# 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?

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Maybe this will help. If we have that $\langle W_1,W_2\rangle = \rho t$ then we can write that $dW_2(t) = \rho dW_1(t) + \sqrt{1-\rho^2}dW_1^\perp (t)$, where $dW_1^\perp(t)$ is a perpendicular brownian motion. –  Jase Dec 12 '12 at 0:56