Norwegian School of Economics Bergen, Spring, 2019
Mean Reversion and Seasonality in Heating Oil Futures
Ola Vold Rennemo
Supervisor: Jørgen Haug
Master thesis, Finance
NORWEGIAN SCHOOL OF ECONOMICS
This thesis was written as a part of the Master of Science in Economics and Business Administration at NHH. Please note that neither the institution nor the examiners are responsible − through the approval of this thesis − for the theories and methods used, or results and conclusions drawn in this work.
Takk til veileder Jørgen Haug for motiverende samtaler og konstruktive innspill. Takk til Jørgen Vold Rennemo for korrekturlesing. Takk til venner og familie for at dere viste interesse for prosjektet og stilte ører til disposisjon.
Contents Introduction ...... 1
Section 1 ...... 3
1.1 Heating Oil ...... 3
1.2 NY Harbor ULSD Futures (HO) ...... 8
Section 2 ...... 10
2.1 The Commodities Literature ...... 10
2.2 Mathematical Representation of the Model ...... 12
2.3 Nested Models ...... 14
2.4 Futures Prices and Volatility Term Structure of Futures Returns ...... 18
2.5 European Call Option on a Futures Contract ...... 20
2.6 Seasonality Function ...... 21
Section 3 ...... 22
3.1 Estimation Method ...... 22
3.2 The Kalman Filter ...... 22
3.3 State Space Representation of a Model Defined By (1), (2) and (3) ...... 25
3.4 Simulation ...... 26
Section 4 ...... 29
4.1 The Data ...... 29
Section 5 ...... 33
5.1 Estimated Parameters ...... 34
5.2 Likelihood-Ratio Tests ...... 35
5.3 Estimated Seasonality Function ...... 36
5.4 Accuracy ...... 38
5.5 Volatility Term Structure of Futures Returns ...... 41
5.6 One-Factor, Three-Factor and Four-Factor Model ...... 42
Section 6 ...... 45
6.1 Practical Implications ...... 45 6.2 Conclusions ...... 49
6.3 Assumptions and Further Works ...... 51
References ...... 52
Appendix ...... 55
Production, Trade and Financial Markets ...... 55
Nested Model Relation Equations ...... 56
1
Abstract
We examine mean reversion and seasonality in heating oil futures prices using an affine N-factor Gaussian model and NY Harbor ULSD futures. We find strong empirical evidence for mean reversion and seasonality in heating oil futures prices.
Introduction
The valuation and hedging of commodity contingent claims are of great importance to commodity producers, commodity consumers, financial intermediaries, and speculators. Consequently, appropriate modeling and estimation of the stochastic behavior of commodity prices are vital for these market participants’ applications of sound investment and risk- management strategies.
In this thesis, we consider an affine N-factor Gaussian model developed by Cortazar and Naranjo (2006), and study its ability to explain the stochastic behavior of heating oil futures prices. In doing so, we determine the parsimonious number of latent unobservable factors to include in the N-factor model, which depends on the existence and scale of mean reverting properties in heating oil’s price process under an equivalent martingale measure. Furthermore, we examine seasonality in heating oil futures prices, which depends on the existence and scale of seasonality in heating oil’s price process under an equivalent martingale measure. Lastly, we examine some implications of our findings regarding the mean reverting properties and seasonality in heating oil futures prices, utilizing model estimated heating oil futures prices and model estimated prices of a European call option on a heating oil futures contract.
The base model for this thesis is Schwartz and Smith’s (2000) two-factor model, which is a two-factor version of Cortazar and Naranjo’s (2006) N-factor model. The seasonal adjustment to this base model is modeled as a deterministic continuous trigonometric function with seasonal frequencies, similar to the seasonal adjustment suggested by Sorensen (2002).
The model parameters are estimated using maximum-likelihood techniques and the Kalman filter. We conduct a simulation study of the parameter estimation procedure, following Andresen and Sollie (2013), to highlight potential econometric issues. We find that the models are robust for use in valuation problems and generally not robust for forecasting 2 purposes, confirming Schwartz and Smith’s (2000) and Andresen and Sollie’s (2013) findings.
In Cortazar and Naranjo’s (2006) study of crude oil price modeling, the authors found grounds for using three- or four-factor models, suggesting two or three mean reverting factors in crude oil’s price process under an equivalent martingale measure. Furthermore, the authors found that the accuracy of model estimated futures prices given by a two-, three- and four- factor model was comparable for futures maturities between 3 months and 3 years, and that for maturities between 3 and 7 years the two-factor model was substantially less accurate than the three- and four-factor models. Crude oil is a major cost driver for heating oil as heating oil is refined from crude oil. Hence, mean reversion in heating oil’s price process under an equivalent martingale measure is probable. Indeed, based on parameter estimates of a two- factor model, fitted on NY Harbor ULSD futures, we find significant mean reversion in heating oil’s price process under an equivalent martingale measure. However, we could not find grounds for including more than one mean reverting factor in the model. The maximal time to maturity of the futures contracts used to estimate parameters in this thesis is approximately 3 years, so we believe the difference in maximal contract maturity is the root cause of the difference in our findings regarding the appropriate number of mean reverting factors. Still, we cannot rule out that the difference in findings reflect differences in the two commodities’ price process under an equivalent martingale measure.
Heating oil’s use as a heating fuel implies that there is an area dependent seasonal distillate fuel oil consumption pattern that is inversely related to the temperature in that area. We find signs of this seasonality in the U.S. distillate fuel oil inventory, and there appears to be a positive correlation between changes in the U.S. distillate fuel oil inventory and changes in the expected development of heating oil prices. Hence, seasonality in heating oil’s price process is probable. Indeed, based on likelihood-ratio tests of the two-factor base model and seasonally adjusted two-factor base models, we find that there is statistically significant seasonality in heating oil’s price process under an equivalent martingale measure. Furthermore, we find that allowing for more than two inflection points to capture this seasonality provided a statistically significantly better fit. In Roberts and Lin’s (2006) study of seasonality in the price process for different commodities the authors found significant seasonality in crude oil’s price process under an equivalent martingale measure, though the authors could not find grounds for allowing more than two inflection points in describing this seasonality. The contrast to our results indicates that there are significant differences between 3 the seasonality of the two commodities’ price process under an equivalent martingale measure.
The outline for this thesis is as follows. In Section 1, we present an overview of heating oil and NY Harbor ULSD futures. In Section 2, we provide a brief overview of the commodity price model literature, present the chosen model, and derive some of its properties, which are important for parameter estimation and valuation and hedging of commodity contingent claims. In Section 3, the parameter estimation procedure is presented and tested in a controlled environment through simulation. In Section 4, the NY Harbor ULSD futures price panel is presented. In Section 5, estimation results are presented and reviewed. Lastly, in Section 6, we provide examples of practical implications of our findings, conclude, discuss assumptions and questions open for further study.
Section 1
In this section we present an overview of heating oil and NY Harbor ULSD futures, and extract some features of these that will be used for appropriately modeling the heating oil price process.1
1.1 Heating Oil
Heating oil is one of the petroleum products produced by refining crude oil (EIA (c), 2019). Heating oil is classified as a distillate alongside other petroleum products with similarities in chemical make-up (EIA (a), 2019). In this thesis, we use the distillate naming convention used by the United States Energy Information Administration (EIA). In EIA’s (a) (2019) naming convention, No. 2 fuel oil (heating oil) and No. 2 diesel fuel are considered as different kinds of No. 2 distillate, and No. 2 distillate is one of the distillates considered as distillate fuel oil. No. 2 diesel fuel is mainly used in diesel engines and No. 2 fuel oil (heating oil) is mainly used in atomizing type burners. To better align this naming convention with heating oil futures and to ease reading we use “heating oil” as a substitute for “No. 2 fuel oil (heating oil)”.
1 Hull (2012), defines a futures contract as a contract that obligates the holder to buy or sell an asset at a predetermined delivery price during a specified future time period with a given settlement scheme. 4
Heating oil is mainly used for commercial and residential space and water heating (EIA (c), 2019). In the United States, the northeastern region accounts for most of the heating oil consumption, and the other regions of the United States typically use Natural Gas for space and water heating (EIA (c), 2019).
There are numerous factors driving fluctuation in heating oil prices, and we will now discuss the most important ones.
Figure 1 shows daily Cushing, OK Crude Oil Future Contract 1 (Dollars per Barrel) prices, daily New York Harbor No. 2 Heating Oil Future Contract 1 (Dollars per Gallon) prices multiplied by 42 and the daily heating oil crack spread from January 2, 2001 to December 12, 2018. The time series are sourced from EIA (2019) which defines Contract 1 as a futures contract specifying the earliest delivery date, often called the front month. As both products have, for each month in a year, a single futures contract with delivery date in that month, the average time to delivery for Contract 1 is approximately 11 business days.2
The heating oil crack spread represents a theoretical profit margin for heating oil refineries. It is defined as the price difference between heating oil futures and crude oil futures with the same delivery month and the same unit (CME Group (b), 2019). As 42 gallons is equivalent to one barrel we derive the crack spread by multiplying the heating oil dollars per gallon futures prices by 42 and subtract crude oil dollars per barrel futures prices.
As futures prices tend towards the spot price when the time to maturity goes to zero (Hull, 2012), the futures prices shown in Figure 1 provides a good approximation for their respective spot prices.