Función de Conteo

Though excellent for a variety of financial applications, geometric Brownian motion in its standard form is ill suited to describe the evolution of energy prices. Particularly, the natural gas market requires a model which describes the presence of mean reversion within the spot prices. This attribute is present due to the pressures of supply and demand on the price caused by the seasonal use of natural gas. High demand during the winter months will put upward pressure on the price of natural gas, and similarly the limited demand and thus increased supply of natural gas during the spring will place downward pressure on the price. Amongst these high and low swings, the price is centered around an average that the market wants to regress to as it moves further away. The mean reversion within natural gas prices can be captured in the stochastic process known as the Ornstein-Uhlunbeck

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process. Using variables related to the topic of natural gas, the process can be described by the following stochastic differential equation

𝑑𝑆𝑡

𝑆𝑡 = 𝛼(𝜇𝑡− log 𝑆𝑡 ) 𝑑𝑡 + 𝜍𝑡𝑑𝑊𝑡. (4.1)

Here, μ(t) is the mean reverting level or price, and α represents the strength of the mean reversion or the rate at which the price will try to revert back to the mean reverting price. The volatility, 𝜍_𝑡, is now a function of time due to its dependence on the changing mean reverting level. The process of obtaining each parameter will be discussed in later sections of this chapter. The other variables retain their identity from the standard GBM process (Eydeland and Wolyniec 2003).

From Equation 4.1, a closed form solution for the distributions of the logarithms of prices can be obtained. Utilizing a simple change of variables

𝑆_𝑡 = 𝑒𝑋𝑡 (4.2)

from which we obtain

𝑋_𝑡 = log 𝑆_𝑡 .

Applying Itô’s Lemma, the process can be written as

𝑑𝑋𝑡 = 𝛼 𝜇𝑡 − 𝑋𝑡 𝑑𝑡 + 𝜍𝑡𝑑𝑊𝑡 ,

where 𝑑𝑋𝑡 is the change in the log price of natural gas. Introducing a new variable

𝑌_𝑡 = 𝑒𝛼𝑡_𝑋

𝑡 (4.3)

and again utilizing Itô’s Lemma, the process of 𝑑𝑌𝑡 can be expressed as

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Since 𝑑𝑌_𝑡 is a normal random variable for every t, the summation of these increments from the current time to the future time will produce the random variable

𝑌𝑡+Δ𝑡 ~ Ν 𝑌𝑡+Δ𝑡+ 𝜇𝑡𝑒𝛼Δ𝑡, 𝜍𝑡 𝑒

2𝛼Δ 𝑡

2𝛼 .

Hence, by the definition of 𝑌𝑡 given in Equation 4.3,

𝑋_𝑡+Δ𝑡 ~ Ν 𝑒−𝛼Δ𝑡_𝑋

𝑡+ 𝜇𝑡(1 − 𝑒−𝛼Δ𝑡), 𝜍𝑡

(1 − 𝑒−2𝛼Δ𝑡₎

2𝛼 (4.4)

which is a closed form solution for the distribution for the logarithms of natural gas spot prices (Eydeland and Wolyniec 2003). However, before simulations can be implemented, parameters α, μ(t), and σ(t) must be determined. The parameters will be estimated using the historical data of Henry Hub Natural Gas spot prices from the years 2004-2009, which can be seen in Figure 18.

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Figure 18 - Henry Hub Historical Natural Gas Spot Prices 2004 - 2009 (Enstor 2009)

Strength of Mean Reversion (α)

The parameter α is the strength or rate of mean reversion, which is the rate at which the price will move toward a certain level. To estimate α, a linear regression is implemented on the historical spot prices. Here, the log return of the spot prices, log⁡(𝑆𝑡+1

𝑆𝑡 ), are regressed against the log of the ratio between the historical average, 𝜇 , of the data and the spot price, log⁡(_𝑆𝜇

𝑡). The slope of this regression is the rate of mean reversion. For this data set, α was computed to be .0063. 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 12/30/03 02/27/05 04/28/06 06/27/07 08/25/08 10/24/09 Pri ce ($/MM Bt u ) Date (mm/dd/yy) Spot Price

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Exponential Smoothing

The mean reverting level for this process, 𝜇_𝑡, is time dependent due to the assumption that the long-term mean will alter over the course of many years. From Figure 18, the average price of natural gas in 2009 appears to be approximately $4.00 per MMBtu, while years prior show an average of $7.00 or higher. Thus, the technique of exponential smoothing is utilized to capture the movement of this long- term mean. Similar to a moving average, exponential smoothing assigns exponentially decreasing weights over time instead of assigning equal weights to past observations allowing current measurements to hold more influence. A simple expression for the exponential smoothing algorithm is

𝜇𝑡 = 𝛽𝑆𝑡+ (1 − 𝛽)𝜇𝑡−1 (4.5)

where β is the smoothing factor between [0,1] (Brown and Meyer 1960). Thus, the smoothed statistic, in this case the mean reverting level, 𝜇𝑡 , is a function of the

weighted average of the new information receive, the spot price 𝑆_𝑡, and the previous smoothed statistic, 𝜇_𝑡−1. Tuning β will change the smoothness of the observations produced form the process. The smaller the β the smoother the observations, and conversely the larger the β the more the observations will track the noise of the original data. For the Henry Hub Natural Gas Spot Prices a β of .002 was chosen in order to capture a gradual shift in the long-term of the spot price. The implementation of exponential smoothing can be seen in Figure19.

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Figure 19 - Natural Gas Spot Prices with Exponential Smoothing, Beta = .002

Volatility

Earlier it was shown that the distribution of 𝑋_𝑡, the logarithms of prices , had a standard deviation of

𝜍 = 𝜍_{𝑡 𝑡} (1 − 𝑒−2𝛼Δ𝑡)

2𝛼 (4.6)

where 𝜍_𝑡 is the volatility seen in the pricing process described by Equation 4.1. Rearranging Equation 4.6, the volatility can be written as a function of the standard deviation of the logarithm of 𝑆𝑡.

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 12/30/03 02/27/05 04/28/06 06/27/07 08/25/08 10/24/09 Pri ce ($/MM Bt u ) Date (mm/dd/yy) Spot Price Exponential Smoothing Price

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51 𝜍𝑡 = 𝜍 𝑡

2𝛼

(1 − 𝑒−2𝛼Δ𝑡₎ (4.7)

The value of 𝜍 can be estimated using either exponential smoothing similar 𝑡

to the form in Equation 4.5 or from a rolling window of historical spot price data by applying the formula

𝜍 = 𝑡 1 𝑚 − 1 log 𝑆_𝑖 − log 𝑆𝑖−1 Δ𝑡 − 1 𝑚 log 𝑆_𝑖 − log 𝑆𝑖−1 Δ𝑡 𝑚 𝑖=1 2 𝑚 𝑖=1

where 𝑆_𝑖 represents the price of natural gas at the closing of each day, which is measured from historical data during a window of m time periods prior to the current time period (Eydeland and Wolyniec 2003). Using this estimate in Equation 4.7, a time dependent volatility for the pricing process can be determined.

Simulation Results

Having selected a mean reversion model and estimated the appropriate parameters based on the historical data, the model expressed in Equation 4.1can be used to simulate the evolution of the spot prices for natural gas. Figure 20illustrates several sample price paths constructed from the simulations of a year of natural gas spot prices.

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Figure 20 - Sample Paths for 1-Year of Simulated Natural Gas Spot Prices