On the Stochastic Properties of Carbon Futures Prices

Abstract

Pricing carbon is a central concern in environmental economics, due to the worldwide importance of emissions trading schemes to regulate pollution. This paper documents the presence of small and large jumps in the stochastic process of the CO$_2$ futures price. The large jumps have a discrete origin, i.e. they can arise from various demand factors or institutional decisions on the tradable permits market. Contrary to the existing literature, we show that the stochastic process of carbon futures prices does not contain a continuous component (Brownian motion). The results are derived by using high-frequency data in the activity signature function framework (Todorov and Tauchen in J Econom 154:125–138, 2010; Todorov and Tauchen in J Bus Econ Stat 29:356–371, 2011). The implication is that the carbon futures price should be modeled as an appropriately sampled, centered Lévy or Poisson process. The pure-jump behavior of the carbon price might be explained by the lower volume of trades on this allowance market (compared to other highly liquid financial markets).

Appendix: Key Properties of Stochastic Processes

In what follows, we provide a brief overview of the key characteristics of stochastic processes, in continuous and jump diffusion settings. By doing so, we build on the notations by Hanson (2007), Knill (2009) and Kroese et al. (2011).

The interested reader may refer to advanced texts in Kannan (1979), Karatzas and Shreve (1997), Rolski et al. (1999), Schoutens (2003), Cont and Tankov (2004), Ghahramani (2005), Stirzaker (2005), Speyer and Chung (2008), Feldman and Valdez-Flores (2010), or Barndorff-Nielsen and Shephard (2012).

1.1 Continuous Diffusion

Let $\Omega$ be a probability space, and let $T \subset \mathbb R$ be time. A collection of random variables $X_t, t \in T$ with values in $\mathbb R$ is called a stochastic process.

If $X_t$ takes values in $S=\mathbb R ^d$, it is called a vector-valued stochastic process (but often abbreviates by the name stochastic process too).

If the sample function $X_t(\omega )$ is a continuous function of $t$ for almost all $\omega \in \Omega$, then $X_t$ is called a continuous stochastic process.

Let us start with the Brownian motion³² as a fundamental example of an important stochastic process which does not feature mean reversion.

1.1.1 Brownian Motion

An $\mathbb R ^d$-valued continuous Gaussian process $X_t$ with mean vector $m_t=E[X_t]$ and covariance matrix $V(s,t)=Cov (X_s,X_t)=E[(X_s-m_s)\cdot (X_t-m_t)]$ is called Brownian motion if for any $0\le t_0 < t_1 < \cdots < t_n$, the random vectors $X_{t_0}, X_{t_{i+1}}-X_{t_i}$ are independent and the covariance matrix $V$ satisfies $V(s,t)=V(r,r)$, where $r=\min (s,t)$ and $s \rightarrow V(s,s)$. It is called the standard Brownian motion if $m_t=0$ for all $t$ and $V(s,t)=\min \{s,t\}$.

A numerical example of a Brownian motion computed for 100 observations, $r=0.02$ with a drift $m_t=\sqrt{0.1}$ is pictured in Fig. 5.

Fig. 5

Numerical illustration of Brownian motion

Define the process $B_t=X([0,t])$. For any sequence $t_1,t_2,\cdot \cdot \cdot \in T$, this process has independent increments $B_t-B_{t-1}$ and is a Gaussian process. For any $x \in \mathbb R$, the process:

$$\begin{aligned} X_t=x+B_t \end{aligned}$$

(5)

is called Brownian motion started at $x$.

Furthermore, if $B_t$ is a Brownian motion, then $X=f(B,t)$ can be written in differential form as:

$$\begin{aligned} d X_t=\alpha X_t dt+ \beta d M_t, \quad X_0=1 \end{aligned}$$

(6)

with $\{\alpha ,\beta \}$ constants, and $M$ a continuous martingale of finite variation. This is an example of a stochastic differential equation (SDE). Unlike ordinary differential equations, one has to use Ito’s formula to integrate when looking for solutions.

Next, we proceed with a generalization of stochastic processes featuring the mean reversion property.

1.1.2 Ornstein–Uhlenbeck Process

The mean reversion process can be considered as a modification of a random walk, where the alterations to the process do not occur entirely independently.

The Ornstein–Uhlenbeck process has the property that it is mean-reverting, i.e. it always tries to come back to its asymptotic mean value. For this reason, it is also called the oscillatory process. The Brownian motion $B_t$ and the Ornstein–Uhlenbeck process $O_t$ are for $t \ge 0$ related by:

$$\begin{aligned} O_t=\displaystyle \frac{1}{\sqrt{2}}e^{-t}B_{e^{2t}} \end{aligned}$$

(7)

The corresponding SDE is obtained by writing:

$$\begin{aligned} d X_t=-\tau X_t dt+ \beta d B_t \end{aligned}$$

(8)

with the parameter $\tau >0$ governing the rate of mean-reversion, and $\beta$ a constant.

An example of the path for the Ornstein–Uhlenbeck process with mean reversion rate $\tau = 0.1$ and diffusion constant $\beta = 0.03$ is given in Fig. 6. The numerical method used here was published by Gillespie (1996).

Fig. 6

Numerical illustration of Ornstein–Uhlenbeck process

1.1.3 Vasicek Process

The Vasicek process is very close to the Ornstein–Uhlenbeck process presented above. This process is a diffusion process, which leads to the following closed form formula:

$$\begin{aligned} dX_t=\kappa ({\theta }-X_t)dt + \beta d B_t \end{aligned}$$

(9)

where the parameters $\kappa , \theta$ and $\beta$ are constants, and the random motion is generated by the Brownian motion $B_t$. An important property of the Vasicek process is that the mean is reverting to $\theta$, and the tendency to revert is controlled by $\kappa$.

To give an idea of what the Vasicek process looks like, we have generated a sample path in Fig. 7 by plugging the values $\theta =0.014, \kappa =0.161$ and $\beta =0.03$.

Fig. 7

Numerical illustration of Vasicek process

1.2 Jump Diffusion

In the pure diffusion stochastic model, there is one obvious missing feature that large market fluctuations or crashes / rallies—which characterize the market’s bullish or bearish trends—are not represented. These discontinuities highlight the statistical importance of including jumps in financial market models.³³ That is why we present below jump diffusion process models.

1.2.1 Poisson Process

A Poisson process $P_t$ with rate $\lambda$ verifies the following property:

$$\begin{aligned} P_t= \text {Number of occurrences in} \ [0,t) \sim P_0 (\lambda t) \end{aligned}$$

(10)

The differential of a simple Poisson counting process satisfies:

$$\begin{aligned} d P_t=\lambda dt \end{aligned}$$

(11)

with $\lambda >0$ and initial conditions $P(0^{+})=0$ with probability one. The simplest approach to view the Poisson processes is to consider these differential stochastic processes as increments, i.e.:

$$\begin{aligned} d P_t = P(t+dt)-P(t) \end{aligned}$$

(12)

for infinitesimal increments in time $dt$. We verify that the Poisson process $P_t$ is quite different from continuous diffusion processes, primarily because of its discontinuity property, and the property that multiple jumps are highly unlikely during small increments of time $d t$.

The sample path of a Poisson process with $\lambda =2$ is displayed in Fig. 8.

Fig. 8

Numerical illustration of poisson process

1.2.2 Bernoulli Process

The Bernoulli process is the discrete time counterpart of the Poisson process. It consists of finite or infinite sequence of independent random variables $X_t, t=1,\ldots ,T$ such that:

$$\begin{aligned} X_t = \left\{ \begin{array}{rl} 1 &{} \text{ with } prob=p \\ -1 &{} \text{ with } prob=1-p \end{array} \right. \end{aligned}$$

In practice, this model corresponds to a regular sampling period $S$ for which observations are missing at random, with failure probability $1-p$. The regular sampling period corresponds to $p=1$. Random variables associated with the Bernoulli process include:

• The number of successes in the first $n$ trials: this has a binomial distribution.

• The number of trials needed to get $r$ successes: this has a negative binomial distribution.

• The number of trials needed to get one success: this has a geometric distribution, which is a special case of the negative binomial distribution.

As an example, the numerical simulation of a Bernoulli process with $p = 0.5$ from a discrete uniform $X_t\sim \mathcal U _d(-1,1)$ is given in Fig. 9.

Fig. 9

Numerical illustration of Bernoulli process

1.2.3 Lévy Process

Lévy processes can be thought of as a combination of a diffusion process and a jump process. Both Brownian motion (i.e. a pure diffusion process) and Poisson processes (i.e. pure jump processes) are Lévy processes. As such, Lévy processes represent a tractable extension of Brownian motion to infinitely divisible distributions. In addition, Lévy processes allow the modeling of discontinuous sample paths, whose properties match those of empirical phenomena such as financial time series.

There have been many efforts to apply Lévy processes, such as the CGMY model (Carr et al. 2003), the variance gamma (VG) model (Carr and Madan 1999), and the Normal Inverse Gaussian (NIG) model (Rydberg 1997).

A $d$-dimensional Lévy process is a stochastic process $\{X_t,t \ge 0\}$ taking values in $\mathbb R ^d$ with the following properties:

• 1. Independent increments: For any $t_1 < t_2 \le t_3 < t_4$, the random variables $X_{t_4}-X_{t_3}$ and $X_{t_2}-X_{t_1}$ are independent.

• 2. Stationarity: The law of $X_{t+h}-X_t$ does not depend on $t$.

• 3. Stochastic continuity: when the process coefficients are not constant, then the process will in general not be stationary, as the preceding condition requires. For many real problems, such as in financial markets, the time-dependence of process coefficients is important (Hanson and Westman 2002).

• 4. Zero initial value: $X_0=0$ almost surely.

A Lévy process can be seen as a continuous time generalization of a random walk process. Indeed, the process observed at time $0=t_0<t_1<t_2<\ldots$ forms a random walk:

$$\begin{aligned} X_{t_n}=\displaystyle \sum _{i=1}^{n} (X_{t_i}-X_{t_{i-1}}) \end{aligned}$$

(13)

whose increments $\{X_{t_i}-X_{t_{i-1}}\}$ are independent. Let $N([0,t]\times A)$ denote the number of jumps of $X$ during the interval $[0,t]$ whose size lies in the ensemble $A$, excluding $0$. Let $\Delta X_t$ denote the size of the jump of the process at time $t$. The measure $\nu$ defined by:

$$\begin{aligned} \nu (A)=N([0,1]\times A), \quad \{t \in [0,1]: \Delta X_t \ne 0, \Delta X_t \in A\} \end{aligned}$$

(14)

is called the Lévy measure of $X_t$. The random measure $N(dt,dx)$ is called the jump measure. We observe that Lévy processes are essentially jump-diffusion processes, but are extended to processes with infinite jump-rates.

A numerical simulation of a Lévy process with $\nu =0.01$ can be observed in Fig. 10.

Fig. 10

Numerical illustration of Lévy process