Why Brownian Cross-Covariances Do Not Multiply

Summary

The document asks whether the instantaneous covariation between two Brownian motions can be found by multiplying their separate covariations with a third motion. It gives a finance context: two stock forwards have stochastic volatility, and the question is how cross stock–volatility correlations relate to same-stock correlations and correlations between stock returns.

The proposed identity is not generally valid. Brownian covariations are entries in a covariance matrix, whose values must satisfy positive semidefiniteness; this imposes bounds and joint constraints but does not usually make one entry the product of two others. Such a product relation can arise under particular dependence assumptions, which would need to be stated and justified. The document supplies no proof, example, or empirical evidence, and it does not specify the dependence structure needed to determine the cross-correlation. Its value is framing a correlation-modeling question rather than presenting a complete derivation.

Key ideas

• Instantaneous Brownian covariations are pairwise entries of a covariance structure.
• A third Brownian motion does not generally make two other motions' covariation equal to a product.
• Valid correlation assumptions must be jointly consistent with a positive semidefinite covariance matrix.
• Cross stock–volatility correlations require additional dependence assumptions to be inferred.

Tags

# On quadratic covariation


# On quadratic covariation












I ran through an equality in a paper I was reading but couldn't check if it is correct.

Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the following holds:

$$\langle dW^1_t, dW^3_t\rangle/dt = \langle dW^1_t, dW^2_t\rangle/dt \times \langle dW^2_t, dW^3_t\rangle/dt$$

If yes, I would like to have some hints to the proof

-- EDIT -- Give more context to the question

Let $F_t^1$ and $F_t^2$ the forward prices of two stocks given by the following SDEs : $\frac{dF_t^i}{F_t^i}=A_i(t,F_t)\sigma^i_t dW_t^{{S},i}$ and let $\sigma_t^i$ follows some simple model such as $\frac{d\sigma_t^i}{\sigma_t^i}=\alpha_t\nu_t dW_t^{{\sigma},i}$.

We seek a relationship between the cross stock/vol correlations, $\langle dW_t^{{S},i}, dW_t^{{\sigma},j} \rangle$ with $i\neq j$, and the other correlations : $\langle dW_t^{{S},i}, dW_t^{{\sigma},i} \rangle$ and $\langle dW_t^{{S},i}, dW_t^{{S},j} \rangle$

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.