Modeling Switching Options using Mean Reverting Commodity Price Models Carlos Bastian Pinto Pontifícia Universidade Católica - PUC Rio de Janeiro
[email protected] +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
[email protected] +55 21 2138-9304 Rua Marquês de São Vicente, 225- Gávea - Rio de Janeiro – Brazil Warren J. Hahn Graziadio School of Business – Pepperdine University
[email protected] 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.
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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 Var [Yt ] =
σ2 (1 − e −2ηT ) 2η
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 Unpredictable Model Predictable Model
Name of Model Geometric Brownian Motion (GBM) Pure Mean-Reversion Model (Mean reverting process) Two and Three Factors Model
More realistic Models
Reversion to Uncertain LongRun Level Mean-Reversion with Jumps
Main References Paddock, Siegel & Smith (80’s) Schwartz (1997, model 1) Gibson & Schwartz (1990), and Schwartz (models 2 & 3) Pindick (1999) and Baker, Mayfield & Parsons (1998) 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 and
5 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 − ≡ Y − ∆tσ (Y , t )
(down move)
q t ≡ 1 2 + 1 2 ∆t
1-qt
µ (Y , t ) σ (Y , t )
(probability of up move)
qt
Yt+
Yt-1 1-qt
Yt-
(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)
q t ≡ 1 2 + 1 2 ∆t
1-qt
η (Y − Yt ) (probability of up move) σ (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 & qt
