JRO_vol_1_2011_p_18-32

Valuing Real Options Projects with Correlated Uncertainties

Luiz E. Brandão1 and James S. Dyer2

1 IAG Business School, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22451-900, Brazil

[email protected] 2 McCombs School of Business, University of Texas at Austin, Austin, Texas 78712, United

States [email protected]

Abstract. Contingent claims are traditionally priced through the use of replicating portfolios, or equivalently, risk neutral valuation. When markets are incomplete, as occurs with many projects and is often the case with claims on real assets when firms are subject to private, project specific risks, these risks cannot be hedged and a replicating portfolio cannot be construed. We proposed a modified approach that enhances the methodology originally developed by Copeland and Antikarov (2003) and provides a practical method for pricing project where management flexibility and correlated risks are present, using the concept of partially complete markets of Smith and Nau (1995).

Keywords: Real Options, Correlated Uncertainties, Decision Analysis

1 Introduction

It is widely recognized that discounted cash flow methods do not adequately value contingent claims, such as options on financial or real assets. The solution to the problem of valuing financial options was pioneered by Black & Scholes [1] and Merton [2] and this approach has been further extended to the valuation of investments in real assets that present managerial flexibility in an approach known as the real options methodology.

Contingent claims are traditionally priced through the use of a market asset or portfolio of marketed assets that replicate the payoffs of the claim in all states and times. Since this replicating portfolio offers the same payoffs and risks as the claim, arbitrage considerations imply that their prices must also be the same. While this analysis is straightforward in the case of complete markets, the markets for real assets are usually incomplete as the number of marketed assets is insufficient to set up the replicating portfolio.

Market risks are due to uncertainties that are market correlated and can be fully hedged by trading in securities. An example is the risk derived from the uncertainty over future oil prices in an oil exploration and development project. For projects

Journal of Real Options 1 (2011) 18-32

ISSN 1799-2737 Open Access: http://www.rojournal.com 18

where a replicating portfolio can be constructed and valued, the market is complete and all project risks can be hedged by shorting this portfolio. On the other hand, even though the basic project uncertainty may be market related, there may not be an active traded market for this particular product or commodity, or there may be project or firm specific uncertainties, such as the uncertainty over the size of the oil reservoir or discrete “lumpy” events that may have a one time effect on the project and are not correlated with the market. In these cases, the project cannot be hedged with market securities and markets are incomplete for projects that bear these types of risks.

When the output of the project is a traded commodity, a standard solution in the real options literature [3-4] is to use this asset to hedge the projects risks and construct the replicating portfolio. Another approach is to treat the project without options as a traded asset, where its present value is assumed to be its true market value, and to use the project to create an underlying portfolio to value options associated with the project [5]. Both approaches address the problem by making assumptions that transform an incomplete market setting into a complete one.

The incomplete market problem can only be addressed directly if we are willing to place restrictive assumptions on the investors’ or managers’ utility functions. Smith and Nau [6] introduce the concept of a partially complete market where the market is complete for market risks, and private events convey no information about future market events. This implies that if m

i and 1p

t are the vectors of all possible market and private states at t and t-1 respectively, then m

i and 1p

t are independent. Under this framework, the market component of the project cash flows is valued using the traditional complete market setting, and the private component may be priced assuming risk neutrality if the investors are sufficiently diversified, or by using a utility function that reflects the investor’s subjective beliefs and preferences otherwise.

There are, however, projects where the distinction between market and private risks is either not so clear, or not a meaningful concept, such as when these two uncertainties are correlated in some way. An example of this is the uncertain change in oil drilling rig rates, which cannot be directly hedged in any existing markets, but nonetheless, are loosely correlated with oil prices.

In this paper we demonstrate how the correlation between market and private risks can be addressed within the framework of a contingent claims valuation when private risks are conditioned to market risk. We solve this problem with a discrete time model based with risk neutral probabilities, and provide a practical computational solution for this approach based on the use of binomial decision trees. This approach is similar to the work of Wang and Dyer [7], who develop a copula based approach for modeling dependent multivariate uncertainties, but differs from the Smith and Nau approach in the sense that Smith and Nau do not explicitly consider the problem of correlation between the market and private uncertainties.

The remainder of the paper is organized as follows: Section 2 introduces the concept decision tree analysis and risk neutral probabilities. Section 3 presents an enhanced approach to project valuation with correlated private uncertainty. In Section 4 we apply the model to solve a sample problem and in Section 5 we conclude with a



summary and discussion of further research issues regarding model formulation and solution procedures.

2 Decision Tree Analysis for Real Option Valuation

Real options derive their value from the managerial flexibility present in a project, which allows firms to affect the uncertain future cash flows of a project in a way that enhances expected returns or reduces expected losses. The flexibility to delay investment in a project, for example, may be viewed as a call option on the project where the required investment is the exercise price. Other typical project flexibilities are the option to switch inputs or outputs or otherwise expand, abandon, suspend, contract or resume operations in response to future uncertainties. Due to the option-like characteristics of management flexibility, discounted cash flow methods cannot be used to capture this value and one must resort to option pricing or decision analysis methods.

Managerial flexibility can be modeled with decision tree analysis (DTA) by incorporating the decision instances that allow the manager to maximize the value of the project conditioned on the information available at that point in time, after several uncertainties may have been resolved. A naïve approach to valuing projects with real options would be simply to include decision nodes corresponding to project options into a decision tree model of the project uncertainties, and to solve the problem using the same risk-adjusted discount rate appropriate for the project without options. Unfortunately, this naïve approach is incorrect because the optimization that occurs at the decision nodes changes the expected future cash flows, and thus, the risk characteristics of the project. As a consequence, the standard deviation of the project cash flows with flexibility is not the same as that of the project without flexibility, and the risk-adjusted discount rate initially determined for the project without options will not be the same for the project with real options. However, real option problems can be solved by DTA with the use of risk neutral probabilities, which implies that we can discount the project cash flows at the risk free rate of return and make any necessary adjustments for risk in the probabilities of each state of nature.

Fig. 1. The Project with Objective Probabilities and a Risk-adjusted Discount Rate

Up state

.50

59.1

Down state

.50

-19.1

Chance Accept 20

Reject 0

Decision 20

Net Payoff59.1 = 120/1.1 - 50

Net Payoff-19.1 = 34/1.1 - 50



We illustrate this concept with an example. Suppose there is a two state project with equal chances of cash flows of $120 or $34 one year from now that has a risk-adjusted discount rate of 10%, and which can be implemented at a cost of $50. The expected present value of the project is $70 = [0.5($120) + 0.5($34)]/1.10 and the NPV is $20, as shown in Fig. 1., where the square represents a decision, the circle represents an uncertainty and the triangles represent path endpoints.

Suppose now that the decision to commit to the project can be deferred until next year, after the true state of nature is revealed, and that the risk free rate is 8%. The original discount rate of 10% cannot be used because the risk of the project has now changed due to the option to defer the investment decision. On the other hand, a set of risk neutral probabilities for the original project can be determined and used to value the project with the deferral option, since the expected cash flows for both problems are the same ($120 and $34).

While the correct risk-adjusted discount rate of a project with options is difficult to determine due to the effect these options have on the project risk, the risk free rate of return can be readily observed in the market. By switching from objective probabilities to risk neutral probabilities, the project NPV with options can then be estimated even without knowing the correct risk-adjusted discount rate. In this example this can be done by solving for the risk neutral probability pr in

$70 ($120) (1 )($34) /1.05r rp p and we obtain pr = 0.4593.

The project with the option to defer has net payoffs of $120-$50=$70 in the up state and zero in the down state as illustrated in Fig. 2. , as there will be no investment if it is known beforehand that the down state will prevail. The net present value of the project with the option to defer is $30.6 = [0.459($66.7) + 0.541($0)] / 1.05, up from $20, which implies that the value of the option to defer is $10.62.

Fig. 2. Project with Risk Neutral Probabilities and Risk Free Discount Rate

The DTA model is based on the idea proposed by Copeland and Antikarov [5], which requires two key assumptions: MAD (Marketed Asset Disclaimer), where the present value of the project assumed to be the best estimate of its market value, and that variations in the project returns follow a random walk. We refer the reader to

Accept 66.7

Reject 0

Decision Up state

.459

66.7

Accept -15.2

Reject 0

Down state

.541

0

Chance 30.6

Net Payoff(120 - 50)/1.05 = 66.7

Net Payoff(34 - 50)/1.05 = -15.2



both Copeland and Antikarov [5] and Brandão and Dyer [8] for a more thorough discussion of these ideas.

While both these assumptions are also subject to a number of caveats, we will adopt this point of view for the purpose of this discussion. Let Vi be the value of a non dividend paying project at time period i and Vi+1/Vi be its return over the time period between i and i+1. Under the random walk assumption, the logarithm of the return

)/ln( 1 ii VV is normally distributed, and we define v and 2 as the mean and variance of this distribution. When the time period length is small, this stochastic model can be expressed as an Arithmetic Brownian Motion (ABM) random walk process in the form dzdtVd ln where dz dt is the standard Wiener process, (0,1)N . Accordingly, changes in Vi will be lognormally distributed, and can be modeled as a Geometric Brownian Motion (GBM) stochastic process in the form

VdzVdtdV where 2 2 . The random walk assumption implies that any number of uncertainties in the model of the project can be combined into one single representative uncertainty, the uncertainty associated with the stochastic process of the project value V, and the parameters of this process can be obtained from a Monte Carlo simulation of the project cash flows.

The value of the underlying project at time i is determined by simply discounting the expected cash flows , = 1, 2, ..., iE C i m at the risk-adjusted discount rate µ,

such that ( )( )m u t i

i iV E C t e dt . If the project pays dividends, then its value will

decrease in each period by the amount of dividends that is paid out. We assume these dividends are equal to the cash flows in each period. The distribution of Vi can be fully defined by the mean and standard deviation of the project returns. Assuming that markets are efficient, purchasing the project at its present value guarantees a zero NPV and the expected return of the project will be exactly the same as its risk-adjusted discount rate. In this sense, the mean return is exogenously defined and is usually set equal to the firm’s WACC. The volatility of the returns can be determined from a Monte Carlo simulation of the stochastic process

lnd V vdt dz where 1 0lnv V V %% and .)~( E The value of the project can

be modeled in time as a GBM stochastic process by means of a discrete recombinant binomial lattice according to the model of Cox, Ross and Rubinstein (CCR) [9]. The pre-dividend value of the project in each period and state is given by

, 0i j j

i jV V u d ,

where tu e and td e are the parameters governing the size of the up and down movements in the lattice. The objective probability of an up movement

occurring is.te d

pu d

, where i = period (i = 0, 1, 2, ..., m) and j = state (j = 0, 1, 2,

..., i). The continuous time stochastic process associated with this dividend-paying project is ( )tdV Vdt Vdz , where δt is the instantaneous dividend distribution

rate at time t. Under uncertainty, the pre-dividend value of the project Vij in period i,



state j, is given by the recursive equation 1

, 01

(1 )i

i j ji j k

k

V V u d

where the

probability Pi,j that the value Vij will occur is , (1 )i j j

i j

iP p p

j

.

The value of the project can be modeled in time as a GBM stochastic process by means of a discrete recombinant binomial lattice according to the CRR model. We choose to model the options as function of the project cash flows rather than on the value, even though both approaches are equivalent. These cash flows, which we will call pseudo cash flows (Ci,j), will themselves be a function of the expected cash flows of the project Ci (i = 1, 2, ..., m), of µ and of the parameters u and d of the binomial mode and can be shown to be defined as:

111,

11, ,

11, 1 ,

11

(1 )2,3,..., 0,1, 2,...

(1 )

j jj

ii j i j

i

ii j i j

i

CC u d i

CC C u

Ci m j i

CC C d

C

(1)

Since we are using risk neutral probabilitiesrt

r

e dp

u d

, these cash flows are

discounted at the risk free rate to arrive at the present value of the project at time t = 0.

3 Correlated uncertainties

Situations where risks are correlated bring an additional level of complexity to the valuation problem. The consolidation of all the market uncertainties that affect the project resulted in the lognormal diffusion process for the project value V, which we defined in the previous steps. We now consider the case where an additional market or private uncertainty is conditional on the consolidated market uncertainties by means of a correlation factor ρ and derive a binomial approximation following the CRR model, but now with conditional probabilities.

Let V and P be the value of the project and of the additional uncertainty at time t, V and P the respective drift rates, σV and σP the volatilities of each process and

and V Pdz dz the standard Wiener processes respectively for V and P. The diffusion

process for these risks is then given by:

where

, (0,1)

V V V V V

P P P P P i

dV Vdt Vdz dz dt

dP Pdt Pdz dz dt N



with correlation , , .V PV P dz dz V PE dz dz dt

Since we will be using risk neutral probabilities to obtain a solution, we must substitute the drift rate for the risk free rate r = r(t), and the process becomes

V VdV rVdt Vdz and P PdP rPdt Pdz . In the binomial mode each asset price

can go up or down at each step, so that we will have four possible states after one time period, each with probability pi, i = 1,2,3,4, as shown in Fig. 3.

V, P

uVV, uPP p1= p(uV,uP)

p2= p(uV,dP)

p3= p(dV,uP)

p4= p(dV,dP)

Probabilities

dVV, dPP

uVV, dPP

dVV, uPP

Fig. 3. States and Probabilities

The size of the up and down movements (uV, uP, dV, dP) and the value of the probabilities pi must be such that the discrete probability distribution converges to the bivariate lognormal distribution when the time period tends to zero. We achieve this by equating the mean, variance and correlation of the binomial and continuous time models, as suggested by Boyle, Evnine and Gibbs [10]. In analogy with CRR, we

make uj dj =1, and j t

ju e , j = V,P. The derivation of the formulas is provided in

the appendix. The conditional probabilities for the uncertainties are:

( | ) (( , ) , ), 1, 2,3,4( )

ki j

j

pP P V j u d i u d k

P V where ( )

r t

u

e dP V

u d

1

( | )4 ( )

V P

V Pu u

u

v vt

P P VP V

1

( | )4 ( )

V P

V Pu d

d

v vt

P P VP V

(2)

1

( | )4 ( )

V P

V Pd u

u

v vt

P P VP V

1

( | )4 ( )

V P

V Pd d

d

v vt

P P VP V



4 Example

We illustrate this approach to the evaluation of real options with a simple four-period project. Due to limitations in the software used to model the problem, the decision tree representation is essentially a binary tree augmented by decision nodes, and it is not recombining like a binary lattice. This results in a large tree due to the unnecessary duplication of nodes, but provides a visual interface and a convenient and flexible modeling tool. We consider initially the case of a project subject to a market uncertainty and next analyze the effects of a private uncertainty on the project value.

4.1 Project subject to market uncertainty

Consider a firm that has just developed a new product and is deciding whether to invest in the manufacturing and marketing of this product. Due to the very competitive nature of this market, the product life is expected to be no more than four years. The spreadsheet with the expected value of the future cash flows and the present value of the project at time zero is shown in Table 1. The risk-adjusted discount rate is assumed to be 10% and the risk free rate is 5%.

0 1 2 3 4

Revenue 1000 1080 1166 1260

Variable Cost (400) (432) (467) (504) Fixed Cost (240) (240) (240) (240)

Depreciation (300) (300) (300) (300)

EBIT 60 108 160 216

Tax Rate (50%) (30) (54) (80) (108) Depreciation 300 300 300 300

Investment (1,200) Cash Flow (1,200) 330 354 380 408

PV0 = 1,157 WACC = 10% Invest = (1,200)

NPV = (43)

Table 1 – Project Expected Cash Flows

The present value of $1,157 is assumed to be the best estimate of the market value of the project and is our base case value. Since the required investment is $1,200, the project has a negative NPV, which indicates that it should not be implemented.

We assume that the project is subject to a single source of market uncertainty, the future value of its revenue stream; although other sources of market uncertainties could be easily incorporated into the model by adding additional uncertainty distributions to the simulation. Suppose the future project revenues R follow a GBM diffusion process with a mean αR = 7.70% (equivalent to a discrete annual growth of 8.0%) and volatility σR = 30%. Using these parameters, a Monte Carlo simulation of the project cash flows may be used to compute the standard deviation of

1 0ln /v V V %% , and to obtain an estimate of the project volatility σ = 24.8%. Finally,



we assume that the project rate of return is normally distributed, so the project value will have a lognormal distribution at any point in time that may be approximated by a binomial lattice.

Modeling the binomial approximation requires that we determine the values of u, d, and the risk neutral probability pr, according to the formulas defined previously. The pseudo cash flows of the project are computed using equations shown in (1), and the value of the project is determined applying the usual procedures of dynamic programming implemented in a binomial tree, and discounting the expected cash flows at the risk free rate of return. Risk neutral probabilities are used to arrive at the project expected value, which is the same as the one calculated with the spreadsheet. Note that the values for , , r and the project expected cash flows Ci can be entered as parameters in a decision tree model, and all the necessary formulae can be incorporated into the tree structure. In effect, tree building can be greatly simplified by developing a standard template for a binary tree for any given number of time periods.

Once the project’s stochastic parameters are determined and the decision tree is structured, the project options can be added with ease. Suppose the project can be abandoned in years two and three for a constant terminal value of $350, and that there exists an option to expand the project by 30% in year 2 at a cost of $100. Given the binary tree representation, these options can be evaluated by simply inserting the appropriate decision nodes in the time period that models the existing managerial flexibility in each year.

The decision tree model is shown in Fig. 4. The project value, computed using the same risk neutral probabilities, increases to $1,280, and the expansion option will be exercised in all states of year 2, except one, while the abandon option will continue to be exercised only in year 3, as can be seen by the lines in bold. Additional options and time periods can be added in a straightforward manner.

4.2 Project subject to correlated uncertainties

Traditional financial theory seeks to obtain market values for assets, and assumes that firms and/or their shareholders are sufficiently diversified so that they become risk neutral in relation to private risks since these risks can be eliminated by an adequate diversification strategy. In this case, the private, firm specific risks must be computed at their expected values and discounted at the risk free rate in order to estimate the market value of a project. On the other hand, for small family and owner operated businesses, for employees that hold large stock investment in the firms they work for or managers that have significant amount of stock options, this may not be the case.



Fig. 4. Decision Tree with Option to Expand and Abandon

To solve this problem we consider the partially complete market concept of Smith and Nau [6] and decompose the project cash flows into their market and private components and value each one separately: option pricing in complete markets for the market risk and use a utility function that reflects the firm’s risk preferences to value the private risk, if we consider an undiversified investor. In our example, the private risk is due to the fact that the project’s cash flows are also affected by the efficiency of the plant, which in turn is correlated with the firm’s manufacturing technology. Since the firm’s technology cannot be hedged in the market, this results in a private risk for which we have no way of determining an appropriate discount rate unless we make restrictive assumptions about the firm’s utility function. We will also assume that the efficiency of the plant is positively correlated with the project value, since a

T4

Continue [2044]

Abandon

302.3

[1688]

Dec 3

High

674.5 .538

[2044]

T4

Continue [1523]

Abandon

302.3

[1424]

Dec 3

Low

410.8 .462

[1523]

T3

Expand

-90.7

[1803]

T3

Continue [1642]

Abandon

317.5

[1119]

Dec 2

High

435.8 .538

[1803]

T4

Continue [1352]

Abandon

302.3

[1254]

Dec 3

High

410.8 .538

[1352]

T4

Continue [1035]

Abandon

302.3

[1093]

Dec 3

Low

250.1 .462

[1093]

T3

Expand

-90.7

[1233]

T3

Continue [1196]

Abandon

317.5

[949]

Dec 2

Low

265.4 .462

[1233]

T2

High

366.1 .538

[1540]

T4

Continue [1209]

Abandon

302.3

[1111]

Dec 3

High

410.8 .538

[1209]

T4

Continue [891.9]

Abandon

302.3

[950.1]

Dec 3

Low

250.1 .462

[950.1]

T3

Expand

-90.7

[1090]

T3

Continue [1053]

Abandon

317.5

[805.8]

Dec 2

High

265.4 .538

[1090]

T3

Expand

-90.7

[801.2]

T4

Continue [764.8]

Abandon

302.3

[879.3]

Dec 3

High

192.4 .538

[879.3]

T4

Continue [616.1]

Abandon

302.3

[804.1]

Dec 3

Low

117.2 .462

[804.1]

T3

Continue [844.6]

Abandon

317.5

[702]

Dec 2

Low

161.6 .462

[844.6]

T2

Low

223 .462

[976.4]

T1

[1280]



greater cash flow stream from higher volumes and prices would generate manufacturing economies of scale and more investment in manufacturing technology.

We assume that the plant operates with a mean efficiency of 80 %, volatility of 0.10 and correlation of ρ = 0.20. The conditional probabilities for the private risk are determined from equations (2) and the private uncertainty is added as a chance node in each period. Fig. 5. shows the model for the base case. If the investor is fully diversified, the we would expect the value of the project to be smaller, since the mean efficiency is less than 100%, and accordingly, solving the decision tree provides a value of $1,058 for the base case, as compared to $1,157.

Fig. 5. Private Uncertainty: Base Case

The project options can be determined by inserting the appropriate decision nodes in each time period in the same manner as before. The decision tree model is shown in Fig. 6. , and the solution to this tree provides the value of $1,181.

Fig. 6. Model with correlated private risk

On the other hand, if the firm or its investors are not sufficiently diversified and this investment represents a significant portion of their wealth, they may be risk averse toward private risks. In this case, assuming an exponential utility function

( ) x RTu x e and a risk tolerance of $200, this level of risk aversion leads to the

High

T2*Priv2/(1+r) 2̂ Low

T2*Priv2/(1+r) 2̂

a

High

Low

Priv2

High

T1*Priv1/(1+r) Low

T1*Priv1/(1+r)

T2

High

Low

Priv1T1

High


T4*Priv4/(1+r) 4̂

High

Low

Priv4

High


T3*Priv3/(1+r) 3̂

T4

High

Low

Priv3

a

T3

Expand

-Invest/(1+r)̂ 2 a

Continue a

Abandon

Abn_Value/(1+r)̂ 2

High

T2*Priv2/(1+r)̂ 2 Low

T2*Priv2/(1+r)̂ 2

Dec 2

High

Low

Priv2

High

T1*Priv1/(1+r) Low

T1*Priv1/(1+r)

T2

High

Low

Priv1T1

High


T4*Priv4/(1+r)̂ 4

High

Low

Priv4

Continue

T4

Abandon

Abn_Value/(1+r)̂ 3

High


T3*Priv3/(1+r)̂ 3

Dec 3

High

Low

Priv3

a

T3



project value decreasing to $1,134. The breakeven point for the risk tolerance factor is $134, below which the optimal project strategy changes. For a further discussion on the use of utility functions for investment valuation see Kasanen & Trigeorgis [11].

5 Conclusions and Recommendations

The method proposed represents a simple and straightforward way of implementing real option valuation techniques using standard decision tree tools easily available in the market. By modeling the correlated uncertainty explicitly, private or otherwise, into the problem, a more accurate estimate of the project value can be obtained that takes into account the possibility the different natures of these uncertainties and their correlation.

Even for a simple model such as this one, the decision tree becomes large very quickly. In most practical problems the complexity of the decision tree will be such that full visualization will be impossible; however, even large problems with literally millions of endpoints for the tree can be solved using this approach. Additional computational efficiencies can be obtained by using specially coded algorithms, although at the cost of having to forgo the simple user interface that decision tree programs offer and the advantage of visual modeling and a logical representation. Suggested extensions include the implementation of recombining lattice capability in current decision tree generating software to cut down on processing time. While a n period recombining binary lattice has a total of n(n+1)/2 nodes, a similar binary tree has 2n+1 -1 nodes, which becomes a significant difference for large values of n. On the other hand, the extension of this model to projects with non-constant volatility can be easily implemented, whereas the effect of changes in volatility cannot be modeled with a recombining lattice.

Perhaps the primary caveat regarding this methodology for the evaluation of projects with real options relates to the assumptions underlying the Copeland and Antikarov [5] approach itself, since the use of decision trees is simply a computational enhancement of their concepts. The use of the Market Asset Disclaimer as the basis for creating a complete market for an asset that is not traded may lead to significant errors, since the valuation is based on assumptions regarding the project value that cannot be tested in the market place. For example, the appropriate choice of the project discount rate for the project without options is left to the discretion of the analyst, and the use of WACC may not be appropriate for all projects. Therefore, it is important to realize that this thorny issue is not resolved by this methodology.



6 References

1. Black, F. and M. Scholes, The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 1973. 81(3): p. 637-654.

2. Merton, R.C., Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 1973(4): p. 141-183.

3. Dixit, A.K. and R.S. Pindyck, Investment under Uncertainty. 1994, Princeton: Princeton University Press. 476.

4. Trigeorgis, L., Real options, Managerial Flexibility and Strategy in Resources Allocation. 1996, Cambridge, Massachussets: MIT Press.

5. Copeland, T. and V. Antikarov, Real Options: A Practitioner’s Guide. 2003, Texere, New York. 368.

6. Smith, J.E. and R.F. Nau, Valuing Risky Projects: Option Pricing Theory and Decision Analysis. Management Science, 1995. 41(5): p. 795-816.

7. Wang, T. and J.S. Dyer, A copula based approach for modeling Dependence in Decision Trees. Forthcoming in Operations Research, 2011.

8. Brandao, L. and J.S. Dyer, Decision analysis and real options: A discrete time approach to real option valuation. Annals of Operations Research, 2005. 135(1): p. 21-39.

9. Cox, J.C., S.A. Ross, and M. Rubinstein, Option pricing: A simplified approach. Journal of Financial Economics, 1979. 7(3): p. 229-263.

10. Boyle, P., J. Evnine, and S. Gibbs, Numerical evaluation of multivariate contingent claims. Review of Financial Studies, 1989. 2(2): p. 241-250.

11. Kasanen, E., & Trigeorgis, L. (1994). A market utility approach to investment valuation. European Journal of Operational Research, 74(2), 294-309.



7 Appendix: Conditional probabilities for a bivariate distribution

To simplify the analysis, we convert to the natural logarithm of the variables, so that xi = ln(Si). It can then be shown through an Ito process that the log of a GBM is an ABM of the form:

ln2

ln , (0,1)2

VV V V

PP P P i i i

dV dx r dt dz

dP dx r dt dz where dz dt N

Discretizing the time steps we have:

V V V V

P P P P

x v t z

x v t z

where 2

2i

iv r

The mean, variance and correlation of the continuous time process are: V V V V VE x E v t z v t

P P P P PE x E v t z v t

2 2V V VVar x E x E x

But since 22 0V VE x v t , we remain with

2 2V V VVar x E x t

2P PVar x t

and finally,

V P V P V PE x x E z z , but V Pt E z z , so we have

V P V PE x x t

The discrete binomial distribution yields: 1 2 3 4( ) ( )V V V V VE x p x p x p x p x

1 2 3 4( ) ( )V V VE x p p x p p x

1 2 3 4( ) ( )P P P P PE x p x p x p x p x

1 2 3 4( ) ( )P P PE x p p x p p x

2 22 2 21 2 3 4V V V V VE x E p x p x p x p x

2 2V VE x x



2 2P PE x x

1 2 3 4V P V P V P V P V PE x x p x x p x x p x x p x x

1 2 3 4( )V P V PE x x p p p p x x

We equate the first and second moments of the discrete distribution to the continuous distribution. Equating also the correlations and making the sum of the probabilities add to one, we arrive at a system of six equations with six unknowns.

1 2 3 4( ) ( )V V Vv t p p x p p x (1.1)

1 2 3 4( ) ( )P P Pv t p p x p p x (1.2) 2 2V Vt x (1.3) 2 2P Pt x (1.4)

1 2 3 4( )V P V Pt p p p p x x (1.5)

1 2 3 4 1p p p p (1.6)

Solution:

V V P Px t and x t

1

11

4V P

V P

v vp t

(1.7)

2

11

4V P

V P

v vp t

(1.8)

3

11

4V P

V P

v vp t

(1.9)

4

11

4V P

V P

v vp t

(1.10)

The conditional probabilities can be obtained from the joint probabilities of the market and private uncertainties.

( )( | ) (( , ) , )

( )i j

i jj

P P VP P V j u d i u d

P V

where ( ) ( , ) , 1, 2,3,4i j kP P V p j u d i u d k



JRO_vol_1_2011_p_18-32

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