Modeling and estimating commodity prices: copper prices

Abstract

A new methodology is laid out for the modeling of commodity prices, it departs from the ‘standard’ approach in that it makes a definite distinction between the analysis of the short term and long term regimes. In particular, this allows us to come up with an explicit drift term for the short-term process whereas the long-term process is primarily driftless due to inherent high volatility of commodity prices excluding an almost negligible mean reversion term. Not unexpectedly, the information used to build the short-term process relies on more than just historical prices but takes into account additional information about the state of the market. This work is done in the context of copper prices but a similar approach should be applicable to wide variety of commodities although certainly not all since commodities come with very distinct characteristics. In addition, our model also takes into account inflation which leads us to consider a multi-dimensional system for which one can generate explicit solutions.

Appendix

1.1 Appendix 1: Approximate solution of the long-term process

The solution of the long-term process is given by,

$$\begin{aligned} x_i^t=x_i^0 \exp \left[ -\left( \mu _i+\frac{1}{2}\sum _{j=1}^J b_{ij}^2\right) (t-t_0)+\sum _{j=1}^J b_{ij} \left( w_j^t-w_j^{t_0} \right) \right] +\mu _i \upsilon _i \int _0^te^{r_i(t,s)ds} \end{aligned}$$

where $r_i(t,s)=-\left[ \mu _i+\frac{1}{2}\sum _{j=1}^J b_{ij}^2\right] (t-s)+\sum _{j=1}^J b_ij \left( w_j^t-w_j^{s}\right)$. We are going to approximate this solution replacing the term $\mu _i \upsilon _i \int _0^t e^{r_i(t,s)ds}$ by its expectation. Then,

$$\begin{aligned} \mathbb {E}\left( \mu _i \upsilon _i \int _0^t e^{r_i(t,s)ds}\right)&= \mu _i \upsilon _i \int _0^t e^{-\left( \mu _i{+}\frac{1}{2} \sum _{j=1}^J b_{ij}^2\right) (t{-}s)}\, \mathbb {E}\left( \exp \left[ \sum _{j{=}1}^J b_{ij} \left( w_j^t{-}w_j^{s}\right) \right] \right) ds\\&=\mu _i \upsilon _i \int _0^t e^{-\left( \mu _i+\frac{1}{2} \sum _{j=1}^J b_{ij}^2\right) (t-s)}\,\mathbb {E}\left( \prod _{j=1}^J \exp \left[ b_{ij} \left( w_j^t-w_j^{s}\right) \right] \right) ds\\&=\mu _i \upsilon _i \int _0^t e^{-\left( \mu _i+\frac{1}{2} \sum _{j=1}^J b_{ij}^2\right) (t-s)} \prod _{j=1}^J \mathbb {E} \left( \exp \left[ b_{ij} \left( w_j^t-w_j^{s}\right) \right] \right) ds \end{aligned}$$

But noting that $\left( w_j^t-w_j^{s}\right)$ is a gaussian process with mean 0 and variance $(t-s)$ we know that,

$$\begin{aligned}&=\mu _i \upsilon _i \int _0^t e^{-\left( \mu _i+\frac{1}{2} \sum _{j=1}^J b_{ij}^2\right) (t-s)} \prod _{j=1}^J \exp \left[ \frac{1}{2} b_{ij}^2 (t-s)\right] ds\\&=\mu _i \upsilon _i \int _0^t e^{-\left( \mu _i+\frac{1}{2} \sum _{j=1}^J b_{ij}^2\right) (t-s)}\exp \left[ \frac{1}{2}\sum _{j=1}^J b_{ij}^2 (t-s)\right] ds\\&=\mu _i \upsilon _i \int _0^t e^{-\left( \mu _i\right) (t-s)}ds\\&=\mu _i \upsilon _i e^{-\mu _i t}\int _0^t e^{\mu _i s}ds\\&=\upsilon _i e^{-\mu _i t}\left( e^{\mu _i t}-1\right) \\&= \upsilon _i \left( 1-e^{-\mu _i t}\right) \end{aligned}$$

Finally, we can approximate the solution of the long-term process to,

$$\begin{aligned} x_i^t=\upsilon _i \left( 1-e^{-\mu _i t}\right) +x_i^0 \exp \left[ - \left( \mu _i+\frac{1}{2}\sum _{j=1}^J b_{ij}^2\right) (t-t_0)+\sum _{j=1}^J b_{ij} \left( w_j^t-w_j^{t_0}\right) \right] \end{aligned}$$

1.2 Appendix 2: Parameter estimation of the short-term process

The SDE that governs the short term process can be written as,

$$\begin{aligned} dS_i^t=\left( \mu _i -\frac{1}{2} \sum _{j=1}^{J}b_{ij}^2\right) dt+\sum _{j=1}^{J}b_{ij}dw_j^t \end{aligned}$$

where $S_i^t=\ln x_i^t$. Then, we know that the $dS_i^t$ follows a gaussian distribution with the following properties (see Dixit [6] and Hull [12]):

$$\begin{aligned} \mathbb {E}\left[ dS_{i}^t\right]&=\left( \mu _i -\frac{1}{2} \sum _{j=1}^{J}b_{ij}^2\right) dt\\ \mathbb {V}\left[ dS_{i}^t\right]&=\sum _{j=1}^{J}b_{ij}^2 dt\\ cov\left[ dS_{i}^t,dS_{k}^t\right]&=\sum _{j=1}^{J}b_{ij}b_{kj}dt \end{aligned}$$

Considering the discrete case we have,

$$\begin{aligned} \mathbb {E}\left[ S_{i}^{t+\Delta t}-S_{i}^t\right]&=\left( \mu _i - \frac{1}{2} \sum _{j=1}^{J}b_{ij}^2\right) \Delta t\\ \mathbb {V}\left[ S_{i}^{t+\Delta t}-S_{i}^t\right]&=\sum _{j=1}^{J} b_{ij}^2 \Delta t\\ cov\left[ S_{i}^{t+\Delta t}-S_{i}^t,S_{k}^{t+\Delta t}- S_{k}^t\right]&=\sum _{j=1}^{J}b_{ij}b_{kj}\Delta t \end{aligned}$$

Then, the easiest method to estimate the parameters of this model is using the fact that $S_i^t=\ln x_i^t$ and historical prices in such a way that,

$$\begin{aligned} \mu _i&=\mathbb {E}\left[ \frac{1}{\Delta t}\ln \left( \frac{x_i^{t +\Delta t}}{x_i^t}\right) \right] +\frac{1}{2} \sum _{j=1}^{J}b_{ij}^2\\ \sum _{j=1}^{J}b_{ij}^2&=\mathbb {V}\left[ \frac{1}{\sqrt{\Delta t}}\ln \left( \frac{x_i^{t+\Delta t}}{x_i^t}\right) \right] \\ \sum _{j=1}^{J}b_{ij}b_{kj}&=cov\left[ \frac{1}{\sqrt{\Delta t}} \ln \left( \frac{x_i^{t+\Delta t}}{x_i^t}\right) ,\frac{1}{ \sqrt{\Delta t}}\ln \left( \frac{x_k^{t+\Delta t}}{x_k^t}\right) \right] \end{aligned}$$

Another way to estimate these parameters is recalling that, for $i\in \{p,r\}$

$$\begin{aligned} \mathbb {E}[x_{i}^t]=x_{i}^{0}e^{\mu _{i} t}, \end{aligned}$$

where $\mu _i$ is the drift and $x_i^0$ the initial value of index $i$.

Then, we estimate $\mu _i, \; i \in \{p,r\}$ and the initial state denoted by $\theta _i, \; i \in \{p,r\}$. Estimating the initial state is very important because in most applications is used the actual spot price as initial condition, forgetting that this also has noise as it is a random variable.

Finally, assuming that the errors in the observations ($x_i^t$) come from white noise around the drift term $\mu _i t$, one has

$$\begin{aligned} x_i^t=\theta _i e^{\mu _i t+\varepsilon _i^t}, \quad t\in T \end{aligned}$$

The main idea of this approach is to minimize the error associated to the estimation. For doing so, we are going to minimize $\sum _{t\in T} |\varepsilon _i^t|^2$, i.e.,

$$\begin{aligned} \left( \hat{\theta }_i, \hat{\mu }_i\right) \in {\mathrm{argmin }} \left( \theta _i, \mu _i\right) \sum _t\left| \mu _i t-\ln \left( \frac{x_i^t}{\theta _i}\right) \right| ^2 \end{aligned}$$

Differentiating with respect to $\theta _i$ and $\mu _i$ we get,

$$\begin{aligned} \frac{d\upsilon _i}{d\theta _i}&=2\sum _t\left( \ln \left( \frac{x_i^{t}}{\theta _i}\right) -\mu _i t\right) \frac{1}{\theta _i}\\ \frac{d\upsilon _i}{d\mu _i}&=2\sum _t\left( \ln \left( \frac{x_i^{t}}{\theta _i}\right) -\mu _i t\right) t\\ \end{aligned}$$

Setting these derivatives equal to 0 we obtain,

$$\begin{aligned} \frac{d\upsilon _i}{d\theta _i}=0 \Rightarrow \mu _i = \frac{\sum _{t\in T} \ln \left( \frac{x_i^{t}}{\theta _i}\right) t}{\sum _{t\in T} t^2}, \qquad \frac{d\upsilon _i}{d\theta _i}=0 \Rightarrow \mu _i = \frac{\sum _{t\in T} \ln \left( \frac{x_i^{t}}{\theta _i}\right) }{\sum _{t\in T} t}. \end{aligned}$$

Solving the system and denoting $a=\sum _{t\in T} t$ and $b=\sum _{t\in T} t^2$ we obtain, for $i\in \{p,r\}$

$$\begin{aligned} \hat{\theta }_i&=\exp \left( \left( a^2-b\eta \right) ^{-1}\sum _t (at-b)\ln (x_i^{t})\right) \\ \hat{\mu }_i&=b^{-1}\left( \sum _{t\in T} t\ln \left( \frac{x_i^{t}}{\hat{\theta }_i}\right) \right) \\ \end{aligned}$$

where $\eta$ is the number of observation, i.e., if we consider just the historical information of the last 12 months $\eta =13$.

Covariance matrix To estimate the covariance matrix with this method we know that,

$$\begin{aligned} cov\left\{ x_i^t, x_j^t\right\} =x_i^0 x_j^0 e^{(\mu _i+ \mu _j)t}\left[ \exp \left( t\sum _{k=1}^{J}b_{ik} b_{jk}\right) -1\right] \end{aligned}$$

Assuming that observations are corrupted by a white noise $\varepsilon _{kl}^t$ that affects $|t|\sum _{j\in \{p,r\}}b_{kj}b_{lj}$ and recalling that $\hat{x}_k^t=\hat{\theta }_k e^{\hat{\mu }_k|t|}$, i.e., for $t=-12,\ldots ,0$,

$$\begin{aligned} \left( x_k^{t}-\hat{x}_k^{t}\right) \left( x_l^{t}-\hat{x}_l^{t} \right) =\hat{\theta }_k\hat{\theta }_l e^{(\mu _k+ \mu _l)t} \left[ \exp \left( |t|\sum _{j\in \{p,r\}}b_{kj}b_{lj}+ \varepsilon _{kl}^t\right) \right] \end{aligned}$$

Then, seeking estimates that minimize $\sum _t |\varepsilon _{kl}^t|^2$, one obtains the estimate $\hat{\beta }_{kl}$ for $\sum _{j\in \{p,r\}}b_{kj}b_{lj}$:

$$\begin{aligned} \hat{\beta }_{kl}=\frac{\sum _t |t| \ln \left[ 1+\frac{\left( x_k^{t}- \hat{x}_k^{t}\right) \left( x_l^{t}-\hat{x}_l^{t}\right) }{\hat{x}_k^{t} \hat{x}_l^{t}}\right] }{\sum _t t^2} \end{aligned}$$

Thus, the estimate for $cov\left( x_p^{t}, x_r^{t}\right)$ is,

$$\begin{aligned} \hat{\sigma }_{pr}^{t}=\hat{\theta }_p\hat{\theta }_r e^{(\mu _p+ \mu _r)|t|} \left( e^{\hat{\beta }_{pr}|t|}-1\right) \end{aligned}$$

and the variance, for $k\in \{p,r\}$,

$$\begin{aligned} \hat{\sigma }_{kk}^{t}=\hat{\theta }_k^2\left( e^{\hat{\beta }_{kk}|t|}-1\right) \end{aligned}$$