Let us say that I want to spot trade a portfolio constituted of a pair of two stocks of respective prices (for example in USD) $S^1_t$ and $S^2_t$, and suppose for example that they co-integrate according to the relation:
$\varepsilon_t$ = $a$ $S^{(1)}_t$ + $b$ $S^{(2)}_t$
where $\varepsilon_t$ is the co-integration factor.
If for example $a$ $=$ $0.6$ and $b$ $=$ $-0.3$, and if $\alpha_t$ units of the portfolio are owned (for simplicity $\alpha_t \in \{-1, 0, 1\})$, the total value owned $\Sigma_t$ writes:
$\Sigma_t = \alpha_t (0.6\ S^{(1)}_t -0.3 \ S^{(2)}_t) + C_t$,
where $C_t$ is the cash in USD.
What does it exactly (mathematically) mean to
1) Short the portfolio,
2) Long the portfolio,
3) Liquidate the portfolio ?
