Decomposing Co-variance of Two Assets Terminal Prices into Forward measures

Question

Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} = E_t[X_T],F^Y_{t,T} = E_t[Y_T]$.

We know then the following equation is True:

$cov_t(X_T,Y_T) = E_t[X_TY_T] - F_{t,T}^XF_{t,T}^Y$

I read in a recent text that it is possible to further decompose this conditional co-variance as:

[EDIT: I misread the original text] $cov_t(X_T,Y_T) = E_t[cov(X_T,Y_T|F_{exp}^X,F_{exp}^Y)]+cov_t(E[X_T|F^X_{exp}],E[Y_T|F^Y_{exp}])$

Where "exp" refers to the expiration/settlement time of the forward. Although the author does not state it explicitly, I presume $t>\text{"exp"}$

My goal is to try to determine if the claim the 2nd equation makes holds true?

Answer 1

I worked through this for a bit, and I think I know how to show this now:

Letting $T^* = \text{"exp"}$ and $t < T^* < T$, if we make the simple structural assumptions $X_{T*} = F^X_{T^*}, Y_{T^*} = F^Y_{T^*}$, then we rewrite the right hand side of the equation as

\begin{align*} E_t[E_{T^*}[F^X_{T,T}F^Y_{T,T}] - F^X_{T^*,T}F^Y_{T^*,T}] + E_t[E_{T^*}[F^X_{T,T}]E_{T^*}[F^Y_{T,T}]] + E_t[E_{T^*}[F^X_{T,T}]]E_t[E_{T^*}[F^Y_{T,T}]] \\ = E_t[E_{T^*}[F^X_{T,T}F^Y_{T,T}] - F^X_{T^*,T}F^Y_{T^*,T}] + E_t[F^X_{T^*,T}F^Y_{T^*,T}]] + E_t[F^X_{T^*,T}]E_t[F^Y_{T^*,T}] \\ = E_t[F^X_{T,T}F^Y_{T,T}] + F^X_{t,T}F^Y_{t,T} \text{ OR we can write this as,}\\ = E_t[X_TY_T] + E_t[X_T]E_t[Y_T] \end{align*}

Which is equivalent to $cov_t(X_T,Y_T)$.

Therefore we can decompose the co-variance of our X and Y process as such. This also shouldn't depend on whether X,Y are continuous Gaussian processes.