Conditional Covariance Decomposition Using Forward Expiration Values
Summary
The document examines whether the conditional covariance of two assets at a later date can be decomposed by conditioning on their forward values at an intermediate expiration. It starts from the usual expression for covariance as the conditional expectation of the product minus the product of conditional means, then presents a proposed decomposition into expected conditional covariance and covariance of conditional expectations. The discussion distinguishes an intermediate forward settlement time from the later terminal date and considers the case where the intermediate time lies between the observation and terminal dates.
The response sketches an argument using conditional expectations and forward prices, with simplifying assumptions that the asset values at the intermediate date equal the corresponding forwards. It concludes that the identity does not require continuous Gaussian paths, despite that assumption in the question. The derivation is informal and contains apparent notation or algebra inconsistencies, including the stated final sign relative to the covariance definition, so the claim should be checked carefully before use. No empirical evidence is offered.
Key ideas
- Conditional covariance can be decomposed using the law of total covariance.
- The proposed conditioning variables are the two assets’ forward values at an intermediate expiration.
- The argument assumes the intermediate asset values coincide with the relevant forward prices.
- The response claims the decomposition does not depend on continuous Gaussian dynamics.
- The displayed derivation has apparent algebraic or notation issues and warrants independent verification.
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Full text
# Decomposing Co-variance of Two Assets Terminal Prices into Forward measures
# Decomposing Co-variance of Two Assets Terminal Prices into Forward measures
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 by Rashad (score 0)
https://quant.stackexchange.com/a/53301
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.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.