Skip to main content
typo in "form"
Link
Richi Wa
  • 13.8k
  • 6
  • 41
  • 91

Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions formfrom prior to posterior

Tweeted twitter.com/#!/StackQuant/status/559039372272885763
Source Link
Richi Wa
  • 13.8k
  • 6
  • 41
  • 91

Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions form prior to posterior

In Black-Litterman we get a new vector of expected returns of the form: \begin{align} \Pi_{BL} = \Pi + \underbrace{\tau \Sigma P^T[P\tau\Sigma P^T+\Omega]^{-1}}_{\text{correction}}[Q-P\Pi] \end{align} where $P$ is the pick matrix and we mix the prior $\Pi$ with the expected value of the views $Q$. $\Sigma$ is the historical covariance matrix and $\Omega$ is the covariance matrix of the views.

Let us assume that $P$ is just the identity matrix and look at the choice $\Omega = \tau\Sigma$, then we see that $$ \Pi_{BL} = \frac12 \Pi + \frac12 Q, $$ thus we have a 50:50 mix and the covariance of the matrix does not affect the posterior at all - it is just a trivial mixture. This is against my intuition. Furthermore optimal weights using this $\Pi_{BL}$ will differ relatively much from optimal weights of the prior (of course depending on $Q$).

If we assume $\Omega = \text{diag}(\tau \Sigma)$ then I can not find a closed form for $\Pi_{BL}$ but appearantly the posterior is more compatible with the prior and the optimal weights are more similar than in the other setting.

My question: how can I choose $\Omega$ best in order to get results that do not deviate too much from my prior? I know that in the literature there are theories (e.g. here The Black-Litterman Model In Detail) but I can't see through. What is used in practice?