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6

The notations in the snapshot are pretty messy. I prefer to proceed as follows. Let $X_t = -\int_t^T f(t, u)du$. Note that \begin{align*} f(t, u) - f(0, u) = \frac{\partial }{\partial u}\left(\int_0^t \frac{\sigma^2(s, u)}{2} ds - \int_0^t \sigma(s, u) d W_s \right). \end{align*} Then \begin{align*} r_t = f(t, t) = f(0, t) + \frac{\partial }{\partial u}\...

2

IIRC, the signs of the PC are meaningless. +/-'ive doesn't itself tell you anything. Rather, the cross-sectional, absolute max of the PCs will tell you which one is most important per item (eg: PC6 looks most important for Beta: M-3). I think 6.6a and 6.6b in Cochrane's asset pricing touch on this (https://www.youtube.com/playlist?list=...

2

It's best to think of the sum of $da_t A_t$ and $da_t dA_t$: $$da_t A_t + da_t dA_t = da_t(A_t + dA_t)$$ which is the cost of rebalancing at the new price $A_{t'} = A_t + dA_t$. You don't rebalance the portfolio at $t$ but at $t'$. And the self-financing condition means you need to finance this cost by rebalancing other assets (including possibly the money ...

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