Modeling Switching Options using Mean Reverting Commodity
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Modeling Switching Options using Mean Reverting Commodity
Price Models
Carlos Bastian Pinto
Pontifícia Universidade Católica - PUC Rio de Janeiro
bastian@pobox.com
+55 21 9496-5520
Rua Alm Sadock de Sa n 69 # 101 – Rio de Janeiro – Brazil
Luiz Brandão
Pontifícia Universidade Católica - PUC Rio de Janeiro
brandao@iag.puc-rio.br
+55 21 2138-9304
Rua Marquês de São Vicente, 225- Gávea - Rio de Janeiro – Brazil
Warren J. Hahn
Graziadio School of Business – Pepperdine University
Joe.Hahn@pepperdine.edu
310-506-8542
24255 Pacific Coast Highway, Malibu, CA 90263, USA
Abstract
Although Geometric Brownian Motion (GBM) stochastic process models are commonly used
in valuing real options, commodity prices are generally better modeled by mean reverting
process. Moreover, the inappropriate use of a GBM model may result in overestimation of the
option value, as well as the deterministic project value itself. Unfortunately, mean reverting
models are not as simple to implement in the discrete lattice format commonly used for option
valuation as are GBM processes. In this paper, we implement a precise and flexible
framework for modeling a one factor mean reverting process via censored probability lattice,
and then extend this approach to a two-variable mean reverting process by using a bivariate
lattice. We then use the latter to value the switching option available to producers of two
commodities which can be chosen as output from one basic source: sugarcane. Prices of the
two commodities modeled, sugar (a food commodity) and ethanol (an energy commodity), are
very well approximated by a mean reverting model. Our model results show that the
switching option has significant value for the producer, however, we also show that this
option is significantly overvalued if we assume GBM commodity price processes, confirming
the importance of stochastic model selection.
Keywords: Real Options; Mean Reverting Model; Switching Options; Commodities Prices.2
1. Introduction
Due to the rise in oil prices, expected long term exhaustion in world oil reserves and projected
increases in demand for energy in the coming years, alternative sources of renewable energy
have become increasingly sought after. One alternative which has gained widespread
acceptance in Brazil is the use of sugar cane based ethanol as automotive fuel, with ethanol
already substituting up to 45% of the country’s gas consumption, compared to 3% in the US.
This development began in the early 1980’s, driven by government subsidies and a mandatory
mix of 20% ethanol to all gas fuel. After two decades and several setbacks, state subsidies
have disappeared, productivity has increased dramatically, ethanol production is on the rise,
and the majority of all new auto sales are flex fuel cars, which can run on any mix of gas and
ethanol.
The main source of ethanol in Brazil is sugar cane, which previously was almost entirely
grown to produce sugar, another commodity in which Brazil is a leading world player.
Currently, ethanol is rapidly gaining the status of a commodity in the world market.
According to the Renewable Fuels Association (RFA)1, in 2005 the largest producers were the
United States with 4,265 million gallons, Brazil with 4,227 million gallons, China with 1,004
million gallons, and India with 449 million gallons.
Brazil’s position in ethanol production is due to the fact that sugar cane based ethanol has a
large price cost advantage over the corn based ethanol, which is the main source of the US
production and which is heavily dependent on government subsidies.
Another advantage of sugar cane based ethanol lies in the switching option available to
Brazilian producers which, depending on the relative prices of sugar and ethanol, can alter the
mix of sugar/ethanol produced in order to maximize profits. Sugar cane can be transformed
into sugar in sugar mills that produce a small quantity of ethanol as a byproduct, or processed
in an ethanol distillery to produce ethanol exclusively.
Another alternative is to invest in a flexible (and more expensive) plant that can produce
either sugar or ethanol. Although this means a larger industrial investment, it appears that this
option is intuitively considered by producers, as most plants currently under construction in
Brazil are flexible (sugar/ethanol) facilities.
1
http://www.ethanolrfa.org/3
The switching option that is available to the producer of these commodities must be modeled
using a bivariate method, since these uncertainties must be kept separate. As noted by
Schwartz (1997) and others, prices of commodities are generally best modeled by a mean
reverting processes, and this applies in the case of sugar and ethanol as well. Therefore, to
facilitate valuation of this option, we use a discrete time approach to model both uncertainties
(sugar and ethanol prices paid to producers) as two mean reverting stochastic processes
combined into a bivariate recombining event tree (bivariate lattice). This approach was then
compared to a Monte Carlo simulation-based approach.
We observed similar results between the two approaches, but found the simulation-based
approach to be less flexible for cases with an option exercise price, for instance. We also
compared the results from the mean reverting model to those obtained from a GBM model,
and confirmed that these yield higher results, not only for the option but to the base case itself.
2. Mean Reverting Modeling of Stochastic Processes
The discrete binomial lattice approach developed by Cox et al. (1979) for valuing real options
has found widespread applications, since it generalizes the Black-Sholes-Merton model
(1973) and addresses some of this model’s restrictions. It is simple to use, flexible, depends
on a limited number of parameters and converges weakly to a GBM, as the time interval
diminishes. But there are instances when the underlying prices modeled do not follow a
stochastic process similar to a GBM. This is often the case where cash flows are dependent on
prices that depend on mean reverting assets, such as non financial commodity prices.
Mean reverting processes are Markov process where the sign and intensity of the drift are
dependent on the current price, which reverts to a market level equilibrium level which we
typically assume is the long-term mean price. Unfortunately, mean reverting processes are not
as simple to approximate by a probability lattice with binomial chance branches as is a GBM.
This is why methods employing Monte Carlo simulation and discrete trinomial trees (Hull,
1999) have been developed for modeling these processes.
The simplest form of mean reverting process is the one factor Ornstein-Uhlenbeck process,
also called Arithmetic mean reverting process, which is modeled as shown in Equation (1):
dYt = η (Y − Yt )dt + σdz t (1)4
where Yt is the log of the commodity price, η the mean reversion coefficient, Y the log of the
long term mean price, σ the process volatility and dz a Weiner process. The log of prices is
used since it is generally assumed that commodity prices are log-normally distributed. This is
convenient since if Y=log(y), then y cannot be negative. The expected value and variance of
the Ornstein-Uhlenbeck process are given by:
E [Yt ] = Y + (Y0 − Y )e −ηT
σ2
Var [Yt ] =
2η
(1 − e −2ηT )
Thus it can be seen that if: T Æ ∞ , then: Var[Yt] Æ σ2/2η, and not to: ∞ , as is the case with
a GBM.
Other mean reverting processes worth mentioning are the Geometric Mean Reversion Motion
(Dixit and Pindick, 1994), where dYt Yt = η (Y − Yt )dt + σdz t , and the Battacharya model
given by dYt = η (Y − Yt )dt + σYt dz t . The logic of a mean reverting process comes from
micro-economics: when prices are low (or bellow their long-term mean), demand for the
product tends to rise while its production tends to diminish. This is because the consumption
of a commodity with low prices will increase, while the lower revenues for the producing
firms will lead them to postpone investments and close down old plants, reducing the
availability of the commodity. The opposite will happen if prices are high (or above their long
term-mean). Empirical studies have shown (Pindick & Rubinfeld, 1991) that with oil prices,
for example, microeconomic logic indicates that the stochastic process has a mean reverting
component. Nonetheless econometric tests only reject the GBM for extremely long series.
Dias (2005) classifies stochastic processes for oil prices in three classes shown in Table 1:
Table 1: Stochastic models
Type of Stochastic Model Name of Model Main References
Geometric Brownian Motion
Unpredictable Model Paddock, Siegel & Smith (80’s)
(GBM)
Pure Mean-Reversion Model
Predictable Model Schwartz (1997, model 1)
(Mean reverting process)
Gibson & Schwartz (1990), and
Two and Three Factors Model
Schwartz (models 2 & 3)
More realistic Models Reversion to Uncertain Long- Pindick (1999) and Baker,
Run Level Mayfield & Parsons (1998)
Mean-Reversion with Jumps Dias & Rocha (1998)
The applicability of these processes to different types of problems is a complicated issue.
Although the GBM is extensively used to model a wide range of uncertain variables and5
offers great ease of use, the mean reverting processes are generally considered better-suited to
model commodities prices and interest rates. On the other hand, we note that it may still be
possible and appropriate to use GBM models, such as in the case of short duration price
series. Moreover, single factor pure mean reverting models (or Ornstein-Uhlenbeck process)
to a fixed price level can be too “predictable” in some instances, and might not perform any
better than a GBM model. In those cases, it would be more realistic to combine a mean
reverting model with a GBM for the equilibrium level, or add a jump process.
2.1. Binomial approximation to mean reverting processes
Nelson and Ramaswamy (1990) produced an approach that can be used under a wide range of
conditions, and is applicable to Ornstein-Uhlenbeck processes. This model is a simple
binomial sequence of n periods of length ∆t, with a time horizon of T: T=n ∆t. A binomial
recombining tree (lattice) can thus be constructed.
Given a general form stochastic differential equation: dY = µ(Y,t)dt + σ(Y,t)dz, and:
Yt + ≡ Y + ∆tσ (Y , t ) (up move) Yt+
qt
Yt − ≡ Y − ∆tσ (Y , t ) (down move)
Yt-1
µ (Y , t )
q t ≡ 1 2 + 1 2 ∆t (probability of up move)
σ (Y , t ) 1-qt Yt-
1-qt (probability of down move)
Substituting with the Ornstein-Uhlenbeck parameters from equation (1), we get:
Yt + ≡ Y + ∆tσ (up move)
Yt − ≡ Y − ∆tσ (down move)
η (Y − Yt )
q t ≡ 1 2 + 1 2 ∆t (probability of up move)
σ
1-qt (probability of down move)
And considering that the calculated probabilities cannot be negative or superior to 100%, it is
necessary to censor the values of qt (and thus of: 1- qt), to the range between 0 and 1:6
⎧1 2 + 1 2η (Y − Yt ) ∆t σ if qt >=0 & qt7
throughout the lattice. In order to construct a bivariate mean reverting process lattice, the joint
probabilities of each of the four outgoing branches at each node of the tree must be
determined (Figure 1). The value of an option C at step n then depends of the four subsequent
nodes at step n+1, multiplied by their respective probabilities.
Figure 1: Bivariate lattice option framework
Step n Pxuyu Step n+1
Pxuyd
Cxuyu
C Cxuyd
Cxdyu
Cxdyd
Pxdyu
Pxdyd
Consider two variables, x and y, such as X(t)=log(x(t)), Y(t)=log(y(t)), each following a
different mean reverting stochastic diffusion process:
dX = η x (X − X )dt + σ x dz , and dY = η y (Y − Y )dt + σ y dz ;
with ∆X = σ x ∆t and ∆Y = σ y ∆t
For these two processes, we can specify the following probabilities:
X going up and Y going up: Pxuyu = (∆X ∆Y+∆Y νx ∆t+∆X νy ∆t+ρxy σx σy ∆t)/4 ∆X ∆Y
X going up and Y going down: Pxuyd = (∆X ∆Y+∆Y νx ∆t-∆X νy ∆t-ρxy σx σy ∆t)/4 ∆X ∆Y
X going down and Y going up: Pxdyu = (∆X ∆Y-∆Y νx ∆t+∆X νy ∆t-ρxy σx σy ∆t)/4 ∆X ∆Y
X going down and Y going down: Pxdyd = (∆X ∆Y-∆Y νx ∆t-∆X νy ∆t+ρxy σx σy ∆t)/4 ∆X ∆Y
where Pxuyu + Pxuyd + Pxdyu + Pxdyd = 1.
These probabilities are dependent on the drift of each variable and their correlation ρxy. The
drifts of the respective processes are given by: ν X = η x (X − X t ) − 1 2 σ x2 , and
ν Y = η y (Y − Yt ) − 1 2σ y2 , but a four branch node of such a joint process cannot be directly8
censored, as the mean reverting model requires. Hahn and Dyer (2006) solve this issue
through the application of Baye’s Rule: p(Xt∩Yt)=p(Yt|Xt) p(Xt). As we can calculate, Pxu and
Pxd (=1-Pxu) from equation (2), censoring as necessary, the conditional probabilities are:
Pyu|xu = (∆X ∆Y+∆Y νx ∆t+∆X νy ∆t+ρxy σx σy ∆t)/2∆Y(∆X+νx ∆t)
Pyd|xu = (∆X ∆Y+∆Y νx ∆t-∆X νy ∆t-ρxy σx σy ∆t)/2∆Y(∆X+νx ∆t)
Pyu|xd = (∆X ∆Y-∆Y νx ∆t+∆X νy ∆t-ρxy σx σy ∆t)/2∆Y(∆X-νx ∆t)
Pyd|xd = (∆X ∆Y-∆Y νx ∆t-∆X νy ∆t+ρxy σx σy ∆t)/2∆Y(∆X-νx ∆t)
In this formulation, Pyu|xu + Pyd|xu = 1 and Pyu|xd + Pyd|xd = 1, which are also censored as
necessary. The corresponding joint probabilities are the result of multiplying these by Pxu or
Pxd . We thus have split the four branch node into marginal and conditional steps, censoring
as necessary (Figure 2). As each node will have four conditional and four joint probabilities,
plus one marginal probability, this is a more involved lattice construction, especially as
compared to a dual GBM approach, which has only four joint probabilities for the entire tree.
Figure 2: Splitting the four branch node into marginal and conditional steps
Commodity X Commodity Y
Pyu|xu Y+∆Y
Y
Pxu X+∆X
Pyd|xu Y-∆Y
X Censor probabilities as
necessary
Pyu|xd Y+∆Y
Pxd X-∆X
Y
Pyd|xd Y-∆Y
3. The Brazilian Sugar/Ethanol Industry
Three factors should be taken into consideration when studying Brazil’s history regarding the
sugar and ethanol industry. First, is Brazil’s "Pró-Álcool" policy during the 1970’s that
transformed ethanol, which up until then was only regarded as a sugar-cane by-product, into a
first order automotive fuel. This drove the creation of the necessary infrastructure and
industrial capacity that transformed ethanol in a high return product for the industry. The
second factor is the end of the Cold War, which caused a significant impact on the sugar9
supply of the old socialist countries by Cuba in the international market, and the raise in
exports from Asia and Africa. Lastly, the end of the state subsidies in the ethanol industry is a
factor which lead to a more efficient and open market (Pretyman, 2005).
Brazil has the largest supply of unused agricultural land in the world, even after adjusting for
increasing amounts of land being set aside for ecosystem perservation. Furthermore, of
Brazil’s available cropland, less than 1% is used for growing sugar cane, despite its
significant energy-related potential; each ton of sugar cane has the energy equivalent of 1.2
barrels of oil (CTC2, 2006). As a renewable crop, sugar cane provides five annual harvests on
average, and is mostly grown in the Southeast Region on approximately 3 million hectares.
The State of São Paulo accounts for 2.6 million hectares, with an average productivity of 79
t/ha (tons per hectare), while the Northeast region, with a little more than 1 million hectares,
has an average productivity of 56 t/ha.
3.1. Sugar panorama in Brazil
Brazil is the world’s largest producer and exporter of sugar, and has been so since the end of
the XVI century, less than 100 years after the beginning of its colonization. Sugar was the
main cash crop for Portugal until the XIX century, and the main reason for the Dutch
invasions in the Northeast. Even after the decline of the demand in the XVIII century, it
remained as an important part of Brazil’s export agenda.
The world sugar market is a mature one, with vegetative raise and a small overproduction. It
is also a highly protected market in the northern hemisphere (US and Europe) with heavy
subsidies to production as well as large entry barriers, which are being questioned in
international forums. In fact, due to local natural conditions, Brazil has the lowest sugar
production cost in the world, representing 34% of that of the European Union – mostly France
- (ÚNICA, 2004) and is the main player in the world sugar market with high market share and
competitivity.
3.2. Ethanol panorama in Brazil
Sugar cane ethanol started to be regarded as an alternative automotive fuel in Brazil during
the 1970’s oil crisis when the country was highly dependent on oil imports and had a large
deficit on its foreign trade balance. Therefore the government inititated a program to create
2
http://www.ctc.com.br/10
not only the supply, but also the demand of automotive ethanol fuel, in the form of 100%
ethanol fueled cars. Production was subsidized by taxes on gasoline sales, and vehicle
manufacturers benefited for some time of tax incentives as well. The program was hugely
successful at first, and by 1986, the vast majority of new automobiles ran exclusively on
ethanol.
But by the end of the 1980s producers exercised their switching option; as oil prices and, by
correlation, ethanol prices dropped and sugar prices increased, producers switched from
ethanol production to sugar, which was in high demand. At the same time, short of cash, the
government began to reduce its intervention in the market and its incentives for ethanol
production. As it became harder and harder to find ethanol at gas stations, consumer
confidence in the fuel collapsed and ethanol car production fell to 13% of the total.
The turning point for the ethanol industry came in 2003, when the first flex fuel automobile
was launched in the market. This vehicle was a result of the research done at Bosh of Brazil,
where the technology was developed that would allow a combustion engine to burn any mix
of ethanol and gas. Flex fuel auto production grew from 2.6% in 2003 to 15.2% in 2004 and
39.3% in 2005, out of a total of approximately 2 million automobiles produced in that year.
As these cars became available in increasing numbers from 2003 on, and the danger for
consumers of running out of fuel simply vanished, ethanol was again competitive with
gasoline and the industry gained large production scale. Additionally, gasoline has today a
mixture of 20% to 25% ethanol, which has replaced MTBE. Alves (2007) uses a real options
approach to calculate the present value of the option available to flex fuel cars owners of fuel
choice and demonstrates that it can represent up to 10% of the car value.
The key to Brazil becoming a leading player in ethanol production is due to a perfect
combination of climate, territorial extension and water reserves. From each ha of planted with
sugar cane 6,800 liters3 of ethanol can currently be produced. In the US where ethanol is
mostly derived from corn, each ha yields 3,200 liters of ethanol.
Thus, ethanol is a market in transition, with high expectations of increased demand and
supply. Major producing countries are organizing themselves as is the case of Brazil and the
US, who created the Interamerican Ethanol Commission4 in December 2006. The comission’s
declared mission is:
3
1 gallon = 3.79 liters
4
http://www.helpfuelthefuture.org11
"Promote the usage of ethanol in the gasoline pools of the Western
Hemisphere" and has the objective of fostering awareness of the
benefits of renewable fuels to economies throughout the Americas.
According to specialists, in less than five years ethanol should reach
the status of commodity, guarantying a sustainable market.”
3.3. Sugar and Ethanol manufacturing process
Sugar-Ethanol producing companies in Brazil are responsible for the processing of sugar cane
into these two products as well as storage. It is both an agricultural and industrial endeavor
that includes choosing sugar cane varieties, planting and harvesting at the appropriate time,
processing and storage. Industrial investments can either be done directly in a flexible plant
(capable of producing either sugar, ethanol or both) or in a single product facility (sugar or
ethanol), which can later be retrofitted to produce the complementary product.
Sugar cane processing plants are highly energy efficient, since the bagasse (crushed sugar-
cane) generated by the process is used as fuel for the plant furnaces and even for generation of
surplus electric power. Comparatively, in the US the process is dependent on energy either
from coal, oil or natural gas. A relatively efficient sugar plant can produce 94 kg5 (207 lb) out
of every metric ton of sugar cane processed, and out of the syrup which is produced together
with this sugar, 10.8 liters of ethanol (2.85 gal) can be obtained. The same ton of sugar cane if
processed in an ethanol plant will produce 70 liters of ethanol (18.50 gal). Therefore
processing of one ton of sugar cane generates:
1 ton of sugar cane = 94 kg of sugar + 10.8 liters of ethanol = 70 liters of ethanol
This relation was utilized to control prices of both products during the early stages of the
"Pró-Álcool" government program, but the equivalence still holds. With this parity defined,
producers can decide which mix of products they will output for every crop without switching
cost once the industrial investment on a flexible plant has been made.
4. Switching Option Valuation Methodology
In this section, we analyze the incremental value due to the flexibility afforded sugar cane
processors to switch production from ethanol to sugar/ethanol or vice-versa, at any given
semester.
5
1 kg = 2.2 lb12
4.1. Data collection
Data on sugar and ethanol prices directly paid to producers was collected from CEPEA6
(Centro de Estudos Avançados em Economia Aplicada, of the Escola Superior de
Agricultura) and are available online. These are the result of daily collection, a work which is
performed by technicians in the covenant between CEPEA, UNICA (União da Agroindústria
Canavieira de São Paulo) and ORPLANA (Organização dos Plantadores de Cana do Estado
de SP). For ethanol, prices are a mean between hydrated and anhydrous alcohol (both
produced in the facilities). The cost for both products (ethanol and sugar) include local taxes.
Although they are only prices paid in the state of São Paulo, this is justified by the fact that
this state produces about 64% of these commodities in Brazil and they are a reference widely
utilized in researche on the ethanol-sugar sector (Pretyman, 2005). Prices are in local
currency, R$ (Reais)7, and for ethanol are given per litter (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 prices were collected from July 2000 to January 2007 on a weekly basis,
resulting in 344 data periods, and were deflated by the mostly widely used Brazilian inflation
indicator: IGP-DI (FGV), also on a weekly basis. In Figure 3 they are plotted together, on
different scales for visual comparison.
6
http://www.cepea.esalq.usp.br/
7
As of February 2007, the going exchange rate was R$ 2,1 / USD.13
Figure 3: Weekly deflated prices of sugar and ethanol – prices paid to producers
60 1,6
Deflated Prices
1,4
50
1,2
40
Sugar: R$/50 kg (bag)
1
Ethanol: R$/litter
30 0,8
0,6
20
Sugar
0,4
Ethanol
10
0,2
0 0
set/2005
set/2006
jul/2000
out/2000
jan/2001
abr/2001
jul/2001
out/2001
jan/2002
abr/2002
jul/2002
out/2002
jan/2003
abr/2003
jul/2003
out/2003
jan/2004
abr/2004
jul/2004
out/2004
dez/2004
abr/2005
jul/2005
dez/2005
mar/2006
jun/2006
dez/2006
All prices are in local currency: R$ (Real), which trades for about: 1.00 R$ ≅ 0.48US $ .
From these series we were able to calculate the parameters needed for the mean reverting
processes used in our model. The long-term mean calculation is straightforward. Volatility
coefficients (σ) were calculated by finding the standard deviation from the log (Pt/Pt-1)
series, obtaining the weekly volatility of the prices. For the semester and annual values, these
were multiplied by 26 and 52 , respectively (26 week in a semester, and 52 in a year).
For the mean reversion coefficients (η), an additional step was necessary. A simple linear
regression was run with log( Pt ) as the dependent variable, and (log(Pt −1 ) − log( P ) ) as the
independent variable. The resulting equation is thus log(Pt ) = β 0 + β1 (log(Pt −1 ) − log( P ) ) . The
coefficient β1 will give us e −η∆t , so η = − log( β1 ) / ∆t . As a further check, we should obtain
β 0 = log(P ) . Plotted regressions can be seen in Figure 4, as well as the results of the
regressions.14
Figure 4: Linear regression on log of sugar and ethanol prices
0,4 4,1
Log(Pt)=β0 +β1 (Log(Pt-1)-Log(Pm))
Log(Pt)=β0+β1(Log(Pt-1)-Log(Pm))
0,2 3,9
y = 0,9832x - 0,0604 y = 0,9884x + 3,6256
R2 = 0,9678 R2 = 0,9771
0
3,7
-0,8 -0,6 -0,4 -0,2 0 0,2 0,4
-0,2
3,5
-0,4 Sugar prices
3,3
Ethanol prices regression
-0,6
regression 3,1
-0,8
2,9
-1 -0,6 -0,4 -0,2 0 0,2 0,4
Results for both commodities show that long term means are consistent with the values
already calculated. Correlation calculated between the two series (Sugar and Ethanol) was
found to be ρSE = 0.78032. All parameters are organized in Table 2.
Table 2: Parameters for mean reverting modeling of sugar and ethanol
Sugar Ethanol
Long-term mean 37.57 R$ / 50 kg bag 0,9424 R$ / liter
Weekly* Semester** Yearly*** Weekly* Semester** Yearly***
Volatility - σ 3.51% 17.91% 25.32% 3.90% 19.91% 28.15%
Mean reversion coef. η 0.01167 0.30336 0.60673 0.01694 0.44051 0.88102
* calculated directly from series - ** 26 weeks - *** 52 weeks
4.2. Model methodology
The model we use measures the income generated from processing one metric ton of sugar
cane per month, either as ethanol or as sugar, with some ethanol as a by product. As ethanol
processing is more cost efficient than sugar processing, a discount of 18% in value is applied
to revenue from the latter (Pretyman, 2005). The time span chosen is five years, in half year
periods (T = 5, n = 10, ∆t = 0.5). We assume that the risk free rate is 6%, a rate which is also
used in similar works (Dias, 2005; Gonçalves, 2005) and is consistent with the present long
term real interest rate of Brazilian government bonds.
Project values are calculated as follows: 1) for pure ethanol processing, the projected ethanol
price (R$ / liter) is multiplied by 70 (liters per ton of sugar cane) and by 6 (semester period);
2) for sugar processing, sugar price (in R$ / 50 Kg bags) is multiplied by 1.88 (94 kg of sugar
from one ton of sugar cane, in 50 Kg bags), plus 10.8 (liters of by product of ethanol per ton15
of sugar cane) multiplied by ethanol price (R$ / liter), and factored by 0.82 (18% discount due
to higher processing costs); and 3) for a flexible plant, the higher of these two values is
chosen.
The deterministic cases are listed in Table 3. As the prices of sugar and ethanol are modeled
to follow a mean reverting process with the parameters in Table 2, they will converge to long-
term calculated mean.
Table 3: Deterministic cases of sugar cane processing (R$)
T0 T1 T2 T3 T4 T5
Mean Reversion Model 1 2 1 2 1 2 1 2 1 2
Ethanol price (R$/L) 0,87 0,89 0,91 0,92 0,93 0,93 0,94 0,94 0,94 0,94 0,94
Sugar price (R$/ 50 kg) 36,63 36,87 37,05 37,19 37,29 37,36 37,41 37,45 37,48 37,51 37,52
Ethanol pure project
1 ton of sugarcane processed / month yields 375,12 382,36 387,09 390,17 392,17 393,46 394,29 394,83 395,17 395,39
Present value in T0 3.330
Sugar project (ethanol by product)
1 ton of sugarcane processed / month yields 388,52 391,11 392,94 394,25 395,18 395,85 396,32 396,67 396,91 397,09
Present value in T0 3.371
In order to build the model, both product lattices were constructed, following the Nelson &
Ramaswamy approach with censored probabilities. These can be seen in Erro! Fonte de
referência não encontrada. and Erro! Fonte de referência não encontrada..
Figure 5: Sugar prices mean reverting process censored lattice
107,3
Sugar prices MRM lattice
100
Censored nodes 89,7
(probability = 0)
80 Long T erm Mean
Expected value from lattice 75,0
62,7
60
R$ /50 kg
52,4
43,8
40
37.6
36,6
30,6
25,6
20 21,4
17,9
15,0
12,5
10,5 8,7 7,3 6,1
0
1 2 1 2 1 2 1 2 1 2
T0 T1 T2 T3 T4 T516
Figure 6: Ethanol prices mean reverting process censored lattice
Ethanol prices MRM lattice 2,35
2,0 Censored nodes
(probability = 0) 1,92
Long term mean
Expected value from lattice
1,58
1,5
1,29
R$ / Litter
1,06
1,0
0,94
0,87
0,71
0,58
0,5 0,48
0,39
0,32
0,26
0,22 0,18 0,14 0,12
0,0
1 2 1 2 1 2 1 2 1 2
T0 T1 T2 T3 T4 T5
These two lattices must be combined according to their correlation to generate the bivariate
values lattice of both commodities prices. This bivariate lattice is large compared to the
univariate case, having 121 nodes at period 10, or 10 ((n+1)2). The marginal probabilities of
an up move in the lattice are dependent on the Log of the price value at each node, according
to Equation 2. Then, as we have split each node into marginal and conditional steps (Figure
2) we generate the conditional probabilities using the expressions given in Section 2.2.
Although this numerically intensive, once done it is easy and straightforward to manage, and
is also easy to audit in an Excel worksheet.
With the resulting bivariate values lattice, at each node a decision will be made regarding the
option available to producer; either to produce ethanol or sugar and some ethanol as a by-
product during this semester. The option corresponds to the following equation:
Maximum (70*Pethanol ; [18.1 * Psugar + 10.2 * Pethanol]*[1-18%]) * 6 (semester) (3)
Once the option is exercised, we discount the tree beginning with step 9 (out of 10) and work
backwards to step 0, as described in Figure 1, calculating the value at each node as the
discounted sum, at the risk free rate, of the four subsequent nodes in the lattice weighted by
the joint probabilities, which are in turn the result of multiplying the marginal probabilities of
ethanol (which was chosen to be the first variable) by the conditional probabilities at each17 node, and adding the value of the node considered. We end up at step 0 with the present value of 1 ton of processed sugar cane per month, during five years, with the semester option of choosing between to two possible outputs. 5. Results 5.1. Results from bivariate lattice The method we described yields a result of R$ 3,501 for every metric ton of sugar cane processed in a flexible plant every month during five years. This compares to R$ 3,330 in an ethanol only producing plant and R$ 3,371 in a sugar producing plant (which produces some ethanol as by product). These results amount to an additional value of 5.14% and 3.87% respectively, and the latter is the value of the switching option (flexible production compared with the highest value base case). It is important to point out that these calculations reflect the true decision making process that producers go through when deciding what their output mix should be. Also these differences are reported for total income. When we compare the resulting option value to the operational margin of a typical producer, which can amount to 25% of his income (Gonçalves, 2005), the option value increases to 38% and 32% of net results, respectively. 5.2. Simulation results Since exercise of the switching option is possible at each period without costs (after the initial cost of investment in a flexible plant) and is independent of all decisions made to this point and of decisions to be made later, we can model these options as a bundle of European options. In this very particular case of managerial flexibility, the same problem can be solved by simulation of the ethanol and sugar prices with the help of a Monte Carlo simulation package. This was done as a way to further check the results obtained in the bivariate mean reverting lattice, using the same input data. The results obtained, after 10,000 interactions with a @RISK® package was R$ 3,516 for the flexible plant value, which is only 0.41% different from the results of the bivariate lattice method. Although this method is easier and faster to model than the bivariate lattice, it is not applicable most real option cases, since it cannot take into account options with exercise costs or which are path dependent, as is the case of most real options, and which need to be calculated backwards from de end of the option expiration time. This approach is also
18
dependent on the availability of a specialized software package with the ability to model
correlated random variates.
5.3. Comparison with GBM process
As we mentioned previously, the GBM diffusion process is very simple and straightforward
to model and implement, but its drawback it that it may not fit the empirical data. We have
shown that for the two commodities analyzed in this article, the mean reverting diffusion
process provides a reasonable fit, whereas a GBM may not be as appropriate. In Figure 7 we
show the projections of ethanol and in Figure 8 of sugar mean prices (as well as the 66%
confidence intervals) derived from both the mean reverting process and GBM.
Figure 7: Ethanol prices projections
1,20
MRM Ethanol prices projection
1,15 66% conf. interval
Long term mean
1,10
GMB
1,05 66% conf. interval
R$/ litter
1,00
0,950,942
0,90
0,867
0,85
0,80
1 2 1 2 1 2 1 2 1 2
T0 T1 T2 T3 T4 T519
Figure 8: Sugar prices projection
50
MRM Sugar prices projection
48 66% conf. interval
46 Long term mean
GMB
44
R$ / 50 Kg
66% conf. interval
42
40
38
37,57
36 36,6
34
1 2 1 2 1 2 1 2 1 2
T0 T1 T2 T3 T4 T5
In order to compare results, the same case was modeled assuming prices followed a GBM
process, and in Table 4 we can see the results of the deterministic case.
Table 4: Deterministic cases of sugar cane processing (R$) GBM model
T0 T1 T2 T3 T4 T5
Geomatric Brownian Motion 1 2 1 2 1 2 1 2 1 2
Ethanol price (R$/L) 0,87 0,89 0,92 0,95 0,98 1,01 1,04 1,07 1,10 1,13 1,17
Sugar price (R$/ 50 kg) 36,63 37,73 38,86 40,03 41,23 42,47 43,74 45,05 46,40 47,80 49,23
Ethanol pure project
1 ton of sugarcane processed / month yields 375,07 386,32 397,91 409,85 422,15 434,81 447,86 461,30 475,14 489,39
Present value in T0 3.650 Diference to MRM model: 9,6%
Sugar project (ethanol by product)
1 ton of sugarcane processed / month yields 396,43 408,32 420,58 433,20 446,19 459,58 473,37 487,57 502,20 517,27
Present value in T0 3.858 Diference to MRM model: 14,5%
The base case shows a difference of 9.4% and 14.4%, respectively for the two projects,
relative to the mean reverting process model, illustrating the effect of the drift in prices
resulting from the GBM model. The bivariate lattice for GBM is indeed simpler to implement,
and follows the Copeland & Antikarov (2001) framework. The value for the option obtained
in this way, using the same volatility and price correlation parameters of the mean reverting
process case, is R$ 4.468. This is 22.4% and 15.8% above the base GBM cases of ethanol and
sugar, respectively, and 24.4% above the flexible mean reverting process case, which
indicates that the use of GBM in this case significantly overestimates the value of the
switching option. Results are listed in Table 5.20
Table 5: Comparison of mean reverting process and GBM results (R$)
Process: M-R GBM Difference
Base cases: Pure Ethanol 3.330 3.650 9,6%
Sugar (ethanol byproduct) 3.371 3.858 14,5%
With option: Flexible plant 3.501 4.420 26,2%
Option Value compared Pure Ethanol 5,14% 21,10%
to base cases: Sugar (ethanol byproduct) 3,87% 14,58%
5.4. Sensitivity to variables correlation
Due to the high correlation between the two uncertain variables (prices of sugar and ethanol)
both methods developed (bivariate lattice and simulation) were used to measure the sensitivity
of the option to this correlation. Both methods yield very similar results with the difference
(mostly with values of negative correlation) being due to the discrete increment in the
bivariate lattice of ∆T=0.5, which may still be considered a relatively large value. Results are
plotted in Figure 9 and show that the option value increases rapidly as the correlation
diminishes, ultimately arriving at a value of 1.07% when there is no correlation (ρ = 0) with
the bivariate lattice.
It is worth noting that the option value is not zero even with both uncertainties totally
correlated (ρ = 1), which might be contrary to expectations. This is due to the fact that this is
the correlation between both Weiner processes of the uncertain variables, and not between the
cash flows, which are also influenced by the volatility of these variables (which are not the
same) and by the prices levels. As a result, the conversion option might still be executed even
with perfectly correlated variables. This last point was confirmed by the Monte Carlo
simulation.
Figure 9: % Value of Conversion Option as a function of correlation ρ
16%
OR % vrs ρ 14% Simulação
Árvore bi-variável
12%
10%
8%
6%
4%
2%
0%
-1 -0,5 0 ρ 0,5 121
6. Conclusions
Ethanol is today being regarded as one of the most promising automotive fuels of the future.
Not only is it less polluting than hydrocarbon based fuels such as gasoline and diesel, it is
derived from renewable sources, is more labor intensive, which is an important consideration
in developing countries with high unemployment rates, but it is now also a technologically
available resource with a capacity to substitute a significant portion of the world’s fossil fuel
use. This paper shows that sugar cane based ethanol producers benefit from a natural hedge
based on sugar market, a long time well established commodity, which acts as a guarantee for
production in view of ethanol’s still developing world market.
We implemented a somewhat computationally intensive but precise and flexible framework
for modeling a two variable mean reverting diffusion process with a bivariate lattice based on
Hahn and Dyer (2006), and applied it to the valuation of a switching option available to
ethanol/sugar producers in Brazil. Although a GBM is much simpler to model as a discrete
binomial lattice relative to the analogous mean reverting diffusion process, several
commodity prices are more realistically modeled by the latter. Moreover, we have confirmed
computationally that using a GBM process in our specific case yields erroneously higher
results than one modeled by a mean reverting process that more closely models a commodity
price evolution.
.
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