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2 Energy Economics 31 (2009) Contents lists available at ScienceDirect Energy Economics journal homeage: Flexibility as a source of value in the roduction of alternative fuels: The ethanol case Carlos Bastian-Pinto a, Luiz Brandão a, Warren J. Hahn b, a Pontifícia Universidade Católica, Rua Marques de São Vicente, 225 Gávea, Rio de Janeiro, Brazil b Peerdine University, Baxter Drive, Malibu, California 90265, USA article info abstract Article history: Received 20 Aril 2008 Received in revised form 11 February 2009 Acceted 12 February 2009 Available online 26 February 2009 JEL classification: C61 D81 O14 O33 Q42 Keywords: Commodity rice models Otion valuation Renewable fuels There is tyically a high degree of flexibility associated with the roduction of alternative fuels due to the ability to source from different inut raw materials or to roduce different outut roducts based on market conditions. In this aer, we consider the articular examle of ethanol and seek to quantify the incremental value from flexibility in its roduction from sugarcane in Brazil. We accomlish this by first jointly modeling the stochastic rocesses for the rices of the two relevant commodities, sugar (a food commodity) and ethanol (an energy commodity) in discrete time as a bivariate lattice. This framework allows us to value the otion to switch outut roducts based on the resective rice signals of the two commodities. However, unlike the usual assumtion of geometric Brownian motion stochastic rocesses, we use the more realistic case of mean reverting commodity rice rocesses. We estimate the arameters for these rocesses by alying a regression-based rocedure to emirical sugar and ethanol data collected during a eriod from 1998 through Our results show that the otion to switch oututs has significant value, even under the assumtion of mean reverting rices, which has imlications for both roducers and olicy-makers alike Elsevier B.V. All rights reserved. 1. Introduction With the recent rise in oil rices, eventual long term exhaustion of world oil reserves and rojected growth in demand for energy in the coming years, alternative sources of renewable energy have become increasingly sought after. One alternative which has gained widesread accetance in Brazil is sugar cane based ethanol automotive fuel. Develoment of this market began in the early 1980's, driven by the government's Pro-Àlcool rogram which included roduction subsidies and a mandatory mix of 20% ethanol to gasoline fuel. After two decades and several initial setbacks, the state subsidies have nowuels, with ethanol currently accounting for 45% of total small vehicle fuel consumtion (UNCTAD, 2005; UNICA, 2007). The main source of ethanol in Brazil is sugar cane, which reviously was almost entirely grown to roduce sugar, another commodity in which Brazil is a leading world layer. Currently, ethanol is also raidly gaining commodity status in the larger world market. According to the Renewable Fuels Association (RFA, 2007), the largest ethanol roducers in 2005 were the United States with 4265 million gallons, Brazil with 4227 million gallons, China with 1004 million gallons, and India with 449 million gallons. Brazil's ethanol roduction industry is suorted by sugar cane based ethanol's cost advantage relative to corn based ethanol, which is Corresonding author. Tel.: ; fax: address: Joe.Hahn@eerdine.edu (W.J. Hahn). the main source of US roduction and which is heavily deendent on government subsidies. In recent years, many Brazilian sugar cane rocessors have invested in flexible lants that can roduce either sugar or ethanol, since sugar cane can be transformed into sugar in a rocess that roduces a small quantity of ethanol as a byroduct, or rocessed in an ethanol distillation rocess to roduce ethanol exclusively. Although such lants require a larger u front investment, it aears that the otential for otion value is intuitively considered by rocessors, as most lants currently under construction in Brazil are flexible (sugar/ethanol) facilities. To value the switching otion that is available to the rocessors, rices of both commodities must be modeled using a joint method that can cature the correlation between the rice movements of each commodity. Aroaches based on the bivariate lattice introduced by Boyle (1988) for modeling dual geometric Brownian motion (GBM) rice rocesses have been used to value these tyes of otions, however, as noted by Schwartz (1997) and others, rices of most commodities are best modeled by mean reverting rocesses, and we show that this alies in the case of sugar and ethanol as well. The main objective of this article is to aly a discrete time lattice based methodology for modeling mean reverting stochastic rocesses in order to quantify the incremental value due to a flexible roduction rocess. We begin with a descrition of the sugar ethanol roduction setting in Brazil in Section 2. Section 3 outlines a bivariate lattice aroach to modeling mean reverting stochastic commodity rice rocesses, and Section 4 follows with the emirical data analysis and /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.eneco

3 412 C. Bastian-Pinto et al. / Energy Economics 31 (2009) stochastic rocess arameter secification, as well as the otion value modeling assumtions and methodology utilized. In Section 5 we resent our modeling results and comare them to both the results from a simulation model and the results when GBM stochastic rocesses are used to model the commodity rices. We conclude in Section 6 with a summary of findings and imlications. 2. The Brazilian sugar/ethanol industry The ethanol roduction industry in Brazil today runs without the benefit of government subsidy, and Brazilian sugar cane based ethanol has become by far the world's most cometitive bio-fuel (Goldemberg, 2007). Aroximately 5.8 million hectares (ha) are currently being used to grow sugar cane in Brazil, of which about 2.9 million hectares are directed for ethanol roduction (Szwarc, 2006), although this number is somewhat variable, due to switching between ethanol and sugar roduction. Each ha lanted with sugar cane can roduce aroximately 6800 l 1 of ethanol. Sugar ethanol roducing comanies in Brazil are resonsible for the rocessing of sugar cane into the two roducts. It is both an agricultural and industrial endeavor that includes choosing the best sugar cane varieties, lanting and harvesting at the aroriate time, rocessing and storage. Industrial investments can either be done directly in a flexible lant (caable of roducing sugar, ethanol or both) or in a single roduct facility (sugar or ethanol), which can later be retrofitted to roduce the comlementary roduct. Sugar cane rocessing lants are highly energy efficient, since the massive bagasse and straw volumes generated by cane rocessing are used as fuel for the furnaces that generate the steam necessary for the rocess. These furnaces tyically also generate surlus electrical ower which is sold to local ower comanies. As a result, the energy efficiency ratio (energy outut divided by energy inut) for sugarcane ethanol roduction is 8.3, which comares favorably to the ratios for the other major ethanol feedstock (Szwarc, 2006). A relatively efficient sugar lant can currently roduce 107 kg (235 lb) 2 out of every metric ton of sugar cane rocessed, along with some syru byroduct which can be converted to 12 l of ethanol (3.17 gal). The same ton of sugar cane, if rocessed in an ethanol lant, will roduce 80 l of ethanol (21.14 gal), which is an increase of aroximately 14% from 1980s levels, due to roductivity gains. Thus, the mass balance for rocessing 1 ton of sugar cane is: 1 ton sugar cane = 107 kg sugar + 12 liters ethanol = 80 liters ethanol ð1þ With this arity defined, and given the rice signals for the two outut commodities, rocessors can decide which mix of roducts they will outut for every cro. Once the investment in a flexible lant has been made, the switching costs are tyically minimal, but they can be factored into the decision making rocess if necessary. 3. Stochastic modeling of ethanol and sugar rices The discrete binomial lattice aroach develoed by Cox et al. (1979) for modeling the stochastic rocess of an underlying asset and valuing contingent otions has found widesread alication, since it generalizes the Black-Sholes-Merton model (1973) and addresses some of that model's restrictions. It is simle to imlement, flexible to use, deends on a limited number of arameters, and converges weakly to a GBM as the time interval diminishes. However, there are instances when the underlying rices modeled do not follow a stochastic rocess similar to a GBM. A common examle is the market rice of many tyes of commodities, which may instead follow a mean reverting stochastic rocess. Mean reverting rocesses are a tye of Markov rocess where the sign and degree of the drift are deendent on the current level of a variable, which reverts to a long-term equilibrium level that we tyically assume is the long-term mean. The simlest form of mean reverting rocess is the one factor Ornstein Uhlenbeck rocess, also called an Arithmetic mean reverting rocess, which has the form shown in Eq. (2): dy t = η Y Y t dt + σdzt ð2þ For commodity rice modeling, in Eq. (2) Y t is the log of rice, η is the mean reversion coefficient, Y is the log of the long term mean rice, σ is the rocess volatility and dz is a Weiner rocess. The log of rice is commonly used since it is generally assumed that commodity rices are lognormally distributed. 3 This is convenient, because if Y=log(y), then y cannot be negative and it also allows future rice movements to be modeled based on the stochastic behavior of returns. The exected value and variance of the Ornstein Uhlenbeck rocess 4 are given by Eqs. (3) and (4): EY ½ t ηt Š = Y + Y 0 Y e Var½Y t Š = σ 2 2η 1 e 2ηT These exressions show that when T, then VAR½Y t ŠY σ 2 2η,as oosed to, as is the case with a GBM. Variations of mean reverting rocesses include the Geometric Mean Reverting model (Dixit and Pindyck, 1994), given by dy t /Y t =η(y Y t )dt+σdz t, and a similar model roosed by Bhattacharya (1978), which is given by dy t =η (Y Y t )dt+σy t dz t, as well as many others. Dias (2005) rovides a survey of some of these different stochastic rocesses and their alication to oil rice modeling. The alicability of the different stochastic rocesses to articular tyes of roblems is a comlicated issue. It may be ossible and aroriate to use GBM models in such cases as short duration rice series. Single factor ure mean reverting models (Ornstein Uhlenbeck rocesses) with a fixed long-term equilibrium level may work better in general, but can also be too simlistic in some instances. In such cases, the best aroach might be to combine a mean reverting model with a GBM for the equilibrium level, as roosed by Schwartz and Smith (2000), although such rocess is more difficult to imlement for valuation uroses. As Dixit and Pindyck (1994) suggest, in order to select an aroriate stochastic rocess for modeling commodity rice or any other variable, the best aroach is to rely on both theoretical considerations, such as equilibrium mechanisms, as well as statistical tests. The logic of a mean reverting rocess comes from microeconomics; when rices are low (or below their long-term mean), demand for the roduct tends to rise while its roduction tends to diminish. This is because the consumtion of a commodity with low rices will tyically increase, while the lower revenues for the roducing firms will lead them to ostone investments and cease roduction, reducing the availability of the commodity. The oosite will haen if rices are high (or above their long term-mean). Emirical studies such as Pindyck and Rubinfeld (1991) have shown that the rices of many commodities follow mean reverting stochastic rocesses. ð3þ ð4þ 1 1 gallon =3.79^l. 2 1 kg =2.2 lb. 3 See Aendix A for the relationshi between the rocesses for Price and Log (Price). 4 The Ornstein^ Uhlenbeck rocess is a continuous version of an AR(1) rocess.

4 C. Bastian-Pinto et al. / Energy Economics 31 (2009) The most common statistical test in for determining whether a GBM or mean reverting rocess is most aroriate is the unit root test (Dickey and Fuller, 1981). For examle, Pindyck (1999) alies a version of the Dickey Fuller unit root test to evaluate several oil, coal and natural gas rice time series. In this aroach, the time series model x t x t 1 =a+(b 1)x t 1 +ε t leads to a hyothesis test with H o :(b 1)=0, or H o :b=1. This null hyothesis osits that a unit root exists, in which case the time series is not stationary. If the null hyothesis can be rejected, then there is suort for the claim of mean-reversion in the time series. The Dickey Fuller test commonly aears in two forms; the standard aroach just described and an augmented aroach that allows for a trend in the series. We alied the Dickey Fuller test to time series consisting of the monthly average ethanol and sugar rices aid to roducers in the State of São Paulo (CEPEA, 2007). We used the standard nonaugmented form of the test, since both of our series are already detrended through the deflation of rices, and obtained the following t-statistic values for the hyothesis test: Ethanol deflated rice: Sugar deflated rice: These can then be comared to the critical values for the test (Wooldridge, 2000): Asymtotic critical values of t-statistic for unit root t-test (no time trend): Significance level 1% 2.5% 5% 10% Critical values The t-value for the ethanol rice series leads to rejection of the resence of a unit root at the 10% level, while the t-value for the sugar rice series does not lead to a rejection of the unit root at any of the significance levels shown above, although one could infer that it might do so at slightly higher levels (e.g., erhas at 20%). We also note that values for both coefficients b (Ethanol: b =0.895, and Sugar: b=0.928) are less than one, which also suggests the resence of mean reversion. In addition to showing some suort for the claim of mean reversion, these results are also illustrative of an issue with unit root tests; that in most instances it is difficult to rule out the ossibility that a GBM might be aroriate with a high (e.g., 90% or higher) degree of certainty. For instance, of all of the time series tested in the emirical work cited above (Pindyck, 1999), only an extremely long time series for oil (96 years, annual average rice) indicated conclusively that there was no unit root, thus rejecting the aroriateness of a GBM rocess. For shorter duration series in that same study, the resence of a unit root could not be rejected, even though the series grahically aear to exhibit mean reversion. For these series, Pindyck oints out that the failure to reject the resence of a unit root does not necessarily rove that a series follows a random walk, but rather it leaves oen the question as to which rocess is most aroriate. Thus, failure to reject of the random walk hyothesis does not necessarily reclude the existence of auto-regression (mean reversion) in the variable of interest. As an alternative, Pindyck suggests that investigating the extent to which rice shocks are ermanent might be more informative than looking for a unit root in checking for random walk or for mean reversion. Under mean-reversion, rice shocks tend to dissiate due the constant force of reversion. This is contrary to the case of a GBM, where rice shocks are ermanent. To test this condition, Pindyck utilizes variance ratio tests, which measure the extent to which the variance of a series grows with the lag of the test. The variance ratio can be exressed as: R k = 1 Var P t + k P t ð5þ k Var P t +1 P t The Var( ) terms in this formula reresent the variance of the series of lagged (by k eriods) differences in the rice series P. In the case of a GBM, as the variance grows linearly with k, the ratio R k should aroach 1 as k grows. On the other hand, under mean reversion, the variance is bounded to a limit as k increases. Therefore the variance ratio above should decay for large values of the lag (k), indicating that rice shocks are not ermanent and that rices are mean reverting. We alied the above variance ratio test to our deflated rice series of ethanol and sugar, with the results shown in Fig. 1 below for both rice and log of rice. As with the series that Pindyck evaluated, the ratios initially grow with lag k, which is consistent with both the GBM and mean reversion assumtions (in this case, variance initially grows and then stabilizes as it reaches an uer limit), but then begin to decrease, eventually stabilizing to values of 0.25 for ethanol and 0.29 for sugar. This attern occurs for both the rice and log of rice series. These relatively low values for the variance ratios are also consistent with the low drift rates for the series (5% for ethanol and 3.4% for sugar, when assessed for GBM arameterization). But more imortant is the low level of variance ratio for both rices, which clearly suorts the assumtion of a mean reverting rocess for modeling these series. Relative to a GBM, a mean reverting rocess is more comlicated to aroximate with a robability lattice with binomial chance nodes. As a result, methods emloying Monte Carlo simulation and discrete trinomial trees (Hull, 1999) have been develoed for modeling mean Fig. 1. Variance ratios for different lag eriods.

5 414 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Fig. 2. Binomial branching node. reverting rocesses. Unfortunately both methods have drawbacks when used for valuation; simulation-based aroaches are comutationally intensive, esecially for roblems with multile concurrent otions, and trinomial trees are more difficult to imlement because they require involved methodologies for secifying valid branching robabilities and lattice cell sizes to ensure convergence to the stochastic rocess Binomial aroximation to mean reverting rocesses In the numerical analysis of otions, simle binomial lattices or trees are often used to aroximate the underlying continuous-time stochastic rocess. As shown in Fig. 2, at each node in a binomial lattice, the variable Y will move u or down by an increment ΔY over a secified time increment Δt. The branching robabilities ( and 1 ) and increment are derived by matching the mean and variance of the binomial node with those of the continuous-time stochastic rocess, thus ensuring convergence as Δt 0. Nelson and Ramaswamy (1990) develoed a general aroach for discrete time aroximations of stochastic rocesses which is alicable to mean reverting Ornstein Uhlenbeck rocesses. Their aroach is a simle binomial sequence of n eriods of length Δt with a time horizon T=nΔt that models the general form stochastic differential equation dy=µ(y,t)dt+σ(y,t)dz as: t uy + ffiffiffiffiffi Δt σðy; tþ ðvalue in u stateþ ð6aþ Y + t uy ffiffiffiffiffi Δt σðy; tþ ðvalue in down stateþ ð6bþ Y t u1 = 2+1= 2 ffiffiffiffiffi μðy; tþ Δt σðy; tþ ðrobability of u moveþ ð6cþ 1 t ðrobability of down moveþ ð6dþ Substituting in the Ornstein Uhlenbeck arameters from Eq. (2) into Eqs. (6a) (6d) yields: t uy + ffiffiffiffiffi Δt σ ðvalue in u stateþ ð7aþ Y + t uy ffiffiffiffiffi Δt σ ðvalue in down stateþ ð7bþ Y t u1 = 2+1= 2 ffiffiffiffiffi η Y Y Δt t σ ðrobability of u moveþ ð7cþ Fig. 3. Bivariate lattice node. This can be summarized in the following nested formula: t = max 0; min 1; 1 = 2+1= 2 ffiffiffiffiffi!! η Y Y Δt t : ð9þ σ In the branching of the binomial lattice, the u and down state sace increments are ΔY + = σ ffiffiffiffiffi Δt and ΔY = σ ffiffiffiffiffi Δt, resectively. If Y is the log of the rice y, then the increments are Δy + = e ffiffiffiffi σ Δt and Δy = e ffiffiffiffi σ Δt. These are the familiar forms utilized in the GBM lattice modeling, and result in a binomial value lattice similar to the one obtained from a GBM (Cox et al., 1979). The calculated robabilities and censoring of these robabilities will yield a model that converges weakly to a mean reverting rocess, as was shown by Nelson and Ramaswamy (1990). It is imortant to note that at each node of the lattice, the robability of an u move ( t ) will change deending on Y t according to Eq. (9), which is what allows the mean reverting behavior to be modeled Converting to a risk neutral rocess There are two ways of discounting cash flows for the urose of valuation: 1) directly using the risk adjusted rate and 2) using a Martingale robability measure with the risk-free rate. The second aroach is tyically used in valuing otions because it is difficult to determine the aroriate risk-adjusted rate when real otions are resent. In the Martingale robability aroach, the drift of the stochastic rocess used for the value of the underlying asset is adjusted so that future ayoffs can be discounted at the risk free rate. As an examle, when a discrete model of a GBM stochastic rocess is used, the robabilities of u and down moves in the lattice are calculated using the risk-free discount rate (Cox et al., 1979). With a mean reverting rocess, a similar adjustment is made to the robability calculations. To 1 t ðrobability of down moveþ ð7dþ Since the calculated robabilities cannot be negative or greater than 1, it is necessary to censor the values of t to the range between 0 and 1, as shown in Eq. (8): 8 ffiffiffiffiffi < 1 = 2+1= 2η Y Y t Δt = σ 0Vqt V1 t = 0 : 1 q t V0 q t z1 ð8þ Fig. 4. Marginal conditional node sequence for two commodities.

6 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Fig. 5. Ethanol and sugar rice data series. Source: CEPEA (2007), UNICA (2007). do this, we substitute Y V= Y π η for Y in Eq. (9), where Y is the riskadjusted long term mean, Y is the unadjusted long term mean, π is the roject risk remium, and η is the mean reversion seed coefficient (Dixit and Pindyck, 1994 and Schwartz, 1997). More detailed discussion of the basis for the risk adjustment can be found in Aendix B Bivariate discrete modeling of mean reverting rocesses The modeling aroach used in this aer is based on a bivariate lattice combining two uncertain variables. The notion of a bivariate lattice was first introduced by Boyle (1988) and was also later discussed by Coeland and Antikarov (2003), who roosed a quadranomial lattice model with two correlated uncertainties, each following a GBM. In order to construct a bivariate lattice, the joint robabilities of each of the four outgoing branches at each node must be determined (Fig. 3). These robabilities reresent the four ossible combinations of u and down moves of the two variables, with the first and second characters in the subscrits for each denoting the direction of the movement for variables X and Y, resectively. To value an otion at any eriod n in such a lattice, we calculate the four ayoffs, which are contingent on the resective values of X and Y at the four subsequent nodes at time ste n+δt, multily by their resective risk-neutral robabilities, and discount back to eriod n using the risk free rate, r. It follows that a dual variable aroach is articularly useful for valuing real otions to switch. Goncalves et al. (2006) analyze a roduction switching otion available to a Brazilian flexible sugar ethanol lant, and model the underlying uncertainties using two distinct GBM diffusion rocesses. However, as noted by Schwartz (1998), Laughton and Jacoby (1993), and others, if commodity rices are indeed mean reverting, then a lognormal geometric Brownian diffusion model can significantly overestimate uncertainty in the resultant cash flows from a roject, and result in overstated otion values. Therefore, the Coeland and Antikarov quadranomial model cannot be directly used for modeling two one factor mean reverting rocesses, since the robabilities at each branch must change throughout the lattice. Kulatilaka (1993) rovides an aroach for valuing a switching otion where the two inut variables are modeled as a single mean reverting rocess for relative rice; however, this aroach is limited to cases with relatively simle value functions that can be written in terms of this tye of variable. To value the otion to switch in this case we need to jointly model the log of two commodity rices, X=ln(x) and Y=ln(y), each following a different mean reverting stochastic rocess of the form in Eq. (2). For these two rocesses, following the convention shown in Fig. 3, we can secify the joint robabilities for X and Y as: uu = Δ XΔ Y + Δ Y m X Δt + Δ X m Y Δt + ρσ X σ Y Δt ð10aþ 4Δ X Δ Y ud = Δ XΔ Y + Δ Y m X Δt Δ X m Y Δt ρσ X σ Y Δt 4Δ X Δ Y du = Δ XΔ Y Δ Y m X Δt + Δ X m Y Δt ρσ X σ Y Δt 4Δ X Δ Y ð10bþ ð10cþ dd = Δ XΔ Y Δ Y m X Δt Δ X m Y Δt + ρσ X σ Y Δt ð10dþ 4Δ X Δ Y ffiffiffiffiffi ffiffiffiffiffi where Δ X = σ X Δt, ΔY = σ Y Δt, and uu + ud + du + dd =1 (Hahn and Dyer, 2008). These robabilities also deend on the drift rate for each rocess, υ X =η x (X X t ) 1/2σ 2 x and υ Y =η y (Y Y t ) 1/2σ 2 y, as well as the correlation ρ between the increments of the two rocesses. Unfortunately, a four branch node like the one shown in Fig. 3 for such a joint rocess cannot be directly censored when necessary, as the mean reverting model requires. Hahn and Dyer (2008) solve this issue by alying Baye's Rule to decomose the joint robabilities into the roducts of marginal and conditional robabilities. To obtain the conditional robabilities, the joint robabilities are divided by the marginal robabilities for X, u = m X Δt ð11aþ 2 Δ X d = m X Δt ; ð11bþ 2 Δ X which leads to the following conditional robabilities for Y: u j u = Δ Xð Δ Y + Δtm Y Þ + ΔtðΔ Y m X + ρσ X σ Y Þ 2Δ Y ðδ X + Δtm X Þ d j u = Δ XðΔ Y Δtm Y Þ + ΔtðΔ Y m X ρσ X σ Y Þ 2Δ Y ðδ X + Δtm X Þ u j d = Δ XðΔ Y + Δtm Y Þ ΔtðΔ Y m X + ρσ X σ Y Þ 2Δ Y ðδ X + Δtm X Þ ð12aþ ð12bþ ð12cþ d j d = Δ XðΔ Y Δtm Y Þ ΔtðΔ Y m X ρσ X σ Y Þ : ð12dþ 2Δ Y ðδ X + Δtm X Þ

7 416 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Fig. 6. Regression to determine ethanol stochastic rocess arameters. Fig. 7. Regression to determine sugar stochastic rocess arameters. Note that in this formulation, u u + d u =1 and u d + d d =1. These robabilities allow the four branch node with joint robabilities to be slit into a marginal conditional sequence (Fig. 4) where all robabilities can again be censored in the manner of Eq. (9). 4. Switching otion valuation methodology In this section, we analyze the incremental value due to the flexibility afforded sugar cane rocessors to switch roduction from ethanol to sugar/ethanol or vice-versa in any given time eriod Estimation of stochastic rocess arameters Data based on daily observations of sugar and ethanol rices directly aid to rocessors was obtained from CEPEA (2007) and are available online. For ethanol, rices are an average between anhydrous and hydrated alcohol (70% of the former, and 30% of the latter, both roduced in the facilities in these roortions). Although we only used rices aid in the state of São Paulo, they were assumed to be reresentative of the general case due to the fact that this state roduces about 64% of these commodities in Brazil and they are a reference widely utilized in research on the ethanol sugar sector Table 1 Regression results for deflated rices of sugar and ethanol. Sugar Ethanol β β R Std error T-statistic (Pretyman, 2005). Prices are in local currency, R$ (Reais) 5, and for ethanol are given er liter (R$/l), whereas for sugar they are in 50 kg bags (R$/50 kg), which is the standard unit in the sector. Both series of rices were collected from May 1998 to December 2008 on a monthly average basis, resulting in 112 data eriods (almost 10 years), and were deflated by the most widely used Brazilian inflation indicator, IGP-DI (FGV), also on a monthly basis. In Fig. 5 they are lotted together on different scales for visual comarison. All rices shown in the figure are in local currency, R$ (Real). The model arameters were estimated using a rocedure based on the methodology outlined by Dixit and Pindyck (1994) which allows simultaneous estimation of all arameters from discrete time series. 6 From these series we were able to calculate the arameters needed for the mean reverting rocesses used in the model. For both series, a simle linear regression was run with log(p t ) log(p t 1 ) as the deendent variable and log(p t 1 ) as the indeendent variable. The resulting regression equation is therefore log(p/p t 1t )=β 0 +β 1 log(p t 1 )+ε. The mean reversion coefficients η can then be obtained from the regression outut as η = log ð β 1 +1Þ Δt, and the volatility and long term rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2logðβ mean are given by σ = σ 1 +1Þ e and P = ex β Δt½ðβ 1 +1Þ 2 0 1Š β + σ 2 1 2η, resectively, where σ 2 ε is the variance of the regression's errors. Plotted regression lines and their corresonding equations for our data can be seen in Figs. 6 and 7. Regression results for both deflated rice series are listed in Table 1, and all stochastic rocess arameter estimates from the regressions are summarized in Table 2. Note that rices for both commodities are currently below their long-term means. 5 As of January 2009, the going exchange rate was R$ 2.20^/^USD. 6 See Aendix C for details on the arameter estimation rocedure.

8 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Table 2 Stochastic rocess arameters for sugar and ethanol. Sugar Ethanol Initial rice R$/50 kg bag R$/l Long-term mean R$/50 kg bag R$/l Year Year Volatility (σ) 34.58% 34.24% Mean reversion coefficient (η) Table 3 Deterministic base case resent value of oerating cash flows (R$000). Stochastic rocess MRM Base cases Pure ethanol 176,143 Sugar (ethanol byroduct) 188,526 We model the interdeendence of the two stochastic rice rocesses through the correlated increments, in the same manner as Gibson and Schwartz (1990), Schwartz (1997), Schwartz and Smith (2000), Tseng and Lin (2007) and others. Thus, we have a rocess of the form shown Eq. (2) for each commodity, where the random increments in the variance terms, dz Ethanol and dz Sugar are standard Wiener rocesses and are correlated as follows: dz Ethanol dz Sugar =ρdt, where ρ is a constant correlation arameter, which was found to be ρ=0.557 for our data set. The rocesses will therefore vary from their exected values in each time eriod in accordance with their individual volatilities and random increments, but the increments are linked together so that as one rocess varies uward or downward, the other rocess will tend to follow suit to the degree secified through the correlation arameter Otion value model methodology The model we use measures the resent value of cash flows generated from rocessing 2,600,000 tons of sugar cane annually, either as ethanol or as sugar, with some ethanol as a by roduct (Gonçalves et al., 2006). This reresents a relatively large rocessing lant, equivalent to 1% of the total Brazilian sugar cane rocessing caacity. The time horizon chosen is five years, in half year eriods (T=5,n=10, Δt=0.5). The roject risk remium (π) of 6% was estimated as the ethanol sugar sector deflated risk remium for comanies listed in the Sao Paulo stock exchange using the CAPM. It is coincidentally very close to the resent deflated risk free rate (r) of 6% in Brazil, a figure which is based on the real interest rate of Brazilian government bonds, as well as the risk free rates used in similar works (Dias, 2005 and Gonçalves et al., 2006). With these arameter values, the risk adjusted long term means are: xv= e X π η ln 0:8409 = e ð 6k ð Þ 1:327Þ =0:8037 RD = liter for Ethanol ð13aþ yv= e Y π η lnð33:86þ 6k 0:9008 = e ð Þ =31:675 RD = 50 kg for Sugar ð13bþ Project oerating cash flow values are calculated as follows: 1) For ure ethanol rocessing, the rojected ethanol rice (R$/l) is multilied by 80 (liters er ton of sugar cane) and then by 0.96% (4% discount due sales tax). This result, minus R$/ton (variable cost), is then multilied by 2600 (yearly sugar cane tons rocessed/1000). The final result is obtained by subtracting fixed costs of R$000 28,726, and multilying by 81% to reflect the income tax. 2) For sugar rocessing, sugar rice (in R$/50 kg bags) is multilied by 2.14 (107 kg of sugar from one ton of sugar cane, in 50 kg bags) and then by 0.84 (16% discount due sales tax), lus 12 (liters of byroduct of ethanol er ton of sugar cane) multilied by ethanol rice (R$/l), and then by 0.96 (4% discount due sales tax). This result, minus R$/ton (variable cost), is then multilied by 2600 (yearly sugar cane tons rocessed/1000). The final result is again obtained by subtracting fixed costs of R$000 28,726, and multilying by 81% to reflect the income tax. 3) For a flexible lant, the higher of these two values is chosen. This follows, because even with the flexibility of choosing any mix between the two oututs, a corner solution is always the otimal one due to the linear relationshi between cash flow and roduct rices. Fixed and variable costs account not only for industrial, but also for agricultural costs, and the latter is the same for both rocesses (Goncalves et al., 2006). These algorithms are summarized in Eqs. (14) and (15). CF Eth = ð½ð80p Eth ½1 4kŠÞ 29:67Š ; 726Þð1 19kÞ ð14þ h i CF Sug = ð 2:14P Sug ½1 16kŠ +12P Eth ½1 4kŠ :94 ð15þ 28; 726Þð1 19kÞ The deterministic cases using these equations and exected values for the two commodity rices are listed in Table 3. To model the variability in roject value, lattices for both commodities were constructed using the aroach described in Section 3.1, Fig. 8. Risk neutral rocess lattice for sugar.

9 418 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Fig. 9. Risk neutral rocess lattice for ethanol. Fig. 10. Sugar rice rojections (non-risk neutral). with the results shown in Figs. 8 and 9. Note that the circled nodes in these figures are reached with robability of zero; therefore the lattices are effectively trimmed on the to and bottom. These two lattices were then jointly modeled according to their correlation using the aroach described in Section 3.3. The marginal robabilities of an u move in the lattice are deendent on the log of the rice value at each node, according to Eq. (9). Then, as the rice movements were slit into marginal and conditional stes, as in Fig. 4, we generated the conditional robabilities using the exressions given in Eqs. (12a) (12d). Although this aroach is numerically intensive, it is relatively straightforward to imlement in an Excel worksheet. In the resulting bivariate rice lattice, beginning one time eriod from the end (in eriod 9 out of 10), we then model the otimal decision-making about whether to roduce ethanol or to roduce sugar and some ethanol as a by-roduct in each eriod, working Fig. 11. Ethanol rice rojections (non-risk neutral).

10 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Table 4 Comarison of results under mean reverting rocess vs. GBM (R$000). Stochastic rocess MRM GBM Δ Base cases Pure ethanol 176, , % Sugar (ethanol byroduct) 188, , % With otion Flexible lant 224, , % Otion value 36, ,812 Flexible lant comared Pure ethanol 27.51% 70.23% to base cases Sugar (ethanol byroduct) 19.13% 68.70% backwards recursively to eriod 0. At each node, we calculate the value as the discounted sum, at the risk free rate, of the four subsequent nodes in the lattice weighted by the joint robabilities, which are in turn the result of multilying the marginal robabilities of ethanol (which was chosen to be the first variable) by the conditional robabilities for sugar at each node, and adding the value of the cash flow at the node considered. We end u at eriod 0 with the resent value of 1 ton of rocessed sugar cane er month, during five years, with the otion in each eriod of choosing between to two ossible oututs. 5. Results 5.1. Results from bivariate lattice model Using the method described in the revious section, we obtained a result of R$ ,592 for the resent value of oerating cash flows for a flexible lant rocessing 2,600,000 tons of sugar cane every year during five years. This can then be comared to the values of R$ ,143 for an ethanol only roducing lant and R$ ,526 for a sugar roducing lant (which roduces some ethanol as by roduct) that are shown in Table 3. These results reresent increases of 27.5% and 19.1% in value, resectively, relative to the non-flexible lants. Thus, the ercent value of the switching otion is 19.1% (flexible roduction comared with the highest value base case) relative to the case with no flexibility, and the absolute incremental value of the switching otion is R$000 36,066. It is interesting to note that even with the sugar only roducing lant having a higher resent value than the ethanol lant, the average mix between the two non-flexible lants was about 51% ethanol and 49% sugar in 2005, with the ethanol fraction increasing (Pretyman, 2005). This is likely due to the growth otential for ethanol; according to industry estimates (UNICA, 2007), sugar roduction is exected to increase almost 3% annually between 2007 and 2020, while ethanol roduction will increase at a 9.7% yearly rate. However, it is not given, ex ante, for any articular eriod which of the non-flexible lants will be more rofitable. Therefore it follows intuitively that rocessors should be aware of the value of flexibility, and invest in flexible lants that allow them to caitalize on either roduct. This is in fact occurring in ractice, since aroximately 64% of the sugar cane rocessing caacity in Brazil is now rovided by flexible lants (EPE, 2008) Comarison with results from a simulation model In rincile, we could have used any of three aroaches as our rimary method of analysis for this roblem; the bivariate lattice aroach we have just shown, Monte Carlo simulation, or finite difference methods. We have chosen to use a bivariate lattice aroach as our rimary method of analysis because 1) it is the most comutationally efficient of the three aroaches, 2) it offers the most straightforward way to model the correlation between the two stochastic rice rocesses, and 3) it is the most robust method with resect to ossible changes in the assumtions we have used. Nonetheless, we can verify our solution by reconstructing the valuation roblem as a bundle of simle Euroean otions, since the sugarcane rocessing switching otion can be exercised in each eriod without costs (after the initial cost of investment in a flexible lant) and since the otion is also indeendent of all decisions made before or after a articular oint. Using this aroach, the value obtained from a simulation with 100,000 iterations was R$ ,115 for the flexible lant, which is 2.9% less than the result from the bivariate lattice method. This small difference is due to the discrete increment in the bivariate lattice of Δt=0.5, and we would exect it to disaear as Δt Comarison with results when GBM rice rocesses are assumed As mentioned reviously, the GBM diffusion rocess is very simle and straightforward to model and imlement, but its drawback is that it may not fit emirical data, esecially for the case of commodity rices. We have shown that for the two commodities analyzed in this aer, the mean reverting diffusion rocess rovides a reasonable fit, whereas a GBM may not be as aroriate. To illustrate the differences, in Figs. 10 and 11 we show the rojections of sugar and ethanol mean rices, resectively, as well as the 90% confidence intervals derived from both the mean reverting rocess and GBM. The volatility arameters for the GBM rocesses for both commodities were estimated by calculating the standard deviation of the log-return of the rice series, while the drift arameters were obtained by adding half of the square of the obtained volatility to the mean of the series log-return. In order to comare valuation results, the same case was modeled assuming rices instead follow a GBM rocess, with the results of the deterministic cases and otion value shown in Table 4. Fig. 12. Value of conversion otions as a function of correlation.

11 420 C. Bastian-Pinto et al. / Energy Economics 31 (2009) The bivariate lattice for two GBM's is indeed simler to imlement following the Coeland and Antikarov (2003) framework; however, the base case results in Table 4 show differences of 19.4% and 12.6%, resectively, for the two rojects, relative to the mean reverting rocess model. Furthermore, the value for the flexible lant obtained in this way, using the same volatility and rice correlation arameters of the mean reverting rocess case is R$ ,046, and the otion value is R$ ,812. These are 70.2% and 68.7% above the base GBM cases of ethanol and sugar, resectively, and 59.4% above the flexible mean reverting rocess case, which indicates that the use of a GBM in this case significantly overestimates the value of the switching otion. This is due to the variance of a GBM, which increases in roortion to t, unlike the bounded variance of a mean reverting rocess Sensitivity of results to correlation between rice rocesses Due to the high correlation between the two uncertain variables (rices of sugar and ethanol) the sensitivity of the otion to this arameter was also investigated. The results are summarized in Fig.12, which lots the value of the switching otion versus the correlation between rice rocesses. The figure shows that the switching otion value increases raidly as the correlation diminishes, ultimately arriving at an otion value of R$000 55,271 (30.24% above the base sugar case) when there is no correlation (ρ=0) used in the bivariate lattice. 6. Conclusions Ethanol is currently regarded as one of the most romising automotive fuels of the future. The basic economic case for ethanol has imroved significantly in recent years, and ethanol is also viewed as being more environmentally friendly than hydrocarbon based fuels such as gasoline and diesel, since it is derived from renewable sources. Also, since its manufacturing rocess is relatively labor intensive, it is viewed favorably in develoing countries with high unemloyment rates. Clearly, ethanol is now a technologically feasible resource with the otential to relace a significant ortion of the world's fossil fuel use. In this aer, we have modeled the rices of ethanol and sugar as mean reverting stochastic rocesses, using a rocedure to estimate the rocess arameters from emirical market data, and have imlemented a comutationally efficient, but recise and flexible framework for modeling the otion value associated with a flexible ethanol roduction rocess. Our results demonstrate that sugar cane based rocessors do indeed derive additional value from a flexible roduction rocess, even under the assumtion of mean reverting rices, and benefit from a natural hedge in the market for sugar, a well established commodity. This is an imortant consideration for otential investors in ethanol's still develoing world market. It is also imortant for olicy-makers considering the degree to which ethanol roduction should be suorted with government subsidies in develoing markets. We have also demonstrated that, although a GBM is much simler to imlement as a discrete binomial lattice relative to a mean reverting rocess, using GBM rice models in this roblem yielded erroneously higher results relative to the case where we used mean reverting rice models, which more accurately deict the rice evolution over time for ethanol and sugar. Finally, we investigated the sensitivity of the switching otion value to the correlation between ethanol and sugar rice rocesses and, as exected, found that the value increased when the two rices moved indeendently, and that it would continue to increase if they became negatively correlated. This imlies that the effect of correlation may become an imortant dynamic, should the rices of sugar and ethanol begin to decoule as the ethanol market continues to develo. Aendix A. Transformation to stochastic rocess for Log(Price) Given that S follows the geometric mean reversion rocess: h i ds = η log S logðþ S Sdt + σsdz; where S is the long term mean rice, we can aly Itô's lemma with Y=log(S): dy = 1 A 2 Y 2 yields AS 2 ds2 + AY AS ds + AY At Substituting in AY AS = 1 S, A 2 Y AS 2 = 1 S 2, AY At =0 and ds 2 =S 2 σ 2 dt dy = 1 S 2 σ 2 dt + 1 h i 2 S 2 S η log S logðþ S Sdt + σsdz ; or "! # dy = η log S σ 2 logðþ S dt + σdz 2η which is the rocess followed by Y=log(S), in terms of the same arameters. By comaring the rocesses for Y and S, we can see the relationshi between Y and S ; Y = log S σ 2 2η ; which is an imortant result for the estimation of stochastic rocess arameters for the Log(Price) rocess from emirical rice data. Aendix B. Adjustment of long-term mean for a risk neutral rocess A standard technique in valuation of commodity-related investments is to aly a risk remium in the form of an adjustment to the drift of the stochastic rocess for the commodity rice. If the rocess can be adjusted so that the drift corresonds to a risk neutral robability measure, then all cash flows can be discounted at the risk free discount rate. 7 This aroach is formally justified by assuming that the state variable is riced according to the intertemoral asset ricing models develoed by Merton (1973) and Cox et al. (1985). In the case of a risk adjusted rocess, the risk adjusted discount rate is µ=α+δ, where µ is the risk adjusted discount rate, α is the rocess drift and δ is the rocess dividend yield. Thus the drift of the risk adjusted rocess can be exressed as α=µ δ. With a riskneutral format, the drift α of the rocess, is relaced by r δ, where r is the risk free rate. If we consider the articular case of the single factor Ornstein Uhlenbeck rocess, which is given by: dy t = η Y Y t dt + σdzt ; then the rocess drift is α=η(y Y). We note that, in contrast to a GBM, the dividend yield is not constant; rather, it is a function of Y. That is, the dividend yield is δ=µ α, or δ=µ η(y Y). With this exression we can write the risk-neutral drift for the mean reverting rocess as r δ=r µ η(y Y). Rearranging terms a few times, we have: r δ = η Y Y ðμ rþ or r δ = η Y ðμ rþ Y : η 7 A rigorous develoment of the risk neutral valuation framework can be found in Duffy (1992).

12 C. Bastian-Pinto et al. / Energy Economics 31 (2009) Then finally, using the relationshi π=µ r, the risk-neutral drift becomes: r δ = η Y π Y ; or η r δ = η Y Y π Note that π=µ r is the risk-remium. Comaring both drifts (risk-adjusted and risk-neutral) shows that the conversion to a riskneutral rocess can be viewed as subtracting the normalized riskremium (µ r)/η, orπ/η, from the long run mean level Y. In other words, in the risk-neutral rocess the rices revert toward a lower level than the real long-run level, and the subtracting term is the normalized risk-remium. Therefore, following the convention used by Schwartz (1995, 1997), we adjust the drift of the rocess by subtracting the risk remium π.thus, the risk-neutral form of the single factor Ornstein Uhlenbeck rocess is: dy t = η Y Y t π dt + σdz; which can in turn be re-written as: dy t = η Y π Y η t dt + σdz This is the basis for the exressions for the risk-adjusted long-term means shown in Eqs. (11a) and (11b) of the aer. Aendix C. Mean reverting model arameter estimation The simlest form of mean reverting model is the single factor Ornstein Uhlenbeck rocess, which has the form shown in Eq. (2), with the exected value and variance given by Eqs. (3) and (4), resectively. In order to determine the values for the arameters for this rocess, we start by writing Eq. (3) for a discrete time interval Δt: Y t = Y + Y t 1 Y Y t Y t 1 = Y 1 e ηδt e ηδt = Y 1 e ηδt + Y t 1 e ηδt ; or + Y t 1 e ηδt 1 Substituting Y t =log [S t ] and Y =log S yields 8 : h i logðs t = S t 1 Þ = log S σ 2 = 2η 1 e ηδt +logs ð t 1 Þ e ηδt 1 Rewriting this equation in the form: σ 2 2η and rearranging logðs t = S t 1 Þ = β 0 + β 1 logðs t 1 Þ; ðiiþ we can the estimate the rocess arameters by simly erforming a linear regression on the rice series S t. From the regression outut, we can then obtain the arameters needed with the formulas below, which were develoed by Dixit and Pindyck (1994) and modified by Dias (2008). From Eq. (i) and (ii), we have β 1 =e ηδt 1, or η = logðβ 1 +1Þ= Δt ðiiiþ 8 See Aendix A for the relationshi between Y and S. ðiþ We can also determine the volatility arameter σ from the variance of errors obtained from the regression, σ ε 2, and Eq. (4). First we equate the two to obtain: σ 2 ε = σ 2 2η 1 e 2ηΔt Then re-writing using the relationshi (β 1 +1) 2 =e 2ηΔt and Eq. (iii), yields σ 2 ε = σ 2 Δt 1 ð β 1 +1Þ 2 2 logðβ 1 +1Þ ; or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 logðβ σ = σ 1 +1Þ ε ðβ 1 +1Þ 2 1 Δt Finally, from Eqs. (i) and (ii), β 0 =[log(s ) σ 2 /2η](1 e ηδt ). With the relationshi β 1 =1 e ηδt, we obtain: β 0 β 1 = h i log S σ 2 = 2η ; or " # S = ex β 0 + σ 2 β 1 2η Alternatively, by substituting the value of η from Eq. (iii), we can rewrite this as: " # S = ex β 0 σ 2 Δt + ; β 1 2ð log ½β 1 +1ŠÞ and further, using the value of σ from Eq. (iv), as: " # S = ex β 0 + σ 2 2Δt log½β 1 +1Š ε β 1 2Δtð log ½β 1 +1ŠÞ ½β 1 +1Š 1 ; so that we have an exression in terms of the regression coefficients: " # S = ex β 0 σ 2 ε + β 1 1 ðβ 1 +1Þ 2 References ðivþ Bhattacharya, S., Project valuation with mean reverting cash flow streams. Journal of Finance 33, Black, F., Scholes, M., The ricing of otions and cororate liabilities. Journal of Political Economy 81, Boyle, P.,1988. A lattice framework for otion ricing with two state variables. Journal of Financial and Quantitative Analysis 23, Center for Advanced Studies in Alied Economics (CEPEA), Indicator of rices. htt:// Coeland, T., Antikarov, V., Real otions: A ractitioner's guide. Texere, New York. Cox, J., Ross, S., Rubinstein, M., Otion ricing: a simlified aroach. Journal of Financial Economics 7, Cox, J., Ingersoll, J., Ross, S., An intertemoral general equilibrium model of asset rices. Econometrica 53, Dias, M., Real otions with etroleum alications. Doctoral thesis, Industrial engineering deartment, PUC-Rio, Rio de Janeiro, Brazil. Dias, M., Stochastic rocesses with focus in etroleum alications, Part 2 mean reversion models. htt://shere.rdc.uc-rio.br/marco.ind/revers.html#mean-rev. Dickey, D., Fuller, W., Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, Dixit, A., Pindyck, R., Investment under uncertainty. Princeton University Press, Princeton, NJ. Duffy, D., Dynamic asset ricing theory. Princeton, Princeton, NJ. Emresa de Pesquisa Energética (EPE), Persectivas ara o Etanol no Brasil. (in Portuguese). htt:// Gibson, R., Schwartz, E., Stochastic convenience yield and the ricing of oil contingent claims. Journal of Finance 45, Goldemberg, J., Ethanol for a sustainable energy future. Science 315, 9. Gonçalves, D., Neto, J., Brasil, H., The otion of switching an investment roject into an agribusiness roject. 10th International conference on real otions. New York.