## 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 $$\pi_t^\top (C^{-1})^\top {\rm diag}(C\xi \xi^{\top} C^\top) \not= \pi_t^\top {\rm diag}\left(\xi \xi^{\top} \right)$$