Since two decades ago, when the early model of oil futures prices developed by Brennan and Schwartz (1985), there has been a vast change in oil markets arising from both economical and political factors, oil markets are more volatile now compared with twenty or even ten years ago. To understand the behavior of the oil market there is a need to understand the stochastic models of oil prices. Over the last two decades different models have been proposed to justify the stochastic behavior of oil futures contracts. The latest model has been the parsimonious three-factor model by Cortazar and Schwartz (2003). The goal of this article is analysis of the parsimonious three-factor modeling of oil futures prices and the solutions to the futures contracts evaluation formula under the risk neutral measure, as well as the volatility term structure model of futures returns.
Before developing a parsimonious three-factor model by Cortazar and Schwartz, some other models had been proposed; however, these models were not satisfactory enough. The first model (Brennan & Schwartz 1985) assumed a geometric Brownian motion and a constant convenience yield but very soon it was replaced by a one-factor mean-reverting model. After that a more realistic model which was a two-factor model with mean-reverting convenience yield was proposed by Gibson & Schwartz (1990), and some other versions of that were proposed in 1997 and 2000 by different researchers; however, these models as Schwartz and Cortazar say adopted rather slowly by practitioners. Difficulties of fitting well model to data for some days showed the need for a three-factor model, thus the parsimonious three-factor model developed by Cortazar and Schwartz in 2003. Before this model, the latest three factor model had been proposed by Schwartz (1997), in that model the third factor was the stochastic interest rate of return r.
The latest three-factor model (2003) is based on the parsimonious two-factor model (Cortazar & Schwartz 2003). The best aspects of this three-factor model are: firstly, all information is from the futures prices in contrast with the three-factor model (1997), secondly, instead of using long-term (unobservable) convenience yield, long term price appreciation is used. In this study, by analyzing the two and three-factor modeling of oil futures prices and using mathematical and financial definitions, analytical solutions to evaluate futures contracts on the commodity have been derived.
In this article, after considering the definitions of some financial and mathematical terms in Section 2, the first part of Section 3 focuses on the two-factor modeling of oil futures prices (Schwartz 1997) as well as the analytical solution for evaluation of futures prices. Afterwards, there is a brief description on the three-factor modeling of oil futures prices (Schwartz 1997) where a stochastic r (the stochastic rate of interest) as a third factor has been included in the model. The third part of Section 3 is an introduction to parsimonious two factor modeling (Cortazar & Schwartz 2003) which is the basis of the three-factor modeling of oil futures prices mentioned in Section 4, and the fifth section of this essay pays attention to the three-factor modeling of oil futures prices as well as futures prices evaluation with analytical results to the related 3-dimensional pricing equation. Moreover, the volatility term structure model of futures returns proposed by Cortazar & Schwartz (2003) will be considered in this section too.
Contents
1 Introduction
2 Definitions of some mathematical terms used in this paper
2.1 Wiener process
- 2.1.1 One-dimensional Wiener process
2.1.2 n-dimensional (independent) Wiener process
2.1.3 n-dimensional (correlated) Wiener process
2.2 Ito’s lemma
- 2.2.1 One-dimensional Ito’s lemma
2.2.2 Multi dimensional Ito’s lemma for independent Wiener processes
2.2.3 Multi dimensional Ito’s lemma for correlated Wiener processes
2.3 Black-Scholes equation
- 2.3.1 One dimensional Black-Scholes equation
2.3.2 Multi dimensional Black-Scholes equation
2.4 Feynman-Kac theorem
2.4.1 Feynman-Kac theorem when our processes are independent
2.4.2 Feynman-Kac theorem when our processes are correlated
2.5 Pricing equation
2.6 Ornstein-Uhlenbeck process (Gauss Markov process)
2.7 Stochastic integrals
2.8 Risk neutral valuation
3 The two-factor modeling of oil futures prices and two alternatives for its improvement
3.1 The two-factor modeling and valuation of oil futures prices with the analytical result
3.2 A brief description on the three-factor modeling of oil futures prices when has a stochastic process (Schwartz 1997)
3.3 The parsimonious two-factor modeling of oil futures prices (Schwartz 2003)
4 The three-factor modeling of oil futures prices
4.1 The three-factor modeling and valuation of oil futures prices with the analytical result (Schwartz 2003)
4.2 The volatility term structure of futures returns
5 Conclusion
References
