Saturday, December 29, 2012

I'm missing something about contragredient transformations and portfolio return

This entry probably just indicates that I am missing something obvious about portfolio return.
The instantaneous return \(\gamma^{\pi}_t\) on a portfolio with weights \(\pi\) decomposes as $$ \gamma^{\pi}_t = \left< \frac{dZ_t}{Z} \right>_t = \pi_t^\top \gamma_t + \underbrace{\frac{1}{2} \left( \pi_t^\top diag(\xi \xi^{T}) - \overbrace{\pi^{T}_t \xi \xi^{T} \pi_t }^{portfolio\ variance} \right)}_{excess\ return\ process} $$ where \(\gamma_t^i\) is the drift of \(\log(X^i_t)\), the log of the \(i\)'th asset and the \(i\)'th row of \(\xi\) represents the factor decomposition of the diffusion into \(n\) independent Brownian motions comprising a vector \(dW_t\). $$ d (log(X^i_t)) = \gamma^i_t dt + \xi dW_t $$ Consider the map taking vectors \(x \mapsto y = \exp(C \log(x)) \) and thereby defining a new asset vector \(Y_t\). That is, \(Y_t\) is related to \(X_t\) by a simple matrix multiplication in log coordinates.
We consider portfolios of the new assets \(Y^i\) with weights \(\varpi_i\) say. And again, the portfolio return is related to the asset return via $$ \frac{dZ^\varpi}{Z^{\varpi}} = \varpi^\top \frac{dY_t}{Y_t} $$ where the fractions indicate coordinate-wise (i.e. pointwise) division. Let us suppose further that any instantaneous portfolio return using assets \(\{X^i\}\) can be replicated by a portfolio using assets \(\{Y^i\}\) only, and vice versa. $$ \varpi^\top \frac{dY_t}{Y_t} = \frac{dZ^\varpi}{Z^{\varpi}} = \frac{dZ^\pi}{Z^{\pi}} = \pi^\top \frac{dX_t}{X_t} $$
By a slightly heavy handed application of the multivariate Ito's Lemma (hey it's good to have it lying around) with \(g(x) = C x\) we have $$ \begin{eqnarray} d (log Y_t) & = & \frac{\partial g}{\partial t} dt + \left(\nabla g\right)^\top d \left( \log (X_t) \right) + \frac{1}{2} \left(d(\log X_t)\right)^\top \left(\nabla^2 g \right) \left(d(\log X_t)\right) \\ & = & 0 + C d \left(\log (X_t)\right) + 0 \\ & = & C\gamma dt + C\xi dW_t \\ \end{eqnarray} $$ so writing the decomposition of portfolio return with \(C\gamma\) in place of \(\gamma\) and \(C\xi\) in place of \(\xi\) we observe $$ \begin{eqnarray} \left< \frac{dZ^{\varpi}_t}{Z^{\varpi}} \right>_t & = & \varpi_t^\top C\gamma_t + \frac{1}{2} \left( \varpi_t^\top diag(C\xi \xi^{\top}C^\top) - \varpi^\top_t C \xi \xi^\top C^\top \varpi_t \right) \\ & = & \pi_t^\top \gamma_t + \frac{1}{2} \left( \pi_t^\top (C^{-1})^\top diag(C\xi \xi^{\top} C^\top) - \pi^\top_t \xi \xi^\top \pi_t \right) \end{eqnarray} $$ if we use contragredient weights \(\varpi_t = (C^\top)^{-1} \pi_t\). But does the contragredient choice actually result in the same drift? The linear and portfolio variance terms are the same, but on the other hand $$ \pi_t^\top (C^{-1})^\top {\rm diag}(C\xi \xi^{\top} C^\top) \not= \pi_t^\top {\rm diag}\left(\xi \xi^{\top} \right) $$

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