability to 0 or 1 as the case may be. This has the effect of guaranteeing a risk-neutral u p - m o v e when the price is very low and guaranteeing a risk-neutral down-move when the price is very high, which is consistent with the notion of mean reversion. If this happens for substantial portions of the tree, however, more steps should be added to the tree. Similarly, consider the mean-reverting process (65). By (73), the risk-neutral expected drift over the interval h is = F,,,+h -- t', =

exp

~4 ( 1 -

e-2Xh)

cr0~fP(l--e-Xh)]--l}.

(101)

As in C R R , we set the up and down moves by (93). From (101) the risk-neutral For general processes, including this one, Carr & Jarrow [1995] also advocate setting the riskneutral probability equal to 1/2. They set the up and down moves (in a transformed process) equal to the instantaneous drift plus and minus cr0~,/h. 39 A quicker but less accurate approximation results from assuming the convenience dividend is constant over the time interval [t, t + h]. In (66), this means that the risk-neutralized expected drift of the diffusion is

and

dr = ( X ( P - P . , ) - ~ o ~ f p ) h - d u-d

Ch. 21. Real Options

677

probability is 40 ( - ~ ) exp[-)~h)-ll exp I cr2~-~(l -- e -2)~h)

cr°°)P(1--e-'~h)l--eC~°~/h)~

= eao~/h _ e-ao~/h (102) In both of these mean-reverting models, the risk-neutral probability is a function of price. The magnitude of the up and down moves does not depend on price, in order that the tree should recombine. Boyle, Evnine & Gibbs [1989] show how to use discrete processes to model multiple stochastic processes that are jointly correlated.

4. Examples 4.1. Interest-rate uncertainty Ingersoll & Ross [1992] investigate the real option to delay the implementation of investment when the sole source of uncertainty is the real rate of interest. By using the martingale pricing operator E[.], they can value an operating project as the present value of the certainty-equivalents of its cash flows. They show that a great deal of the practical analysis can be simplified to the simple case of project with a balloon payment to be received at time T after the implementation of the project. This case tends to generalize to projects that pay a stream of cash inflows because they find that the optimal deferral strategy is not sensitive to the project duration T, as long as the interest-rate uncertainty is not too large. Suppose the spot real rate of interest r follows a square root process with no drift: dr = cro~/Tdco

(103)

where ~o follows a Wiener process. Consider an interest-rate contingent claim with a value W(r, t). The claim pays a cash dividend of D(r, t) per unit time. Cox, Ingersoll & Ross [1985b], or CIR, show that the price of such a claim, satisfies

i2 Wrr~r~r + Wt + )~rWr + D = r W , 40We could also use the quicker, but less accurate approximation from (68):

and calculate #=

h [). (In/5 + a2/2)~ _ In P) - ooOfp] + 1 - e ,5o4h e~o,A _ e - C ~ o ~ / h

(104)

678

G. Sick

where - ~ is the price of interest-rate risk. This can be obtained by substituting r for P in (34) and making two extra observations. First, in (103), c~(r,t) = 0. Second, we are setting the interest-rate risk p r e m i u m to be - L r = or(r, t)(~f(f, t ) p [ w , a)f]. Given that the volatility in (103) is or(r, t) = cr0~/~, one might have expected the interest-rate risk p r e m i u m to be -,k~/F, for some constant )~. However, the interest rate variable u p o n which we price the option is a systematic risk variable for the whole economy. Thus, the c o m p o n e n t of the volatility of the market pricing factor that is related to interest-rate risk 41 is also proportional to ,,/7. This gives the extra factor ~/7 for the risk-premium term. For example, for a pure discount bond that matures at time T, set the price P = W and the c o u p o n dividend D = 0. Since Pr < 0 for a bond, examination of (104) establishes that )~ > 0 corresponds to a positive term p r e m i u m in the sense that the expected rate of return on the (long-term) b o n d exceeds the riskless rate of return by -)~rPr > 0. C I R show that the price of the pure discount T-period b o n d in this situation is P(r, T) = e -b(r)'-

(105)

where 2(e ×T 1) (g - )O(e×T -- 1) + 2V I

b ( T ) ----

and

g -- ~/~-z+Zcr{~

(106)

Suppose the m a n a g e r has the perpetual real option to select the time v at which a project is implemented. Consider, first, a simple project with an investment 1 and a single cash inflow of $1 to be received at time T after the investment. At the time ~: of the investment, the value of the real option is the net present value of the project, which is the price of a discount b o n d with maturity T periods hence, when the spot interest rate is r = r(~). That is, the project value at time v when the spot interest rate r = r ( r ) is: W ( r ) = W ( r ( v ) , T, I) = P ( r ( v ) , T ) - I.

(107)

The incentive to exercise early in this problem is the fact that the a d o p t e d project gives'a predetermined dollar payoff ($1) that does not grow with a delay in starting the project. By adopting early, the owner of the option can invest the N P V of the project in the bank and earn a market rate of interest. Thus, the market rate of interest to be earned by investing the proceeds of the N P V of the project is the dividend that motivates the owner of the option to exercise early. N o t e that the interest rate process is Markov with the spot interest rate as the only state variable. Thus, the value of the real option to adopt the project does not 41 In the martingale valuation or the consumption CAPM, there is only a single factor used for pricing. Arbitrage pricing theory decomposes this into multiple factors, one of which could reflect interest-rate risk. That is, if we project the interest rate r onto the market factor f, we get the interest rate variable itself. The CIR model is built on power or logarithmic utility and a single state variable Y that locally has volatility proportional to -,/~. Since the only source of uncertainty in this problem comes from interest-rate risk, we can rederive the whole pricing theory for these problems using interest-rate risk as the pricing factor, and take Y to be proportional to r.

679

Ch. 21. Real Options

depend on the time t, although it does depend on the duration of the project, T. The optimal adoption policy specifies the boundary point(s) r* for the adoption region in interest-rate space. The project is adopted the first time the interest rate hits r* from outside the adoption region. If NPV is declining in the interest rate, this occurs at a random stopping time that depends on the interest-rate path: {(r*) =- inf {t [ ?(t) _> r*}.

At this boundary, the value of the option satisfies (107): (108)

W(r*) = P(r*, T) - I

Thus, the real option is an interest rate contingent claim that satisfies (104) with W¢ = C = 0. At the boundary it satisfies (108). To eliminate a spurious term from the solution to the differential equation (104), we specify that the value of the option falls to zero as the interest rate becomes arbitrarily large: limr--+c~ W ( r ) = 0. The general solution is W ( r ) = A e -vr

(109)

where A =--- e ( v - b ( r ) ) r * -

[ e vr*

and

v

--=

+ Y > 0.

(110)

Maximizing the value of the option is equivalent to selecting r* to maximize the value of A. This is equivalent to the high-contact condition. This occurs when the critical interest rate is set at l(v-b(T)) r* = r * ( T , I ) = b - ~ In

-vff

"

(111)

The value of the option to adopt the project is, by (109) and (110): W ( r ) = a e -vr = (e [v-b(T)lr* - I e ~ r * ) e -~r = e-U(r-r*)(e b(T)r* _ 1) %

(112) The second factor is the net present value of the adopted project at the time of exercise W ( r * ) = e -b(r)'* - I. Thus, the first factor is the risk-neutralized expected present value factor of waiting to adopt the project until the time {(r*) when the interest rate first drops to r*, namely

e-V(r-r*)

=

E0

E exp

-

~(s) ds 0

)1

(113)

The manager invests the first time that the interest rate r falls below the hurdle value r*. In contrast, the internal rate of return of the project is calculated from

G. Sick

680 (105) and (107) as I R R = I R R ( T , I) -

ln(1)

(114)

b(T)"

Thus, r* - I R R =

l(v-b(T))

b(r~ In

v

< 0

(115)

so that the real option is exercised at a lower interest rate than the IRR, which provides a break-even value of the NPV. However, Ingersoll & Ross make an interesting observation about the relationship between this difference r* - I R R and the duration of the project T and the interest-rate volatility parameter o'0. (Recall that interest-rate volatility is locally c~0~/r.) For small volatility levels (up to an annual standard deviation of the real rate of return of 20 to 30 basis points), the difference r* - I R R is almost invariant to the project duration T. This leads them to suggest a practical method of calculating the approximate hurdle interest rate for adoption of projects that return a stream of cash flows, rather than just a single cash flow: 1. Calculate the internal rate of return of the project. 2. Calculate the duration T of the project. 3. For a project that pays a single cash flow at this duration, calculate the difference r* - I R R in (115), noting that the difference is invariant to the investment level. 4. Add this difference to the I R R to get a good approximation to the hurdle real interest rate for the adoption of the project that has multiple cash flows. This approximation can be improved by using the following technique for projects that have a net present value N(r) that is declining in the interest rate. Using (112) and (113), the value of the real option to adopt the project when the critical interest rate is ? is

e-v(r-P) N (?) = e -~r (eVeN(?).

(116)

The task then reduces to one of selecting ? to maximize the factor in brackets. This can be done analytically or numerically. Ingersoll & Ross analyze the value of the option to adopt the project as a function of the term premium X. As X increases, long-term interest rates increase, and the value of an adopted project decreases. On the other hand, the higher (long-term) interest rates tend to increase the value of the option by increasing the benefit of deferring the date of incurring the investment cost. They show that the former effect dominates, so that the value of the option to invest in a project falls as the term premium increases and the yield curve steepens. They also study a related question: Will an overall decrease in the level of interest rates stimulate an economy by, causing real options to be exercised that would otherwise be deferred? If NPV is a declining function of the discount rate and investment decisions are made using the NPV rule, a decrease in interest rates will cause some projects to be accepted that were formerly rejected. Now, when we consider the option to delay a project, this answer does not always obtain. Ingersoll

Ch. 21. Real Options

681

& Ross show that a simple nonstochastic project with a declining term structure of interest rates can have the property that a decrease in short- and long-term interest rates makes it preferable to defer investment that otherwise would have been undertaken immediately. Intuitively, the declining term structure makes the NPV at the time of adoption more valuable when the project is deferred, which can tend to offset the loss due to discounting the NPV back to the present. When both long and short interest rates decline, the decline is proportionally more significant for the lower long rate, so the benefit of deferral becomes relatively more preferable. They go on to argue that this result is not simply an artifact of the declining term structure. In a stochastic model, they suggest that increasing risk over time could have the same effect. Ingersoll & Ross also address the question of estimation error of the parameters and its resulting effect on-estimation error of the hurdle rate r*. The loss in option value for selecting the wrong hurdle is asymmetric for over- or under-estimating r*. When interest rates are driven by the square-root process, significant losses occur when the project is adopted at too high an interest rate, while the losses are not so substantial for a policy of waiting too long and adopting at too low an interest rate. On the other hand, when they modify the analysis to the interest-rate process dr = cr0r3/2doJ, the asymmetry reverses direction, so the result is sensitive to the choice of interest-rate process. In traditional capital budgeting, multiple IRRs, which are caused by reversals in signs of future cash flows, can create multiple project acceptance regions. That is, there can be unconnected sets of interest rates for which the project NPV is positive. Since the value of a real option to a project converges toward the NPV as the volatility of the interest rate falls, or as an expiration date approaches. Thus, with alternating cash flow signs, there can be multiple interest-rate acceptance regions. The regions are not characterized by the NPV graph crossing zero, as in the traditional I R R analysis, but they are caused by multiple high-contact points between the N P V graph and the graph of the value of the real option, as a function of interest rate. This requires that the NPV graph have sufficiently large swings that it comes into contact with the graph of the value of the option as shown in Exhibit B. Ingersoll & Ross show how to use the existing bond-pricing literature to evaluate real options with more general risk-neutralized interest rate processes that have the general form: dr = Kr#(rc - r ) d t + cror~do) where K, st, fl > 0, and or0 > 0 are parameters. The procedure is to evaluate the risk-neutral expected present value of a claim to $1 to be received at {(r*), which is the first time the interest rate falls to r* from above. This is the general version of the right hand side of (113):

~b(r,r*) --= E0

exp

-

E(s) as 0

/ 1 r(O)

r

(117)

G. Sick

682 Value NPV V

a

l

u

I\

e

of real option

t\

Exhibit B. Multiple internal rates of return occur when the NPV graph crosses the zero axis. Multiple acceptance rates for real option to adopt project occur when the graph of option value is tangent to the NPV graph. This is induced by large swings in the NPV graph.

This is multiplied by the NPV at the time of exercise to get the value today of the real option to adopt the project the first time the interest rate falls to r*:

Wlr, -~ dp(r, r*)NPV(r*).

(118)

In general NPV(r) depends on the cash flow stream and the pricing structure of discount bonds for the chosen interest-rate process. The value of r* is chosen to maximize the value of the real option Wit*. This can be done numerically or analytically, in some cases.

4. 2. Further examples 4.2.1. Abandon and exchange The option to adopt or abandon a project are options to exchange one asset for another. In the call option to adopt, an asset of certain value (construction cost) is exchanged for an underlying asset of uncertain value. In the put option to abandon, the risky underlying asset is exchanged for a certain scrap value, or a reduction in the present value of fixed operating costs. These two options can be generalized to options to exchange one risky asset for another. In general, this will involve two state variables and necessitates a multivariate analysis, such as used by Gibson & Schwartz in Section 3.5.2 on stochastic convenience value. However, it is often possible to calculate a sufficient statistic that combines the two state variables into a single variable that is sufficient to characterize optimal exercise. This stems from the idea of Margrabe [1978] that one could cast the value of one of the assets in terms of the value of the other as a numeraire, thereby reducing the problem to the traditional option analysis with a single state variable.

Ch. 21. Real Options

683

McDonald & Siegel [1986] find that the sufficient statistic is the ratio of the two asset prices if they are joint log-normally distributed. Sick [1989b] shows that the sufficient statistic is the difference of the two asset prices if they are joint normally distributed.

4.2.2. Options to develop and redevelop property Paddock, Siegel & Smith [1988] develop a three-stage model of exploration and development in the petroleum industry. At the first step, the firm is determining the value of a petroleum lease or right to explore for oil and gas. The probability distribution of the quantity of reserves, has a spike of mass at zero reserves (quantity of petroleum) and a continuous distribution for positive reserve levels. At the next step, the firm has proved up a quantity of reserves and determined a rate of exponentially declining production that can be sustained, and the uncertainty is about the price of oil and gas. The final step is the ownership of a developed operating petroleum property. At both of these transitions, the owner must analyze a real option. They also compare their estimates of the value of an exploration lease to that of the United States Geological Survey and the highest and mean bid posted by oil companies for several offshore leases. The average (across leases) highest bid was significantly larger than both the USGS and option valuation estimates. 42 Only at a high price of gas (compared to the prevailing spot price) was the option value greater the USGS discounted cash-flow valuation. The authors wondered whether their low estimates of option values were a function of the poor data they were forced to use on estimates of potential reserves. The petroleum companies would have more refined reserve estimates than the USGS reserve estimates used for the option analysis. Capozza & Sick [1994] model an urban area as having uncertain rent at the center, with net rental revenue per unit density declining as distance increases from the center. Developers have the option to convert agricultural land (which earns rent at a possibly stochastic rate) into urban land by paying a fixed development cost. The optimal conversion policy determines the location of the urban boundary. An interesting aspect of the model is that the risk in the rental rate has systematic and unsystematic components. As systematic risk increases, the value of developed property falls, but the value of undeveloped property may either rise or fall. It will rise if the increased risk sufficiently increases option value, for any given level of value for the developed property. Up to this point, option models have always derived option value for a given value of the underlying asset, and had not explored the possibility that risk may be good or bad for the value of the option asset. 42For each lease, the mean bid is less than the highest bid, and is more in line with the estimate of option value. However, the best estimate of value is the winning bid, since each bidder should revise downward his or her estimate of the lease value to adjust for the 'winner's curse'. If the bidders have independent unbiased estimates of lease value, and bid according to their estimates, the winning bid will be an overestimate of lease value, conditional on winning, because the winner is more likely to have had positive valuation errors. This tendency for the winner to overestimate is the winner's curse.

684

G. Sick

One interesting use for models of this sort is to determine the option value lost to a farmer whose land is placed in a green belt by a government. Capozza & Sick [1991], Capozza & Li [1994] and Williams [1991] examine another dimension of the option exercise decision: the choice of density of development. The value of a developed property equals the density times the price per unit of density. The development decision is made in two steps. First, for a given rental rate, the optimal development density is determined in a model of increasing costs to develop a unit of density. This gives a value of optimally developed property for each rental rate, which becomes the underlying asset in an option model. Second, given this payoff to optimal scale of development, the real option problem is solved to determine the development criterion in terms of a hurdle price for a unit of land. The choice of process for the asset price and the functional cost of developing a unit of property is important in these models. For example, with log-normal asset prices and development cost that is a power function of density, the optimal density is often undefined, because setting a higher and higher density level, along with a higher and higher critical unit price of developed land delays development more and more into the future. The value gains in waiting exceed the loss in time value of money. Williams obtains a finite solution by imposing a maximum density (as in a zoning bylaw). Capozza & Sick get finite solutions by switching to an additive diffusion in underlying asset value. The Capozza-Sick paper uses this variable density model to measure the loss in value to a lease 'ownership' contract (compared to a perpetual or fee-simple ownership contract) that arises from the distorted decisions to redevelop the property. That is, since redevelopment benefits are lost at the termination of the lease, the owner of a leasehold property has incentives to redevelop earlier and to a lower density than the owner of an otherwise identical fee-simple property. This model can be used to explain the significant discounts in the value of leased property without having to resort to high growth rates that lead to unrealistic rent multipliers. The Williams paper also allows for stochastic development costs as well as income to a unit density of developed property. By expressing all prices and costs relative to a numeraire of the development costs, the model becomes the same as the log-normal model with one stochastic variable. H e also examines the option to abandon property. 4.2.3. Production options Various authors have modelled the flexibility a firm has to alter production rates as a real option. Brennan & Schwartz assume that there are three states of a mine or petroleum property: abandoned, shut in and operating. A transition cost (like an exercise price) is paid by the asset owner to go from one state to another. Only while operating does the property generate a positive net inflow or dividend. While shut in, the property incurs a cash outflow (to maintain the property). When abandoned, the property generates no cash flow. The manager must decide the critical points (in terms of spot commodity price, for example) at which to go from

Ch. 21. Real Options

685

one state to another. An important characteristic of these optimal decision points is that they exhibit hysteresis. That is, the commodity price at which one would shut in an operating property is lower than the price at which one would put a shut-in property back on stream. This is analogous to setting the critical hurdle price of a real call option higher than the exercise price. If there are no costs of opening or closing the operating property, then it is optimal to open when revenue exceeds cost and optimal to close otherwise. This decision can be made on a daily, weekly or annual basis, so it makes sense to view the flexible operating property as a portfolio of European call options on production, in which the exercise price is the operating cost. This approach was taken by McDonald & Siegel [1985]. In general, the net revenue from flexible production is some nonlinear function of an underlying risky variable (such as a commodity price). The function need not be piecewise-linear, as in an American call option. McDonald & Siegel [1985] show that a Cobb-Douglas production function leads to net profit function that is a power function of the commodity price. More generally, Sick [1989b] observes that the valuation of the claim to production for a given year in the future is the present value of the expected payoff under the martingale distribution. This can be determined by explicitly computing the expectation, or by approximating the payoff by a Taylor polynomial and using tabulated functions for the noncentral moments of the martingale distribution. Alternatively, the analyst could determine approximate payoff function values corresponding to ranges of values for the underlying risky variable. The present value of the risk-neutral probability of these ranges then becomes a price of a simple ,4xrow-Debreu claim, and the value of the property can be determined by state-space pricing methods. This method is developed by Breeden & Litzenberger [1978] and Banz & Miller [1978]. Mason & Baldwin [1988] also discuss the flexibility of the manager to alter short-term production as a feature of a larger model designed to assess the value of a government subsidy that is designed to encourage a nearly deployment of a project. In effect, the government is distorting the investment incentives of corporations in order to increase current economic activity or secure access to some critical national resource. By analyzing the situation as a real option, the government can determine the minimum subsidy needed to induce early development. The analysis will also provide a valuable measure of the cost of the distortion, so that policy makers can decide whether it was an effective way of developing jobs or achieving other social goals. Cortazar & Schwartz [1993] model a firm as a two-stage production process in which the first stage has a bottleneck or maximum capacity constraint, and the second stage has no capacity constraint. Each stage has a constant unit production cost, and the only source of uncertainty is the unit selling price of output from the second stage, which follows a log-normal process. There is no cost of starting or stopping production. In this situation, production decisions are either all or nothing. If production is optimal at the second stage, all of the available output (including inventory) from the first stage is immediately processed, since there is no production constraint. Otherwise, nothing is processed. Similarly, at the first

686

G. Sick

stage it is optimal to either produce nothing (the lower bound) or at capacity (the upper bound). The second stage provides a speculative opportunity to accumulate inventory, which is a perpetual option to sell a unit of commodity in exchange for the fixed production price, which works like an exercise price. The solution is the same as in Section 3.4 for the perpetual American call option on a log-normal asset. The first stage of production involves the production of these second-stage options. They find that it may be possible to have any of the following three situations: 1. the operation is closed, with no first- or second-stage production, 2. first-stage production occurs with inventory accumulation because no secondstage production occurs, or 3. both first- and second-stage production occurs, at a capacity constrained by the first-stage bottleneck. They find that the model has some interesting properties. First, it is optimal to operate first-stage production at prices below marginal first-stage cost, because of the value of acquiring the speculative inventory option. Similarly, the optimal perpetual-option exercise decision means that the firm should wait until price substantially exceeds marginal second-stage cost before commencing second-stage production. Second, an increase in the risk-free interest rate or in the commodity price volatility 43 increases the value of the firm, and it also induces more firststage output and less second-stage output. With sufficiently high interest rates and volatility, it is optimal to accumulate speculative inventories. Thus, optimal response to external economic variables is complex. 4.2.4. Option to use excess capacity

The traditional approach to assessing the opportunity cost of using excess capacity is the equivalent annual cost (EAC) method. The E A C is the annual rental payment that has the same present value as the capital cost of an asset. Suppose a capital asset is projected to have sufficient capacity to support the current (status quo) operations until year T, and suppose that a new project will use some of that excess capacity to the extent that it is expected that a new capital asset will be needed in year t < T. Then, the E A C approach charges to the project the opportunity cost of EAC for each of years t, t + 1 . . . . . T. By reviewing the rhetoric of the previous explanation of the EAC approach, we can see that the calculation requires a forecast of the starting and ending years of the period of binding capacity arising from the new project. But, these forecasts are subject to revision and the firm does not really have to make a decision until some time in the future. Thus, the E A C method overstates opportunity cost to the extent that it misses the flexibility option in determining years t and T. Moreover, the existing projects could be temporarily shut down, in the sense of the 43 In their model, interest rates and volatilities are nonstochastic. Thus, a change in these variables would have to be a once-and-for-all surprise shock to the economy. In general, both of these variables are stochastic, and a full treatment would require combining the stochastic interest rate model of Ingersoll-Ross in Section 4.1 with a model of stochastic commodity prices and volatility.

Ch. 21. Real Options

687

production options discussed in the previous section. It may be that the capacity is not needed if it is optimal to not operate some of the current status quo projects in some of the future years in which capacity was thought to be binding. Again, the EAC method ignores this flexibility option and tends to overstate opportunity cost. McLaughlin & Taggart [1992] suggest calculating the opportunity cost of using excess capacity with the formula T

Opportunity cost = ~

W~ + I r - I0,

(119)

r=0

where Wr is the value of a European option to produce the status-quo products in year r with the equipment, Ir is the value of the option to invest the capital in T, and I0 is the value of the option to invest the capital immediately. Using this approach, they assess the value of the opportunity cost of using excess capacity in a variety of situations, and find that the EAC method does generally overstate the true opportunity cost by understating the value of flexibility, which mitigates cost. In more general situations, investing in a project now creates and destroys future investment options, and these options all interact with each other as discussed in Trigeorgis [1992]. Thus, the value of the portfolio of these options may not always be a simple sum of option values, as in (119). 4.2.5. Real options and industrial organization While the fundamental risk that creates a real option is typically exogenous, other firms besides those of the analyst may also be responding optimally to this risk. They may be adopting projects that compete with or are synergistic to those of the analyst's firm. The analyst's firm may have market power, so that adopting a project affects the profitability of projects for the other firms. Baldwin [1982], McDonald & Siegel [1986], Dixit & Pindyck [1994] and Trigeorgis [1995] discuss various aspects of these oligopolistic and game-theoretic models.

5. Concluding remarks In this paper we have summarized the basic techniques and ideas of real options analysis, as it extends the capital budgeting literature beyond the discounted cash flow technique. We reviewed the basic techniques in asset pricing, with a focus on the risk-neutral martingale and generalized CAPM approaches. These techniques are based on very broad economic principles, and are likely to stand the test of time. The only debate will concern the fine details of the market pricing factors or determination of an appropriate martingale measure. Little emphasis was placed on arbitrage analysis, because real assets often cannot be traded or sold short, as in financial option analysis. Indeed a real option can derive risk from an economic variable such as an interest rate that does not correspond to an asset price. We have also advocated discounting the certainty-equivalents in real options by a tax-adjusted riskless bond return.

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We have studied real options from both a continuous-time-and-space approach, as well as from a discrete-time-and-space approach. Both approaches are useful and compatible with each other. The continuous-time approach is useful for modelling stochastic processes and for obtaining analytic solutions to option valuation problems in special limiting cases. It is based on a fundamental partial differential equation for the option value, coupled with boundary conditions that describe exercise payoffs amongst other things. An important boundary condition is the high-contact condition that characterizes optimal exercise or development policy. We applied these techniques to present the classic solution to the perpetual real American call option to develop an underlying project that has a log-normally distributed value. We advocate an economic approach to option valuation that involves inferring price and convenience dividend information from futures markets by pricing futures contracts for common stochastic processes for spot prices. We also show how to relate the certainty-equivalents of futures markets to the expected cash flows and risk-adjustment techniques that are commonly used in discounted-cashflow analysis. This imposes an extra consistency requirement that improves the quality of value estimates even for discounted cash-flow analysis. The discrete-time approach is useful for analyzing finite-lived options or when the stochastic process has special characteristics such as mean reversion that make analytic solutions impossible to find. We have shown how to determine discrete processes (with a lattice or tree-like time and space structure) that approximate the economic characteristics of the continuous-time processes. We concluded with a discussion of various real option models. The first model was of interest-rate uncertainty, which shows how the underlying economic variable need not be the price of a traded asset. In this model, the unusual (by classical economic standards) result can obtain in which a reduction in interest rates can deter rather than encourage the adoption of projects. A series of examples from the literature were then discussed of applications of real options analysis in the real estate and mineral industries. Real options can be used to analyze abandonment decisions, production decisions and to determine the opportunity cost of excess capacity. Real options analysis will become a standard tool of capital budgeting analysis over the next one or two decades. Firms that ignore it will be missing out on valuecreating opportunities. Moreover, they will find themselves trading these opportunities through asset restructuring with firms that do know their true value. The knowledgeable firms will use real options analysis to acquire undervalued assets and divest over-valued assets. The other side of these trades will be taken by firms that ignore real options analysis, and which will slowly but surely shrink in value.

Acknowledgements The author gratefully acknowledges the research support of the Social Sciences a n d Humanities Research Council of Canada. Helpful comments have been

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p r o v i d e d by J a m e s B u r n s , and by p a r t i c i p a n t s Connecticut and Baruch College.

in w o r k s h o p s

at U n i v e r s i t y o f

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