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Estimating Cross-Commodity Shock Correlations in Multi-Factor Futures Models

Article Quant Q&A · Author: user1141785
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Summary

The document describes two-factor models for electricity and gas futures, with each commodity's forward-price changes represented by volatility loadings on two Brownian shocks. Principal component analysis is used to estimate each market's volatility structure, yielding independent factors within each commodity. The modeling challenge is to extend the framework across commodities by estimating correlations between their shocks.

The author considers inferring two latent shocks from one observed return per commodity and asks whether their standard normal and independence assumptions make that possible. The text offers no solution or empirical evidence. In particular, it does not explain how factor alignment, sign ambiguity, or estimation uncertainty would affect cross-market shock correlations. It is a problem formulation about connecting separately estimated factor models, rather than a demonstrated estimation procedure.

Key ideas

  • Each commodity's futures returns are modeled as a weighted combination of two Brownian shocks.
  • Principal component analysis supplies within-market factors that are treated as independent.
  • Cross-commodity modeling requires estimating correlations between the electricity and gas shocks.
  • A single return observation does not directly identify two underlying shocks without additional structure.

Tags

FuturesCommoditiesStatistics
Full text
# Estimating two normal random numbers with one equation


# Estimating two normal random numbers with one equation












Subtitle: Estimating the correlation of the shocks driving two commodities in two multi-factor models

I am fitting two 2-factor models to electricity and gas futures, respectively.

In order to extend this framework to a multi-commodity model I have to integrate correlation.

The models are (HJM-type model from forward interest rate models):

$\frac{dF_{E} (t,T)}{F_{E} (t,T)} =σ_{1,E} (t,T) dW_{1,E} (t)+σ_{2,E} (t,T) dW_{2,E} (t) $

$ \frac{dF_{G} (t,T)}{F_{G} (t,T)}=σ_{1,G} (t,T) dW_{1,G} (t)+σ_{2,G} (t,T)dW_{2,G} (t) $

where $dW$ are Brownian Motion increments and $F(t,T)$ are the futures prices at $t$ with maturity $T$ for electricity (E) and gas (G). That is, the logarithmic return on the futures is modeled (fraction on the left side of equation).

I estimate the volatility functions σ via Principal Component Analysis, so $dW_{1,E} (t)$ and $dW_{2,E} (t)$ are independent by definition. The same applies for the two Brownian Motions driving the futures price changes for gas futures.

But since electricity and gas futures prices are correlated, I would like to model this correlation by correlating the Brownian Motions.

I am looking for 4 correlation coefficients:

$dW_{1,E} (t) $ to $ dW_{1,G} (t) $

$dW_{1,E} (t) $ to $ dW_{2,G} (t) $

$dW_{2,E} (t) $ to $ dW_{2,G} (t) $

$dW_{2,E} (t) $ to $ dW_{1,G} (t) $

Now I wanted to estimate which shocks $dW$ led to a specific original (observed) return series $\frac{dF(t,T)}{F(t,T)}$ and then calculate how these shocks are correlated. Then I would just use BMs with that correlations in my model. The problem I have is that I have two shocks to estimate, for electricity $dW_{1,E} (t)$ to $dW_{2,E} (t)$, but only one return data point and thus only one equation. Nevertheless, I know $dW_{1,E} (t)$ to $dW_{2,E} (t)$ are independent standard normally distributed. Does that help?

Does anyone have any ideas how to solve that equation for the shocks?

Or any ideas how else the correlation could be integrated into this model?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.