JRO Vol 1 2011 Complete

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

Volume 1, Issue 1, 2011

Editor: Mikael Collan

From the editor Mikael Collan

Research Articles

Real Option Valuation of Offshore Petroleum Field Tie-ins, 1-17 Stein-Erik Fleten, Vidar Gunnerud, Øystein Dahl Hem, and Alexander Svendsen

Valuing Real Options Projects with Correlated Uncertainties, 18-32 Luiz E. Brandão and James S. Dyer

Estimating Changing Volatility in Cash Flow Simulation-Based Real Option Valuation with the Regression Sum of Squares Error Method, 33-52

Tero Haahtela

Contact information

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[email protected]

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Journal of Real Options

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24100, Salo, Finland

Submission information

Submissions to the Journal of Real Options are exclusively done through the on-line submission system that can be found at the journal homepage at www.rojournal.com.

Author guidelines with information about paper formatting with templates, and all other information about publication in the journal is available on the journal homepage.

All published articles go through at least double-blind peer review and are required to adhere to a high academic standard of writing and to present new and interesting contributions in the field.

All accepted papers are published on-line and are available for free download.

Editorial Board

Editor-in-Chief, Mikael Collan, Lappeenranta University of Technology, Finland Co-Editor-in-Chief, Christer Carlsson, Åbo Akademi University, Finland Editorial Board Members: Robert Fullér, Eötvös Loránd University, Hungary

Mario Fedrizzi, University of Trento, Italy

Yuri Lawryshyn, University of Toronto, Canada

Richard de Neufville, Massachusets Institute of Technology, United States of America

Gill Eapen, Charles River Associates, United States of America

Frode Kjaerland, University of Nordland, Norway

Markku Heikkilä, Åbo Akademi University, Finland

Peter Majlender, Stockholm University, Sweden

Joszef Mezei, Åbo Akademi University, Finland

Marco Antonio Dias, Petrobras, Brazil

Farhad Hassanzadeh, Sharif University of Technology, Iran

Michael Flanagan, Manchester Metropolitan University, United Kingdom

Makoto Goto, Hokkaido University, Japan

Lauri Frank, University of Jyväskylä, Finland

Scott Mathews, The Boeing Company, United States of America

Luiz Brandão, Pontificia Universidade Catolica, Brazil

Peter Linquiti, George Washington University, Washington DC, USA

Tero Haahtela, Aalto University, Finland

Contents

Volume 1, Issue 1

From the editor Mikael Collan

Research Articles

1-17 Real Option Valuation of Offshore Petroleum Field Tie-ins,

Stein-Erik Fleten, Vidar Gunnerud, Øystein Dahl Hem, and Alexander Svendsen

18-32 Valuing Real Options Projects with Correlated Uncertainties,

Luiz E. Brandão and James S. Dyer

33-52 Estimating Changing Volatility in Cash Flow Simulation-Based Real Option Valuation with the Regression Sum of Squares Error Method,

Tero Haahtela

From the Editor

Mikael Collan

Lappeenranta University of Technology, Business School, PO BOX 20, 53850 Lappeenranta, Finland

[email protected]

Welcome to the first issue of the Journal of Real Options! It has already been a long time since research on real options merits a dedicated journal and now the time has come. In the modern world of academia it is challenging to start a new journal. I have still decided to go against the grain and have set up the Journal of Real Options to be a channel of academic publication for research on real options. These efforts are backed up by the board that composes of many international top names in real options research today.

In this inaugural issue we have three academic research papers: In the first paper Stein-Erik Fleten, Vidar Gunnerud, Øystein Dahl Hem, and

Alexander Svendsen examine the valuation of real options related to offshore petroleum production. The problem is to look at the analysis of investing in adding a smaller tie-in oil field into an already existing larger field.

In the second paper Luiz E. Brandão and James S. Dyer present a

straightforward way of implementing real option valuation using standard decision tree tools, while taking the correlation between private and market risks into consideration. Considering the correlated risk brings an additional level of complexity into the modeling. The paper demonstrates how the correlation can be addressed in a practical way.

In the third paper Tero Haahtela presents a volatility estimation method for

simulation based real option valuation under changing volatility that can also handle negative asset price. The paper discusses how uncertainty decreases as the future is revealed and shows how the method presented is a good example of “rolling” valuation with respect to new more accurate information.

It is my sincere hope that this journal will become a place where researchers on

real options will be able to share their research results and their thoughts about the application and theoretical issues of real options. And remember, the Journal of Real Options welcomes also short case reports, discussion papers, method notes, state-of-the-art articles, and book reviews in addition to high quality academic research articles.

Real Option Valuation of Offshore Petroleum FieldTie-ins

Stein-Erik Fleten1,?, Vidar Gunnerud2, Øystein Dahl Hem1, and Alexander Svendsen1

1 Norwegian University of Technology, Department of Industrial Economics and Technology,Trondheim 7491, Norway

2 Norwegian University of Technology, Department of Engineering Cybernetics, 7491Trondheim, Norway

Abstract. We value two real options related to offshore petroleum production.We consider expansion of an offshore oil field by tying in a satellite field, andthe option of early decommissioning. Even if the satellite field is not profitable todevelop at current oil prices, the option to tie in such satellites can have a signifi-cant value if the oil price increases. Early decommissioning does not have muchvalue for reasonable cost assumptions. Two sources of uncertainty are consid-ered: oil price risk and production uncertainty. The option valuation is based onthe Least-Squares Monte Carlo algorithm.

Keywords: Investment uncertainty, satellite fields, petroleum development, oil fields,energy commodities

1 Introduction

We explore the flexibility related to investment timing in offshore oil exploration andproduction. Offshore oil production can require large investments in infrastructure, off-shore and onshore facilities and well-drilling costs. These costs are to a large degreesunk once the investment has been made. Since 2000, oil prices have been increasinglyvolatile, thereby creating uncertainty about whether marginal projects can deliver a suf-ficient return on the investment.

During the financial turmoil in 2008/2009 the development of several smaller fieldson the Norwegian Continental Shelf were postponed due to uncertainty related to whetherthey could deliver a sufficient return, among others the satellite field Alpha connectedto the Sleipner field. This should make the problem of optimal investment timing in-teresting for practitioners assessing investment opportunities and both government andresearchers forecasting the future level of investment in petroleum production.

The most critical decisions in a petroleum production project with regards to prof-itability is when and if the field should be developed and the largest part of the in-vestment is made. Depending on the field and the technology used to produce it, the

? Corresponding author, [email protected], Phone: +47 73 59 12 96, Fax: +47 7359 10 45

Journal of Real Options 1 (2011) 1-17

ISSN 1799-2737 Open Access: http://www.rojournal.com 1

operator might also have choices available after the field has started to produce. Oftenthere are smaller reservoirs surrounding the main field. Generally too small to warrantan independent production unit, these reservoirs can be developed via the productionunit at the main field. The operator also has to decide when to abandon the field anddecommission the production unit by taking into account future production as well asequipment lifetime and operating cost. Once the field is abandoned, restarting produc-tion is in most cases not realistic.

Real options valuation (ROV) has been applied to petroleum projects for a long timeas they have many attributes that make them suitable for this valuation. These projectsoften involve a large initial investment, the output is a risky and easily traded commod-ity, and management have many choices available related to timing, production tech-nology and size. Siegel, Smith and Paddock [27] assess investing in offshore petroleumleases. Cortazar and Schwartz [7] use a Monte Carlo model to find the optimal timingof investing in a field with a set production rate that declines exponentially and withvarying, but known, operating costs. With this predetermined production rate, the valueof the field becomes a function of the oil price, which is modeled as a two-factor modelwhere the spot price follows a geometric Brownian motion and the convenience yieldfollows a mean-reverting process. Smith and McCardle [20] consider the timing of in-vestment, the option to abandon and to vary the production rate by drilling additionalwells. Both prices and production rates are modeled as stochastic processes, where theprice follows a geometric Brownian motion. Ekern [12] uses a ROV model to valuethe development of satellite fields and adding incremental capacity using a binomiallattice model. He finds that satellite fields that are currently unprofitable can have anoption value. Lund [18] considers an offshore field development by using a case fromthe North Sea field Heidrun. The model used is a dynamic programming model, andtake into account the uncertainty regarding both reservoir size and well rates in addi-tion to the oil price. The paper models the price as a geometric Brownian motion, anduse a binomial valuation model to find the optimal size of the production rig and in-vestment timing. Armstrong et al. [2] uses information from production logging and acopula-based Bayesian updating scheme for real options valuation of oil projects. Diaset al. [11] use Monte Carlo simulations together with non-linear optimization to find anoptimal development strategy for oil fields when considering three mutually exclusivealternatives. Chorn and Shokor [6] combine dynamic programming and real optionsvaluation to value investment opportunities related to petroleum exploration. Dias [10]provides a more thorough review of ROV related to petroleum exploration and produc-tion.

The contribution of our paper is that we consider the decision to add a known(smaller) tie-in field to an existing one, taking into account possible abandonment, pricerisk and technical risk. We use a real options approach where the valuation and optimalexercise is found using the least-squares Monte Carlo (LSM) algorithm presented byLongstaff and Schwartz [17]. We do not consider the problem of initial investment inthe main field, as this problem has been considered both in petroleum production and



other industries before, see e.g. Kort et al. [15]3. Furthermore, the value of deferring aninvestment is generally low in petroleum production [18]. Instead, we focus on deci-sions being made as the field is in production, where we model expanding productionby tying in surrounding satellite fields as well as the option to abandon the field earlyand selling the production equipment.

In Section 2 we present the data used in this work, and discuss its properties. Further,in Section 3 we study the option to expand the production with a tie-in field as well asabandoning the field early. We then apply these models in a case study of two such realoptions. Finally, in Section 4 we conclude and offer suggestions for further work.

2 Data

To estimate the long term behavior of the oil price and to find a suitable time-seriesmodel, we have used the real price of crude oil denominated in 2008 USD from ReutersEcoWin [24]. The series can be seen in Fig. 1 and consist of the US average price inthe years from 1861 to 1944, then Arabian Light posted at Ras Tanura from 1945 to1985, and Brent spot since 1985 to today. It has 148 annual observations going back to1860. We could have used more high-frequency data for the latter years, but these arenot available prior to 1946 for monthly data and 1977 for daily data. To avoid mixingthe different series, only the annual observations have been considered.

To obtain risk-neutral growth and the oil lease rate we use forward prices at time texpiring at time T , Ft,T , from Wall Street Journal [29] for Light Crude Oil, as seen inFig. 2. The series have contracts for each month till December 2014 and semiannualcontracts expiring as late as December 2017. We find an estimated risk-neutral longterm growth of 2.70%. The longest duration for the forward contracts used, T , is eightyears, but we assume that the growth indicated by these, ln F0,T

S0is a good estimate for

the growth in our twenty-year period. We use the expected growth together with anestimate for the risk-free rate, r f to find the oil lease rate, δ , by using (1). This puts theoil lease rate at 1.6%:

δ = r f −1T

lnF0,T

S0(1)

For a market-based estimate for the volatility, we have used implied volatility fromoptions quoted at ICE [14] for Brent oil options. The series have options expiring at 17different dates with several options set to expire at each date. The longest time to expiryis 3 years. We have used an average value over all strike prices available to find a meanimplied volatility for each date. The implied volatility is falling with longer expirationtime, implying that the 3-year forecast might not be valid for the long-term real options.Even so, we use the implied volatility for the 3-year option as the oil price long termvolatility, with an implied volatility of 29.5%. This is quite a lot higher than the historic

3 They study what influences the choice of developing the whole project at once versus devel-oping it in gradual steps.



average for the last 148 years, but close to the volatility in the last 40 years of 28.8%.It is also higher than the 20% that Pindyck [21] found when estimating volatility fromhistorical data. Costa Lima and Suslick [8] refer to Pindyck [21] and also argue that thevolatility has been stable around 20%. We use the implied volatility as an estimate forthe long term volatility. One reason for the difference between the market view and theconclusions of Costa Lima and Suslick [8] and Pindyck [21] could be the increase inoil price volatility in the last years.

To estimate the USD-denominated risk free rate, we have used 20-year US Treasurybonds from [24] as an estimator for the risk free rate. The risk free rate is estimated tobe 4.3%.

Fig. 1: Real price adjusted Brent spot price, USD 2008

Fig. 2: Light crude oil forward prices with increasing time to maturity. Observation date2009–09–11



3 Real Option Valuation

3.1 Flexibilities in Petroleum Production

In this section we consider two cases where the operator has flexibility, and developvaluation models for this flexibility. We include both input (resource) uncertainty andoutput price uncertainty, as in Bobtcheff and Villeneuve [3]. Unlike their analysis, weignore capacity choice issues.

The Value of Including Satellite Fields We assume that the search and explorationphase has been completed; see e.g. Martinelli et al. [19] for a bayesian network analysisof which location to drill a prospect well. In many situations, the operator knows of asmaller and nearby field that can be produced through the main production platform.These smaller fields will often have higher per-barrel costs due to economies of scaleand are more interesting to consider in a real option model than ordinary fields sincethey are not necessarily economical to develop. Typically, such fields will not be largeenough to warrant an independent platform, but it can be profitable to tie the fields to ex-isting platforms. Tying in a small field will increase the produceable reserves connectedto the platform, but will require an investment. The deterministic NPV of tying in sucha satellite field can be calculated by using the reservoir model presented in Sect. 3.2and valuing the incremental production from the satellite, given the capacity constraintsand the time of connection. Given that the increased costs by adding the satellite arefixed, the value of extra production will vary only with the price of oil and the time ofconnection. If the satellite field is connected before the production declines, then it willnot increase the production from the platform until the main field is off its plateau, sincethe plateau is given by the platform’s maximum production rate. Further, if the satelliteis connected near the end of the platform’s life time, much of the extra fields reserveswill be left in the ground unless one extends the lifetime of the platform, which mightnot be possible depending on the availability of infrastructure etc. Developing a satellitefield can require a large initial investment, and it is assumed that any extra operationalcosts are included in the investment cost. Since these are modeled as deterministic cashflows, the NPV of the future costs are simply added to the investment. Thus, the valueof being able to include a satellite field takes the form of a call option to acquire theextra production by paying the investment cost.

The increase in production is the difference between the line and the dotted line inFig. 3. We can calculate the net present value of increased production when connectingthe tie-in at time t by (2):

NPVS,t = ST

∑j=t

Prod∆j e−δ j − I (2)

S represents the price of oil, Prod∆i the extra production from the satellite in period

j, δ the convenience yield and I the present value of the investment and operationalcosts.



The Timing Dimension of Including a Tie-in Field The process of valuing a projectwith a fixed end date is different than for an ordinary stock. Even with uncertainty inthe output, one is certain that the tie-in will be worthless at the time the main platformis decommissioned. In the case study we have used a production profile from Robinson[25] to calibrate the model of Lund [18] in order to get a representative productionprofile.

The Value of Early Shut Down Some offshore production units can be moved if thevalue of the remaining production is low, and the production unit is not near the end ofits life. This can be the case if the true field reserves are lower than estimated. To modelthis, we have used the same price and reservoir model as in the expansion case, but nowit is the whole project value that is relevant. Thus, the value of ending the productionprematurely can be calculated by using (3):

NPVS,t = Kt −ST

∑i=t

(Prod∆i e−δ i −Cie−ri) (3)

This states that the value of decommissioning the field early is the income fromselling the production unit, Kt , less the future expected profit, stated as remaining pro-duction less the operational costs, Ci. It is assumed that it is possible to sell the uniteither to another project or another company for a positive price. We have assumed thatthe unit depreciates linearly and that the income from a sale follows this value, and thatit has a planned lifetime equal to the the field’s lifetime. The strike will take the form:

Kt = K0T − t

T(4)

The Effect of Uncertain Production In Sect. 3.2, we model the production uncertaintyas a mean-reverting process. Unlike a Brownian motion, the expected value of a mean-reverting process at time t is dependent on both its current value and its equilibriumvalue.

E(γ) = α +(γ0 −α)e−λ t , (5)

where γ represents the production level, α the mean index level, and λ the speed ofmean-reversion.

Finding a Suitable Model for the Oil Price One of the most significant factors invaluing a potential oil field is the price of oil. Like the price of other tradeable itemsthe oil price is governed by supply and demand. The theoretical ideal model would takeinto account all the factors that affect supply and demand and produce a forecast ofthe oil price based on this information [21]. Several such models have been developed,among others the Hubbert model of supply [13] and the LOPEC model [23]. Thesemodel the price development by looking at the underlying factors that drive supply andto some extent demand. There are two major obstacles for implementing such a modelfor generating long term forecasts. First, identifying all of the factors affecting the oil



price is in itself a difficult task. Second, producing good forecasts for all of these factorsmight be just as difficult as producing a forecast for the oil price. A time series modelthus seems like an attractive alternative model formulation. Pindyck [21] finds that theoil price can be modeled both as a mean-reverting model and a geometric Brownianmotion. Postali and Picchetti [22] shows that a geometric Brownian motion is a goodapproximation for the oil price movements in the long run. We model the oil price as ageometric Brownian motion because we are interested in the long-term behavior. Thegeometric Brownian motion is described by (6).

dPP

= αdT +σdZ (6)

3.2 Reservoir Model

The field production profile is useful when valuing real options, since it provides infor-mation on volume and time of production. A realistic model of reservoir performanceis challenging to create and to calculate, because of the need to model many parametersin a 3D-setting with many non-linear relations. In this work, a simple zero-dimensionalmodel of Wallace et al. [30] is used. This models the reservoir as a tank with a uniformfluid and with uniform properties in the whole reservoir. Thus, it does not account fordifferences in permeability in different areas or local differences in pressure caused bythe well flow as the areas surrounding the producing wells empties. It is, however, asimple model that has great computational advantages compared to a more complexreservoir model, and it does reflect the form of reservoir production profiles of severaltypes of petroleum fields [18].

Table 1: Reservoir parametersPw,0 - Initial reservoir pressurePw,t - Reservoir pressure at time tPmin - Abandonment pressureR0 - Initial reservoir volumeRt - Reservoir volume at time tqr,t - Maximum reservoir depletion rate at time tqw - Maximum well rateqmax - Maximum capacity, or plateau productionqramp−up,t - Maximum production during field developmentNt - Number of wells producing at time t

The reservoir pressure follows the following relation:

Pw,t = Pw,0 −R0 −Rt

R0(Pw,0 −Pmin) (7)

The reservoir pressure provides the maximum well flow, which decays exponen-tially with time with continuous production if there are no other constraints on the well



flow. The maximum well rate is based on the capacity of the wells installed.

qr,t = NtqwγtPw,t −Pmin

Pw,0 −Pmin(8)

Together, (7) and (8) becomes the simple equation

qr,t = NtqwRtγt

R0(9)

This is the maximum production from the field, given that there is no water injec-tion or other types of pressure maintenance performed. It is rarely optimal to constructthe production unit so that it can produce at the maximum rate qr,t , because of highinvestment costs. When the field has a maximum processing capacity that is lower thanthe field maximum production, the production profile will have a flat region where theproduction is equal to the capacity maximum. This level is called the plateau produc-tion. The optimal plateau level is mainly a function of investment cost, production andrequired rate of return, since it is a trade off between investment cost and the ability toget the oil quickly out of the ground. There might also be technical reasons to limit thecapacity. We have included a ramp-up period of three years, which is similar to the casefound in Robinson [25]. During this ramp-up period we have assumed that the produc-tion grows linearly to capacity maximum over the three year period. The backgroundfor such a ramp-up period is among other topics well drilling. It will not be possibleto drill all wells at the same time, and connecting the streams to the platform will alsorequire some time. The actual production thus becomes the minimum of qr,t , qmax andqramp−up,t .

Production Profile with a Tie-in Field To model the increase in production by a tie-insatellite field, the new reserves, Rnew are added to the initial reserves. This increasesboth the initial reserves, R0 and the reserves at the connection time, Rt . The effect ofthis increase is dependent on when the new field is built. If the satellite is connectedbefore the field goes into decline, then the plateau production will be maintained longeras seen in Fig. 3a.

Uncertainty in Production Production volumes are often uncertain as wells can pro-duce more or less than planned. Lund [18] models this by a changing well capacity. Thewell capacity is modeled as a simple stochastic function, where the well can either havea high or a low well rate. The probability of one of the wells changing regime from ahigh rate to a low or opposite is 0.1 per period of 6 months. Each well capacity will behighly random, but with a large number of wells the process resemble a mean-revertingstochastic process. The variance of the field production will be very dependent on thenumber of wells connected to the field. McCardle and Smith [20] take a different ap-proach by modeling the decline rate as a geometric Brownian motion. This might beappropriate when the field is in decline, but it does not take into account the effect ofthe production capacity limit and it does not clarify which fundamental property that



(a) t=5 (b) t=10

Fig. 3: Production profiles with tie-in at t=5 and t=10

varies. We consider changing well rates as the main source of uncertainty, as in the theswitching model in Lund [18]. We do not model each well individually, however, in-stead we consider the whole field production by assuming a number of wells. This isimplemented as a production factor for the whole field, γt , as a mean-reverting process.We believe that this aggregate production factor is more versatile than the model ofLund [18], as operators can create historic production factors from current and previousfields and easily take into account other risk factors like technology development orunscheduled maintenance. The production factor follows:

γt = γt−∆T +λ (α − γt−∆T )dT +σdZ (10)

where γt is the well production factor at time t, and λ , α and σ are mean rever-sion parameters from the regression. The parameter values can be seen in Table 2. Theparameter values are found by Monte Carlo simulations from the model used by Lund[18], and regressing the simulation results to find a mean-reverting model.

Table 2: Production factor mean reversion parametersParameter Value

α 0.665λ 0.218σ 0.050

3.3 Valuation Framework

There are mainly two ways of calculating the present value of future cash flows. Onesolution is using risk-adjusted rates of return and real expected growth rates. The other



is risk-neutral pricing.

The rate of return used in the valuation of real options have a significant influenceon both optimal exercise policy and option value. Especially with long term valuations,like many real options, a slight change in the rate of return can make a substantialdifference due to the compounding effect. Using an appropriate discount rate is thusimportant to obtain correct results.

Another procedure of obtaining a valuation is to price the cash flows using othersecurities with similar risk profiles that are traded in the market. By replacing the realprice growth with the risk-neutral price growth obtained from traded forward-contracts,one can use the risk-free rate to obtain the value of the project and connected options.This treats risk in a consistent manner compared to the market, avoiding biases that canoccur otherwise (Laughton [16]). This is commonly called risk-neutral valuation. Sinceall parameters are estimated from financial markets, which are assumed to be efficient,this leads to an accurate valuation of the project.

Using risk-adjusted rates has the advantage of being familiar to decision-makersin most firms today, and is perhaps the most intuitive of the two approaches. We dohowever choose to use risk-neutral pricing, since this ties the valuation of the riskycash flows directly to observed prices of this risk. The risk-neutral method is also themost common approach when valuing options. One issue with using risk-neutral pric-ing is that the risk-neutral method can underestimate capital costs when risk of defaultis present (Almeida and Philippon [1]). This can lead to inaccurate valuations when thecost of distress is high. This was the case during the financial crisis in 2008/2009, whenthe risk-free rates went down but the cost of capital for firms increased. Thus, the riskneutral valuation would advice firms to invest more in a time where firms’ capital costsincreased, which is clearly the wrong advice. However, in more stable conditions thedistortions related to the risk of default should be low, specially when considering largepetroleum companies.

3.4 Case Study

For valuing finite-maturity American call options one must use numerical methods.Common approaches include lattice methods, a la Cox et al. [9], or finite differencemethods, see Brennan and Schwartz [4]. However, these methods are cumbersomewhen there are multiple and possibly heterogeneous sources of uncertainty. In such sit-uations, approaches based on Monte Carlo simulation come to the fore; see [5,28,17].

Input Data In this section, we use the model developed in previous sections to valuetwo real options connected to an offshore oil project with the Least Squares MonteCarlo algorithm developed in Longstaff and Schwartz [17]. First and second degreemonomials of the forward price of the underlying asset as presented in (2) and (3) areused as regressors in the LSM calculation. We use risk neutral pricing.



Table 3: Financial parametersS0 - Current oil price - USD 60r f - Risk free rate of return - 4.3%δ - Lease rate - 1.6%σ - Annual volatility oil price - 29.5%KE - Expansion option strike/Total cost tie-in field - MUSD 600KA - Early decommissioning option strike/Initial production unit sales price - MUSD 500

Table 4: Reservoir parametersR0 - Initial reservoir reserves - 300 MMbblRtie−in - Initial tie-in reserves - 15 MMbblqw - Maximum well production - 66 MMbbl/yrqp - Platform production capacity - 33.17 MMbbl/yrT - Field life time - 20 YearsITot - Total Investments - MUSD 2,228TRamp−up - Production Ramp-up time - 3 Years

Expansion Option The option to invest in a tie-in field takes the form of a call option,as discussed in Sect. 3.1. To acquire this option the operator might have to invest in extradeck-space or other forms of extra capacity today, denoted Ctie−in. This will be the costof obtaining the real option, and should not be confused with KE which is the investmentneeded when the tie-in is connected. Using the input data in the previous section andtaking the price growth into account, we find that the maximum static NPV is obtainedat T = 8 which is the last year of plateau production. However, after deducting invest-ment costs the NPV is MUSD −176 at the optimal investment time discounted back tot = 0. In a deterministic setting, it does not pay off to produce the satellite and basedon this the operator should not invest in excess capacity in order to have the opportunity.

When we add price uncertainty the answer changes. By valuing the investment op-portunity as an American call option on the incremental production, the option to investis estimated to be worth MUSD 150. This implies that if the investment needed today,Ctie−in, is less than MUSD 150, the operator should invest in order to have the option.This helps explain why operators frequently invest in extra capacity, since having theopportunity of producing nearby satellite fields creates valuable real options.

Adding further uncertainty by introducing uncertainty in production, the optionvalue is still in the same range as before with an option value of MUSD 161. The lowercontribution is not surprising, as the variation in production is lower compared to pricevariation and the production follows a mean-reverting process rather than a Brownianmotion.

Sensitivity Analysis As we can see from Fig. 4a, the option value increase with in-creasing initial oil price. Unlike a static NPV calculation the option value increasesnonlinearly with low initial oil prices, but the growth becomes linear at higher prices.



Table 5: Monte Carlo parametersN - Number of realizations - 100 000M - Number of time points - 100

This is natural, as the tie-in is almost certain to be developed at high prices, and theextra value from the option is low. In this case, the option value is almost equal to astatic NPV. However, unlike the static NPV the option value is never negative. Becausethe operator has the choice but not the obligation to develop the tie-in, it will never bedeveloped if it has a negative NPV.

Another important variable is oil price volatility, and the option sensitivity to thisvariable can be seen in Fig. 4b. The option does not have any significant value forvolatilities below 5% per year, and this confirms the conclusion that the project wouldnot have positive NPV in a static valuation method. That the value of a project shouldincrease with larger volatility is contrary to common intuition. The crucial differencebetween real option valuation and a discounted cash flow approach is that the projectowner has the option to not exercise the option. Thus the owner is protected from thecase where the price falls, since the satellite field will not be developed in this case.High volatility increases the value because it increases the probability of a very highpayoff, without increasing the probability of a large loss. However, higher volatilitywill increase the optimal exercise price and delay the investment time as seen in Fig.5b. This is because one needs to have a price high above the break-even price to becertain that the price will not drop to a level where the project has a negative NPV whenthe volatility is high. Also, we observe that the volatility has less effect on the optionvalue than the initial oil price.

(a) Initial oil price (b) Oil price volatility

Fig. 4: Expansion option value sensitivities



Another important output from a ROV is the optimal oil trigger price that triggersthe investment. For the option to develop the satellite field, the development of thetrigger price can be seen in Fig. 5. The trigger price is defined as the smallest price thattriggers investment in the LSM-algorithm. As expected, the trigger price increases withincreasing volatility and with decreasing satellite size.

(a) Volatility (b) Satellite size

Fig. 5: Trigger price sensitivities

Early Decommissioning Option The opportunity of decommissioning the field pre-maturely could be a response to lower production volume than expected, or very low oilprices. The operational costs of an oil project are often low compared to the investmentcost, and the value of being able to prematurely abandon the field is believed to be low.

When disregarding uncertainty in reservoir reserves, making price risk the onlysource of uncertainty, the option value is MUSD 4.4. Adding uncertainty in the reser-voir reserves, we obtain an option value of MUSD 4.5. We conclude that the option ofabandoning the field prematurely is not very valuable, and that the flexibility relatedto being able to sell the production unit can be disregarded when choosing productiontechnology.

Sensitivity Analysis Since the decommissioning option is similar to a put option, weexpect the option value to decrease with rising oil prices. This is also the case, as can beseen in Fig. 6a. Unlike a regular put, the option is worth more than the strike price as theoil price approaches zero. This is because as the project is abandoned the operator alsoavoids the operating costs. The option value of abandoning is high when the oil price islow, but since the project as a whole will have a negative NPV it will not be built in thefirst place. Also, we have assumed that the value of the production unit is deterministic.



A more realistic assumption would be that the sales price is positively correlated withthe oil price, as few new projects will be initiated if the price is low. This will furtherreduce the value of early decommissioning. For initial prices close to todays price theoption value is negligible compared to the investment. The option value is sensitive tothe price volatility, as seen in Fig. 6b. If the price volatility should continue to increasein the future, decommissioning options could become valuable.

(a) Initial oil price (b) Oil price volatility

Fig. 6: Abandonment option value sensitivities

When considering the trigger prices, we find that the oil price will have to fall below40 USD per barrel if early decommissioning is to be considered. Compared to historicaloil prices this is not an unrealistic situation. Early exercise is however most likely at theend of the production unit’s lifetime when the expected sales price is low. We also notethat the oil price volatility does not have a large impact on the exercise trigger price.

Fig. 7: Abandonment option trigger price



4 Conclusion

In this paper we study the flexibility related to investment timing in offshore oil explo-ration and production. The oil price is the main source of risk that influence the value ofreal options related to the project. It is shown that the option to abandon by moving theproduction unit is not significant compared to the cost of developing the field. The op-tion to expand the production by adding new fields adds value and the value of makinginitial investments in order to be able to connect such satellite fields in the future canbe large even when the current NPV from the satellite fields are negative. As expected,both options increase in value when faced with increased volatility.

For further work, exploring if other price models, e.g. the two-factor model pre-sented by Schwartz and Smith [26], leads to different option valuations would be aninteresting extension. Another extension related to the option value framework wouldbe to introduce a stochastic process governing when and if a tie-in field is found. Thiswould be more general than our assumption that the operator knows from the start ifthere is a nearby field.

Acknowledgements

We would like to thank Afzal Siddiqui for comments, and Marta Dueas Diez of Rep-sol YPF for her advice related to the issues in petroleum production. We acknowledgethe Centre for Sustainable Energy Studies at the Norwegian University of Science andTechnology (NTNU), and are grateful for support from the Center for Integrated Oper-ations in the Petroleum Industry at NTNU, and from the Research Council of Norwaythrough project 199908. All errors are solely our responsibility.



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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



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



Estimating Changing Volatility in Cash Flow Simulation-Based Real Option Valuation with the

Regression Sum of Squares Error Method

Tero Haahtela

Aalto University, School of Science and Technology P.O. Box 15500, FI-00076 Aalto, Finland

[email protected]

Abstract. This paper presents a volatility estimation method for cash flow simulation based real option valuation with changing volatility. During cash flow simulation, the present values of future cash flows and their corresponding cash flow state variable values are recorded for all time periods. Then, for each time period, regression analysis is used to relate the present value to the cash flow state variables of the same time period. Each regression equation provides an estimate of the expected present value as a function conditioned on the resolution of all uncertainties up to that time. Then, basic regression statistics of Pearson’s correlation R2 and the sum of squares error (or standard error) for each equation provides all the information required for estimating how the standard deviation and volatility of the stochastic process change over time. The method requires only one pass of simulation runs, allows negative underlying asset values, and is easy for a practitioner to apply. Keywords: Real options, cash flow simulation, volatility estimation

1 Introduction

Real options analysis is a framework for valuing managerial flexibility under uncertainty. It has adopted advanced methods from financial derivatives valuation and made valuation of projects with several sequential and parallel decision alternatives more accessible. Difficulty in volatility estimation has been one of the reasons for the slow acceptance of the new valuation framework [1, 2]. Volatility is probably the most difficult input parameter to estimate in real options analysis [1], and this is also the case with financial options. However, volatility estimation in the case of financial options is easier because of past observable historical data and future price information. With real options, especially if related to R&D, such information is not necessarily available [3, 4]. Therefore, volatility estimation has to be based on some other method.

One alternative is to use Monte Carlo simulation for the gross present value and volatility estimation [1, 2, 5-15]. In this approach, forecast data for future cash values with probabilities is converted into an estimated underlying asset value and volatility.


ISSN 1799-2737 Open Access http://www.rojournal.com 33

The cash flow model is usually a gross present value (i.e. a conventional net present value of the completed project's expected operating revenues less investments) where uncertainties related to parameters are presented as objective or subjective distributions with different correlations, time series and other constraints. After the simulation, the mean and the standard deviation of the rate of return, i.e. volatility, are calculated.

Monte Carlo simulation on cash flow calculation consolidates a high-dimensional stochastic process of several correlated variables into a low-dimension process. The most common assumption is the univariate geometric Brownian motion (gBm) process. Firstly, the Monte Carlo simulation technique is applied to develop a probability distribution for the rate of return. Then, volatility parameter (or other parameters for stochastic processes other than gBm) of the underlying asset can be estimated with alternative approaches, of which the most common is presumably calculating the standard deviation of the simulated probability distribution for the rate of return.

Several authors have suggested different approaches for applying Monte Carlo simulation on cash flow calculation to estimate the volatility. The existing cash flow simulation-based volatility estimation methods are the logarithmic present value approach of Copeland and Antikarov [6] and Herath and Park [7], the conditional logarithmic present value approach of Brandão, Dyer and Hahn [8, 9], the two-level simulation and the least-squares regression methods of Godinho [10], and the generalized risk-neutral volatility estimation over different time periods [2, 13-15]. All these methods are based on the same basic idea. The Monte Carlo simulation technique is applied to develop a probability distribution for the rate of return. Then, the volatility parameter of the underlying asset is estimated by calculating the standard deviation of the rate of return.

The previously mentioned methods have different strengths and qualities. Different aspects related to them are theoretical correctness, computational efficiency, capability to handle negative underlying asset values, capability to estimate changing volatilities, ability to separate ambiguity from volatility, and ease of use from the intuitive aspects of understanding the logic behind the volatility estimation as well as ease of use in terms of available software to do the estimation. Luckily, not all of these aspects are relevant in all practical valuation cases. Therefore, usability of the methods depends heavily on the case in which they are applied.

The approach presented in this paper can be considered a mixture of the previously mentioned methods. The purpose is to sustain and combine many of their good qualities while keeping the procedure as straightforward as possible for a practitioner. In the method presented in this paper, the present value of the cash flows and the cash flow state variable values are recorded for each time period during cash flow simulation runs. Then a regression analysis is run for relating the PV for each year to the corresponding cash flow state variables. Each regression equation provides an estimate of the expected present value as a function conditioned on the resolution of all uncertainties up to that time. Then, the basic regression statistics of Pearson’s correlation R2 and sum of squares error (or standard error) for each equation provide all the information required for estimating how the standard deviation and volatility



change over time. The method is computationally efficient as it requires only a single pass of simulation runs, it is easy to understand and use, and it can handle changing volatility and negative underlying asset values.

After introductory Section 1, Section 2 discusses in detail earlier cash flow simulation based methods, because the procedure presented in this paper is based heavily on their ideas and results. Section 3 presents the intuition behind the suggested method, the idea of the proportion of unsolved and solved uncertainty during the project’s timespan. The main contribution of this paper is in Section 4 which describes the regression sum of squares error and R2 based method for estimating changing volatility in cash flow simulation-based real options valuation. Section 5 illustrates the use of this method in a case example. Finally, Section 6 concludes the paper.

2 Cash Flow Simulation Based Volatility Estimation Methods

2.1 Logarithmic Present Value Approach of Copeland and Antikarov

Copeland & Antikarov [6] (henceforth C&A) presented the first detailed explanation of the use of Monte Carlo simulation for volatility estimation based on cash flows. This logarithmic present value approach in terms of Mun [1, 16] is based on several assumptions. Firstly, it relies on the marketed asset disclaimer and Samuelson’s proof [17] that the correctly estimated rate of return of any asset follows random walk regardless of the pattern of the cash flows. Secondly, the approach is based on the idea that an investment with real options should be valued as if it were a traded asset in markets even though it would not be publicly listed. Thirdly, the present value of the cash flows of the project without flexibility is the best unbiased estimate of the market value of the project were it a traded asset. This is called the marketed asset disclaimer assumption. Therefore, the simulation of cash flows should provide a reliable estimate of the investment’s volatility.

The idea of C&A’s logarithmic present value approach is – similarly to most other consolidated volatility approaches – analogous to stock price simulations where the theoretical stock price is the sum of all future dividend cash payments, and with real options, these cash payments are the free cash flows. The sum of free cash flows’ present value PV0 at time zero is the current stock price (asset value), and at time one, the stock price PV1 in the future. As the stock price at time zero is known while the future stock price is uncertain, only the uncertain future stock price is simulated. As a result, the C&A approach uses the Monte Carlo simulation on a project’s present value to develop a hypothetical distribution of one period returns. On each simulation trial run, the value of the future cash flows is estimated at two time periods, one for the present time and another for the first time period. The cash flows are discounted and summed to the time 0 and 1, and the following logarithmic ratio is calculated according to Equation (1):



z ln (1)

where PV1 means present value at time t = 1, FCF1 means free cash flow at time 1, and PV0 project’s present value at the beginning of the project at time t = 0. Then, volatility is calculated as the standard deviation of z.

Modifications to this method include duplicating the cash flows and simulating only the numerator cash flows while keeping the denominator value constant. This reduces measurement risks of auto-correlated cash flows and negative cash flows [1]. Whereas simulating logarithmic cash flows gives a distribution of volatilities, and therefore also a distribution of different real options values, this alternative gives a single-point estimate.

Although the fundamental idea of C&A’s volatility estimation approach is correct, it has one significant technical deficiency, as noted by Smith [18] and Godinho [10]. The method would be an appropriate volatility estimate if the PV1 were period 1’s expected NPV of subsequent cash flows and this volatility would reflect the resolution of a single year’s uncertainty and its impact on expectations for future cash flows. However, in C&A’s solution, this PV1 is the NPV of a particular realization of future cash flows that is generated in the simulation, and therefore the calculated standard deviation is the outcome of all future. Therefore, the approach overestimates the volatility [10, 18].

2.2 Logarithmic Present Value Approach of Herath and Park

Herath and Park’s [7] volatility estimation is very similar to C&A’s and is based on the same Equation (1). However, whereas in C&A only the numerator is simulated and the denominator is kept constant, Herath and Park [6] apply simulation to both the numerator and denominator with different independent random variables: “…both PV0 and PV1 are independent random variables. Therefore, a different set of random number sequences has to be generated when calculating PV0 and PV1”. However, this alternative has the same over-estimation problem as the original C&A, and by having a non-constant denominator, the approach actually worsens the situation and over estimation of the volatility.

2.3 Conditional Volatility Estimation of Brandão, Dyer and Hahn

Other authors have resolved the original problem of C&A’s approach. The conditional volatility estimation of Brandão, Dyer, and Hahn [8, 9], based on comments of Smith [19], and similarly to Godinho [10], suggest an alternative where the simulation model is changed so that only the first year’s cash flow FCF1 is stochastic, and the following cash flows FCF2,…,FCFn are specified as expected values conditional on the outcomes of FCF1. Thus, the only variability captured in PV1 is due to the uncertainty resolved up to that point. The method works well if the conditional future values are straightforward to calculate or estimate. Then, the



standard deviation of the following Equation (2) is used to estimate the volatility of the rate of return:

,…, |

(2)

The deficiency with the method is that it may be hard to compute the expected

future values given the values simulated in earlier periods. This is true especially for both auto- and cross-correlated input variables in cash flow simulations.

2.4 Two-level Simulation of Godinho

The two-level simulation of Godinho [10] is also based on the idea of conditionality in expected cash flows given stochastic C1. In comparison with the conditional volatility estimation, it works also in situations where conditional outcomes given C1 cannot be calculated analytically. Firstly, the simulation is done for the project behavior in the first year. Secondly, project behavior given the first year information is simulated for the rest of the project life cycle. Thirdly, average cash flows after the first year are used to calculate PV1, which is then used to calculate a sample of z. Finally, the volatility of z (standard deviation) is calculated. The method provides correct estimation of volatility, but it suffers mostly from the required computation time. This is because the calculation is iterative, meaning that after each first year simulation, a large number of second-stage simulations is required. Therefore, the total number of simulations is the product of first and second stage simulation runs. In practice, whereas other methods compute the volatility within a few seconds even with larger models, this procedure requires at least several minutes of computation time with present computers and algorithms.

2.5 Least Squares Regression Method of Godinho

Inspired by Longstaff and Schwartz [19]1, Godinho [10] presents the least squares regression method for volatility estimation. This procedure consists of two simulations. In the first simulation, the behavior of the project value and the first year information is simulated. Then, PV1 is explained with linear regression with first year information as follows according to Equation (3):

⋯ (3)

Then, in the second simulation round, volatility is calculated as the standard

deviation of z:

1 Who got their idea from Carriere [20]



(4)

Often a good and straightforward approximation is to use the first year’s free cash flow FCF1 as the explaining variable with intercept term. Then, in the second simulation round, only the first year cash flow is simulated, and the estimation model is used to calculate the expected value of PV1 for calculating the sample z. Similarly to conditional volatility estimation [9] and two-level simulation [10], this method also estimates volatility correctly. This least-squares volatility estimation approach is computationally fast with only two simulation rounds required and it is also much easier to apply in practice than conditional volatility estimation.

2.6 Generalized Risk-neutral Valuation Approach

Another straightforward volatility estimation approach is based directly on the assumptions and qualities of the geometric Brownian motion and its lognormal underlying asset value distribution. Smith [18] suggests that correct parameterization for the mean value and volatility could be found by changing the volatility until the underlying asset mean and standard deviation match the simulated cash flow and its standard deviation. However, if common gBm assumptions hold, this can actually be solved analytically [2, 13-15]. Given that PV0 is known, and it is possible to simulate future cash flows, distribution of the cash flows in future can be simulated. As well as discounting all the cash flows to the present value, they can also be undiscounted to their future value. Because the present value of cash flows is known (PV0), as well as their simulated undiscounted future value with standard deviation, it is possible to find the volatility parameter analytically without any unnecessary additional computations and simulations. It is known that in a risk-neutral world, asset value increases with time according to Equation (5), and that the standard deviation of the process increases according to Equation (6).

(5)

1 1 (6)

Therefore, if the standard deviations of the underlying asset process at certain time

points are known, it is possible to compute the average volatility for each time period. Starting from the beginning of the process, each i can be calculated according to Equation (7) as

∑

(7)



The information required is the length of time period T for volatility estimation, value of asset S0 at the beginning, interest rate , and standard deviation of the asset value std(S) at the end of the volatility estimation period. Interest rate does not change the volatility estimation as long as the same interest rate is used both in cash flow simulation and when computing the volatility. Therefore, even if the risky expected return is used in volatility estimation, the option valuation still follows risk-neutral pricing with risk-free interest rate used.

2.7 Least Squares Regression Method with the Risk-neutral Approach of Haahtela

Haahtela [14] further extends the idea of Godinho [10] and properties of generalized risk-neutral valuation for estimating changing volatilities for different time periods2. The approach uses ordinary least squares regression for estimating PVt with cash flow simulation state variables as explaining input parameters. However, instead of the common approach of directly estimating volatility as the standard deviation of the rate of return as z = ln(St+1/St), PV regression or response surface estimators are used directly for estimating both the underlying asset value and its standard deviations at different time points. Figure 1 illustrates this. Based on this information, volatilities are estimated for different time periods. After cash flow simulation and constructing the regression equations, the actual procedure for the standard deviation calculation is presented as follows:

Calculate the estimated expected value for from the cash flow parameters Xi,k generated in the simulation trial run using the regression equation.

Calculate the differences between values predicted by the regression estimator and the values actually observed from the realized simulations.

a) Square the differences, b) add them together, c) divide them by the number of observation, and d) take the square root.

After that, volatility for different time periods is calculated period by period

according to Equation (7). While the method provides sound volatility estimation, it is not very usable or comfortable in practice. If there are changes in any cash flow simulation parameters, the whole volatility estimation procedure has to be repeated again for all the time periods. The advantage of the approach is that it also allows negative underlying asset values and use of the displaced diffusion process instead of the commonly assumed geometric Brownian motion.

2 Volatility estimation is not the main topic of the paper, it is only presented as an example of

that it is possible to estimate changing volatilities with regression equations based on simulated cash flows



Fig. 1. Response surface equations are used to estimate underlying asset value for different time periods. This information can be used to estimate volatility according to the generalized

risk-neutral valuation approach assuming geometric Brownian motion.

3 Intuition Behind the Regression Sum of Squares Error Method: Proportion of Solved and Unsolved Uncertainty

Before going to the actual regression solution procedure, it is good to illustrate the idea behind the calculation. Usually, uncertainty reveals itself over time and we have a better understanding and knowledge about the situation and the investment viability of a project during its later stage. Also, proportional uncertainty is likely to decline over time, especially in the case of R&D and learning. Therefore, volatility is often time changing and declining. Finally, all the uncertainty has been revealed and there is no uncertainty and volatility left anymore.



Fig. 2. Unsolved uncertainty and solved uncertainty during investment. Usually uncertainty resolves faster in the early stages of a project.

Figure 2 illustrates this situation, and the same is demonstrated in the forth-coming example. In the beginning, we estimate the uncertainty of the investment project (in terms of variance) to be 12 000. During each time period of the process, more and more of the uncertainty has been revealed. At the end, all the uncertainty has been revealed. Now, if we were able to construct good forward-looking value estimators PV1…PVn for each time period, we could estimate the overall uncertainty solved (and unsolved) for each time point, and this information could then be used to calculate the volatility for each time period according to Equation (7).

For simplicity, we assume in this example use of arithmetic Brownian motion, and the interest rate is assumed to be zero. Arithmetic Brownian motion is normally distributed with mean and variance 2. The sum of normally distributed variables is

∑ , , ⋯ , ⋯ , ⋯ (8)

The standard deviation of the arithmetic Brownian motion is:

Σ (9)

The example has only four annual cash flows, each with an expected value of 100.

These cash flows are all normally distributed with an expected value of 100 and the standard deviation for time periods 1…4 are 80, 60, 40, and 20. Therefore, based on the properties of normal distribution, the sum of these cash flows is 400 and the variance is 12 000, which is the same as our present value of cash flows because the interest rate is assumed to be zero.



Table 1: Example calculation of how uncertainty reveals itself over time.

Cash flows, variance and resolving of uncertainty during time periods

Time period 1 2 3 4

Cash flow N(100; 802) N(100; 602) N(100; 402) N(100; 202) Variance (variation, uncertainty) of the time period

6 400 3 600 1 600 400

Solved (explained) uncertainty after time period

6 400 10 000 11 600 12 000

Unsolved (unexplained) uncertainty after time period

5 600 2 000 400 0

Proportion of total variance of 12 000

0.533 0.300 0.133 0.033

Cumulative proportion of solved uncertainty 0.533 0.833 0.967 1.000

Proportion of unsolved uncertainty 0.467 0.167 0.033 0.000

Volatility of the time period 20% (80/400)

15% (60/400)

10% (40/400)

5% (20/400)

If we consider this cash flow calculation period by period, we can recognize how

the variance uncertainty reveals over time. Table (1) shows this. After we know the realization of the first cash flow, N(100; 802), actually 6 400 (or 53,3 %) of the overall variance and uncertainty of 12 000 has been resolved, leaving 5 600 of the uncertainty unsolved. During the second time period, 3 600 of the total variance is solved, and thus the cumulative portion of solved uncertainty is 10 000, leaving 2 000 left as unsolved uncertainty. After the third time period, 1 600 more is solved, and in the final stage, the remaining 400 is resolved. As a result, the amount of solved uncertainty increases over time until all the uncertainty is solved, while the amount of unsolved uncertainty diminishes. This is precisely the same as demonstrated in Figure 2.

When the variance and standard deviation changes between time periods are known, this information can be used for the volatility estimation. Starting from the last time period 4, and using Equation (9) with the given parameters Std(S4) = 20, S0 = 400, r = 0 %, and t4 = 1, we get 4 = 0.05. This means that standard deviation is 5 % of the expected value. This is also quite self-evident, given that we already knew that standard deviation (20) is 5% of the expected value (400). Then, having this information available, we can step backwards into time period 3 and knowing also Std(S3), calculate the volatility 3. This is 0.10, which is also consistent with standard deviation of 40 being 10% of the mean value 400. With the same backward logic, respectively, we can calculate 2 = 0.15 and 1 = 0.20.



4 Regression-Based Volatility Estimation

As mentioned in the example of the previous section, if we are able to construct good forward-looking value estimators PV1…PVn with information of their expected values and standard deviations for each time period, we can estimate the overall uncertainty solved (and unsolved) for each time point, and this information can then be used for volatility estimation. An alternative way to do this is to use ordinary least squares regression equations.

In statistics and econometrics, ordinary least squares (OLS), or linear least squares, is a method for estimating unknown parameters in a linear regression model. The method is a standard approach to the approximate solution of overdetermined systems, i.e. sets of equations in which there are more equations than unknowns. The most important application is in data fitting. The OLS method minimizes the sum of squared distances, also called residuals or squares of errors, between the observed responses in the dataset, and the responses predicted by the linear approximation model. The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side.

Regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. Regression analysis helps us understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables — that is, the average value of the dependent variable when the independent variables are held fixed. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.

Regression divides the sum of squares (SS) in Y, total variation, into two parts: the sum of squares predicted due regression (SSDR) and the sum of squares error (SSE), so that SS = SSDR + SSE. SSDR is the sum of the squared deviations of the predicted scores from the mean predicted, and the SSE is the sum of the squared errors of prediction. The SSDR therefore describes the variation that the regression model can explain and the SSE is the unexplained variation. The square root of the SSE is the standard error.

A common measure of the regression model is a coefficient of determination, R2. It is the proportion (percentage) of the sum of squares explained due to regression divided by the total sum of squares, i.e. R2 = SSDR/SS. R2 is the same as Pearson’s correlation measure between the predicted and observed values. As a result, if we know R2 and standard error, we can also calculate the explained variation, the square root of SSDR, according to:

. . (10)



As a result, we can use regression equations to estimate the expected value and

standard deviation of PV, time period by time period, by linking the explaining cash flow variables to the explanatory estimators of PVt. These explaining variables can be any parameters in the cash flow calculation. Often sufficient explanatory variables for estimating PVt are free cash flows of the given and the previous years3, i.e. FCF0,…,FCFt. As a result, we can calculate PV0 according to Equation (11). This increases for the following time periods with the risk-free interest rate according to Equation (12).

∑ (11)

(12)

Then, we can construct the regression estimators as suggested for each time period

according to the following equations:

, (13)

, , (14)

:

, , ⋯ , (15)

Now we can use Monte Carlo simulation on our example cash flow model. Before

the cash flow is simulation run, we set the free cash flows of the time periods (FCFt) and the present values of project (PVt) as output parameters. These parameter values are then saved during the simulation run so they can be used in regression calculations after the simulation. Then we use regression Equations (13 - 15) for forecasting the expected values and standard errors of PVs.

There are many different alternatives to conduct the simple OLS regression analysis. Even the basic tools provided with the common spreadsheet programs (e.g. MS Excel or OpenOffice Calc) are sufficient for this purpose4. As an example of this, the following Table 2 is taken from the Excel summary output regression statistics report for the PV’s of the previous example. As we can see from the results in Table 2, the numbers of R Square are very close to those of Cumulative proportion of solved uncertainty in Table 1.

3 Haahtela [14] has more details about constructing good regression estimators. 4 In practice, majority of the potential users would rather use other software (e.g. SPSS, SAS,

R, Statistix, Matlab) for the data analysis, because more advanced software is required anyway for many other common tasks related to data analysis and other financial analysis.



Table 2: Regression statistics of the example.

SUMMARY OUTPUT

Regression Statistics year 1 year 2 year 3 year 4

Multiple R 0.730 0.912 0.983 1.000

R Square 0.532 0.831 0.966 1.000

Adjusted R Square 0.532 0.831 0.966 1.000

Standard Error 74.48 44.78 20.00 0.000

Observations 5000 5000 5000 5000

The only information needed of the summary output regression statistics is the

correlation coefficient R Square and Standard Error used in the calculation of the explained variation according to Equation (10). These results are presented in Table 3. The explained variation is indeed the standard deviation of the underlying asset process at the particular time point, i.e. how much of the uncertainty has been revealed until that time point from the earlier time points. With this information, we can calculate the volatilities for the different time periods according to Equation (7). Also SSDR, the square of the explained variation, and SSE, the square of standard error, are presented in the table to compare the simulated results with the calculated results presented earlier in Table 1.

Table 3: Explained variation and the volatilities for different time periods.

Explained variation 79.47 99.30 107.09 (108.9)*

SSDR 6315 9860 11467 (11860)*

SSE 5547 2005 400 0

Volatility 19.87 % 14.89 % 10.02 % 4.96 %

* observed directly from the simulation results

As we can see from the numbers in Table 3, results of the SSDR are very similar to

the results of Solved (explained) uncertainty, and the results of SSE are close to the numbers of Unsolved uncertainty in Table 1. If more simulation runs are used, the regression based results converge gradually to the accurate results of Table 1. The results thus also show that the logic and intuition behind the procedure is in line with the idea of resolving uncertainty during the project timespan presented in Section 3.

In the following section the use of the regression based volatility method is illustrated with an actual case example. We can’t directly calculate comparative results such as those presented in Table 1 because of the serial- and cross-correlations among the cash flow input parameters; therefore, modeling the forthcoming cash flow parameters conditional on earlier realizations of several input parameters is impossible in practice. The suggested regression-based method is not sensitive to that. It can also be adjusted for stochastic interest rate. Another advantage of the approach



is that it can be used with different stochastic processes (e.g. with arithmetic, geometric, or displaced diffusion process).

5 Case Example

The following example illustrates the use of the approach. The input parameters (unit sales price, sales quantity, and variable unit costs) are presented as normal distributions, and they also have correlations with each other. For example, if the sales price is low in 2011, it is also likely to be lower in the years ahead. Also fixed costs have some normal variation. These uncertain input parameters of cash flow calculation are presented in italic in Table 4. As such, this is a typical simplified cash flow calculation that has several partly correlated parameters and some uncertainty.

Table 4: Cash flow calculation of the case example.

Year 2011 2012 2013 2014 2015 2016 2017 Unit sales price 2.00 1.80 1.62 1.46 1.31 1.18 Sales quantity 25 000 37 500 37 500 28 125 21 094 15 820 Variable unit costs 1.70 1.36 1.16 0.98 0.88 0.80 Cash flow calculation Revenue 50 000 67 500 60 750 41 006 27 679 18 683 Variable costs -42 500 -51 000 -43 350 -27 636 -18 654 -12 591 Fixed costs 5 000 5 000 5 000 5 000 5 000 5 000 Depreciation -833 -833 -833 -833 -833 -833 EBIT 11 667 20 667 21 567 17 538 13 192 10 259 Taxes -4 667 -8 267 -8 627 -7 015 -5 277 -4 104 Depreciation 833 833 833 833 833 833 Cash flows (CFt) 7 833 13 233 13 773 11 356 8 748 6 988 Discounted CF's (11%) 7 057 10 740 10 071 7 480 5 192 3 736

Present value 44 276

PVt(Project) PV0 PV1 PV2 PV3 PV4 PV5 PV6 (rf = 5.0 %) 44 276 46 546 48 933 51 442 54 079 56 852 59 767

The ordinary cash flow calculation model requires one additional row that

describes how the expected present value of the discounted cash flows PV0 increases over time according to the risk-free interest rate. Before the cash flow simulation, the free cash flows of the time periods (FCFt) and the present values of the project (PVt) are set as output parameters. These parameter values are saved during the simulation run so they can be used later in regression calculations.



After the simulation, we can investigate the overall uncertainty of the project. Figure 3 shows the simulated underlying asset terminal value distribution discounted to the present value. We know the expected underlying asset value in a risk-neutral world increases according to the risk-free interest rate, but we do not know how the uncertainty changes over the time periods. Based on the assumption of generalized risk-neutral valuation and assuming the stochastic process to follow geometric Brownian motion5, we can calculate according to Equation (7) the annualized volatility to be 18.4 %6. However, we want to know how the uncertainty and thus the volatility evolves over time; therefore, we use the regression-based volatility estimation presented earlier.

Fig. 3. Probability distribution of the case example present value PV0. Basic statistics of mean and standard deviation are also presented as the parameterization for the shifted lognormal

distribution.

Then we use the regression approach similarly to the approach presented in previous Section 4. The regression estimator Equations (13-15) were used with each PVt as an explanatory variable, and the free cash flows of the corresponding and previous years as the explaining variables for each case.

After that, the regression analysis is done, and we get the following regression analysis summary output results presented in Table 5. Based on the R Square (proportion of uncertainty solved) and Standard Error (uncertainty left), we calculate the explained variation for each time period according to Equation (10). After that, volatility for each time period is calculated according to Equation (7). The method and calculation of the results can be confirmed by using the approach of Haahtela [14] for volatility estimation. The results given by both methods are exactly the same when calculated using the same simulated data set. However, using the approach presented in this paper is much easier and faster for a practitioner to apply.

5 Later, this assumption of gBm is also omitted and displaced diffusion process is assumed

instead. 6 Given S0 = 44 321, t = 6, Std(S)0 = 20 981, and rf = 5%.



Table 5: Summary output of the regression analysis and calculation of explained variation, volatility and shifted volatility.

SUMMARY OUTPUT

Regression Statistics 2011 2012 2013 2014

Multiple R 0.794 0.943 0.981 0.994

R Square 0.630 0.890 0.961 0.988

Adjusted R Square 0.630 0.890 0.961 0.988

Standard Error 13419 7698 4788 2858

Observations 10000 10000 10000 10000

Explained variation 17506 21875 23895 25506

Volatility i 36.34 % 22.32 % 11.47 % 7.39 %

Shifted volatility dd,i 16.89 % 10.75 % 5.59 % 3.62 %

However, as the distribution of the present value (Figure 3) shows, the shape of it

is between a normal and a lognormal distribution, and there are also negative underlying asset values. Therefore, the common assumption of the geometric Brownian motion does not hold. One viable alternative is to use the displaced (also called shifted) diffusion process suggested by Haahtela [15]7 that has the evolution of the underlying asset S out to a time horizon T given by equation

,½ √ ∼ 0, 1 (16)

where is the displaced mean parameter,d is volatility of the displaced diffusion and is shifting parameter. The corresponding underlying asset value distribution is a displaced (shifted) lognormal distribution. This process is similar to the displaced diffusion process of Rubinstein [21] and D-binomial process of Camara and Chung [22]. The displaced diffusion process of Equation (16) with different values of S (mean) and (shift) is capable of modeling stochastic processes that are between multiplicative (lognormal) and additive (arithmetic) processes, and it also allows negative underlying asset values.

6 Discussion and Conclusions

This paper presented a practical volatility estimation method for cash flow simulation based real option valuation cases with changing volatility. During cash flow simulation runs, the present value of the cash flows and the corresponding cash

7 This diffusion process is used also in Haahtela [11], but [15] discusses the topic more

thoroughly.



flow state variable values are recorded for each time period. Then a regression analysis is run for relating the PV for each year to the cash flow state variables. Each regression equation provides an estimate of the expected present value as a function conditioned on the resolution of all uncertainties up to that time. Then, basic regression statistics of Pearson’s correlation R2 and sum of squares error (or standard error) for each equation provide all the information required for estimating how the standard deviation of the stochastic process and the volatility change over time.

The method is computationally very efficient as it requires only one pass of simulation runs regardless of the number of time periods. Straightforward calculation of required regression statistics and their availability in any statistical software and even in spreadsheet programs make this approach very easy for a practitioner to apply. Also, the intuitive logic behind the procedure and the capability to handle negative underlying asset values can be regarded as strengths of the method. As such, the approach is sufficiently robust for a cash flow simulation-based volatility estimation.

The method presented in this paper is already quite good in practical terms. It takes into account the stochastic time-dependence of volatility and the level of the underlying asset’s price. The latter of these means the model is able to take into account the common characteristic of the underlying asset to have lower volatility with higher underlying asset values. The next logical step for improving the accuracy of volatility estimation is to use a complete state-space dependent volatility structure. This is very close to the logic of applying an implied volatility term structure method where simulated cash flows determine the underlying asset price state-space. However, this approach requires more computation and forward-looking estimation and is not that easy to implement in practice in comparison with the method this paper suggested.

One alternative between the two previously mentioned alternatives (time-varying volatility for the displaced diffusion process and the implied volatility term structure approach) is to model the change in volatility as a function of several underlying parameters. One alternative to determining these functions would be to use statistical software on simulated data to investigate what kind of response surface functions could be feasible. However, finding a good function (or a set of functions for different time periods) for describing state-dependent volatility may be somewhat difficult. On the other hand, this approach allows direct linking of the cash flow calculation parameters to the parameters describing the underlying asset value process. This may be a useful feature for sensitivity analysis purposes if we want to investigate how changes in cash flow calculation parameters affect the project value with real options.

However, similarly to the complete state-space implied volatility approach, this approach is not a significant improvement over the approach suggested in this paper. This is mostly because cash flow simulation follows the law of large numbers. According to the central limit theorem, the sum of a sufficiently large number of independent random variables of fairly finite mean and variance, is approximately normally distributed. If the process is multiplicative, i.e. the random variables are multiplied, the result is lognormally distributed. Typical cash flow calculation has both of these properties i.e. they are sums of several cash flow variables, with some of



these variables having multiplicative correlated properties over the time periods. Therefore, when we sum up a large number of cash flows over multiple simulation runs, the terminal value distribution is often very close to the shape of the displaced lognormal distribution. Therefore, the method presented in this paper, especially when applied to the displaced diffusion process, is a sufficiently accurate volatility estimation method, given the realities of accuracy of any consolidating cash flow simulation-based volatility estimation method8.

Another question is how much we are interested in knowing the volatility changes over time. The consolidated cash flow simulation-based volatility measure is not an observable market variable that could be used for hedging purposes. As such, the volatility parameter may not even be needed in valuation. If the approach based on the simulated implied value distribution approach is used with an implied binomial lattice, calculating the volatility parameter for each node does not provide significant advantages over the approach of directly estimating the risk-neutral implied probabilities for the transitions9. For example, Wang and Dyer [23] use this approach without calculating the volatility parameter for each node. Another advantage of this approach is that it offers a very flexible distribution assumption for project values. The shortcoming of this approach is that it is more complex to apply in comparison with this paper’s method. On the other hand, such approach is better for such cases where the underlying asset is clearly tractable, which is the case for many projects related to natural resource investments. If this assumption does not hold, the approach presented here gives similar results with less effort.

However, even if the procedure presented here may be technically feasible, it still has many non-computational drawbacks common to similar cash flow simulation-based methods. Firstly, we are consolidating a high-dimensional simulated process into a very low-dimensional process. As also discussed in Brandão et al. [8] and Smith [18], the applicability of such approaches is usually case dependent. Secondly, the high-dimensional process is based on a simulated cash flow, not on a real observable process. As such, it may have highly subjective estimates, especially when estimating the correlations between different cash flow variables. Thirdly, the whole valuation may appear as a black box for the decision-maker after the consolidation, because it is harder to monitor and understand how different parameters and their changes affect the consolidated process. This may significantly reduce the capability to exercise the real options optimally. The use of the sensitivity analysis and linking the primary cash flow calculation parameters to the option valuation parameters with regression equations may mitigate this problem [24].

There are several computational issues that could be engaged to technically improve the approach presented here. One suggested alternative for future research is to take into account the level of the underlying asset value in more detail than 8 This does not mean that the reality would follow this kind of pattern. It is just a consequence

of how sampling based methods work. 9 If we know the risk-neutral transition probabilities between the nodes and assume some

stochastic process (most often geometric Brownian motion), we can easily calculate the implied volatilities for each node.



provided by the displaced diffusion process. Also, it would be of high practical relevance if the link between the single cash flow calculation parameters and the stochastic process properties of mean and standard deviation could be better modeled. However, from both the academic and practitioners’ viewpoint it, would be far more interesting and relevant to investigate, based on several real life cases from different industries, how and when the consolidated approach-based methods are of practical use in comparison with some other valuation methods.

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