An application of Ornstein-Uhlenbeck process to commodity pricing in Thailand
- Nattiya Chaiyapo^{1, 2} and
- Nattakorn Phewchean^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1234-y
© The Author(s) 2017
Received: 8 February 2017
Accepted: 7 June 2017
Published: 23 June 2017
Abstract
In this paper, we examine an application of Ornstein-Uhlenbeck process to commodity pricing in Thailand. Prices of Tapioca Starch, Ribbed Smoke Sheet no. 3, and Thai Hom Mali Rice are investigated. We use three parameter estimation methods: least squares estimation, maximum likelihood estimation, and jackknife estimation in order to find the best estimation for the model. Jackknife technique is the most appropriate estimation for our commodity pricing model, which provides the least sum-squared error of commodity prices.
Keywords
Ornstein-Uhlenbeck process stochastic process parameter estimation commodity pricing1 Introduction
In the economics, agricultural commodity prices have an important role due to the cost of production and services. Bayramoglu [1] studied the relationship between agricultural prices and agricultural employment in Turkey by using the VAR method. Results show that there is a relationship between agricultural prices and agricultural employment. Qiangand and Ying [2] investigated the relationship between China’s oil markets and other commodity markets. The results show that China’s fuel oil market is influenced by international oil market and has effect on China’s other commodity markets. Price of given commodity can represent the supply and demand for that commodity, for example, the demand of rice will be low when the price is high. Thus the mathematical model used to analyze the relationship should reflect this difference [3].
In recent years, the commodity markets are rapidly expanding and more interesting to many investors in the financial world. The variety of the future constructs and underlying commodities are alternative choices for investors. There are some important characteristics of commodity price; for example, spot and future prices are mean reverting [4]. Some behaviors of economic variables may be described by mean-reversion process. Since the process suggests that the price or returns usually moves back toward the mean or average in the long run.
The most popular stochastic process that describes the characteristic of the process to drift toward the mean is the Ornstein-Uhlenbeck process [5]. Here, we pay attention to study the Ornstein-Uhlenbeck process and its applications. Many researchers study this area. Ribeiro and Hodges [6] introduced a new model by adding two factors, spot price and convenience yield. Paschke and Prokopczuk [7] constructed the continuous-time autoregressive moving-average (CARMA) model in which the convenience yield follows an Ornstein-Uhlenbeck-type process of pricing the crude oil future market. In this paper, we investigate the Ornstein-Uhlenbeck process behaviors affecting commodity pricing and applying the Ornstein-Uhlenbeck model to pricing the Thai commodity market. There are three types of agricultural future contracts that we are investigating: Tapioca Starch (TS), Ribbed Smoke Sheet no. 3 (RSS3), and Thai Hom Mali Rice 100% grade B (BHMR).
In this research, the analysis of parameters of the Ornstein-Uhlenbeck process are focused upon. The parameter estimation methods we are applying are least squares estimation, maximum likelihood estimation, and maximum likelihood with jackknife estimation.
In this research paper, the content is organized as follows: in the next section, we describe the Ornstein-Uhlenbeck process. Then we apply the parameter estimation technique. After that, we discuss the simulation results of the Ornstein-Uhlenbeck process and parameter estimations. The last section includes conclusion and discussion of the future work.
2 The Ornstein-Uhlenbeck process
3 Parameter estimations
To estimate the parameters of an observed Ornstein-Uhlenbeck process, we use three techniques: least squares estimation, maximum likelihood estimation, and jackknife technique, which may be described as follows.
3.1 Least squares estimation
3.2 Maximum likelihood estimation
3.3 Jackknife technique
4 Simulation result and discussion
Value of parameters used in simulation
λ | μ | σ | |
---|---|---|---|
BHMR | 3.00 | 29.39 | 1.8565 |
RSS3 | 0.60 | 66.95 | 14.8473 |
TS | 4.05 | 8.4639 | 1.0381 |
Then we simulate the future price with the Ornstein-Uhlenbeck process by using a Matlab code written by Smith [10]. The results are shown in the figures below.
Parameter estimation for BHMR
Known parameters | |||
---|---|---|---|
λ | μ | σ | |
Actual | 3.00 | 29.3900 | 1.8565 |
Estimation | |||
---|---|---|---|
\(\boldsymbol{\hat{\lambda}}\) | \(\boldsymbol{\hat{\mu}}\) | \(\boldsymbol{\hat{\sigma}}\) | |
Least squares regression | 3.6724 | 29.7853 | 4.6176 |
Maximum likelihood | 3.6724 | 29.7853 | 4.6128 |
Jackknife technique | 3.0107 | 29.5938 | 4.6865 |
Parameter estimation for RSS3
Known parameters | |||
---|---|---|---|
λ | μ | σ | |
Actual | 0.60 | 66.9500 | 14.8473 |
Estimation | |||
---|---|---|---|
\(\boldsymbol{\hat{\lambda}}\) | \(\boldsymbol{\hat{\mu}}\) | \(\boldsymbol{\hat{\sigma}}\) | |
Least squares regression | 0.7434 | 75.1161 | 16.7193 |
Maximum likelihood | 0.7434 | 75.1161 | 16.6971 |
Jackknife technique | 0.5772 | 78.6778 | 17.5704 |
Parameter estimation for TS
Known parameters | |||
---|---|---|---|
λ | μ | σ | |
Actual | 4.05 | 8.4639 | 1.0381 |
Estimation | |||
---|---|---|---|
\(\boldsymbol{\hat{\lambda}}\) | \(\boldsymbol{\hat{\mu}}\) | \(\boldsymbol{\hat{\sigma}}\) | |
Least squares regression | 3.5025 | 8.0599 | 1.0948 |
Maximum likelihood | 3.5025 | 8.0599 | 1.0926 |
Jackknife technique | 5.1518 | 8.7526 | 1.1069 |
4.1 Behavior with weak mean reversion
We have simulated the stochastic behavior of commodity price with mean reversion equal to 1 (\(\lambda=1\)) to observe the behavior of weak mean reversion. The result is shown below.
4.2 Simulation results with the parameter estimations of λ
4.2.1 BHMR
4.2.2 RSS3
4.2.3 TS
For parameter estimation, least squares regression and maximum likelihood estimation give the same mean reversion value up to 4 decimal places (λ). The tendency of mean reversion process depends on the value of λ. When the value of λ is high, the prices show higher tendency to revert the drift toward the mean.
Sum squared error
Parameters | Least squares regression | Maximum likelihood estimation | Jackknife technique |
---|---|---|---|
BHMR | 1,345.9561 | 1,345.9561 | 1,255.5176 |
RSS3 | 277,508.3873 | 277,508.3873 | 70,589.1459 |
TS | 1,143.7425 | 1,143.7425 | 1,522.6408 |
5 Conclusion
We have presented the use of Ornstein-Uhlenbeck process in pricing Thai commodity and the parameter estimations with least squares estimation, maximum likelihood estimation, and jackknife technique. The pricing models simulated by Matlab shows the trend of the commodity prices toward the mean. So, we can predict the commodity price in the future market by using the method of the Ornstein-Uhlenbeck process. In the parameter estimation, the jackknife technique can be used to reduce the bias of λ estimation. We discover that, in TS product, parameter estimations are close to the real values, but parameter estimation in other products are not very good. For future studies, to improve the methodology, we will consider the influence of economic factors, such as inflation rate, and develop the Ornstein-Uhlenbeck process that incorporates these factors.
Declarations
Acknowledgements
Centre of Excellence in Mathematics, Bangkok, Thailand, and Matlab code supports from William Smith are acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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