# Spot trading: exact mathematical definition of the positions for a portfolio

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 ?

• The position is only determined once you specify a "scale factor" $\lambda$ (which you can choose based on how much money you have/how much risk you want to take). The position consist of $\lambda a$ shares of stock 1 and $\lambda b$ shares of stock 2. Usually a and b will be of opposite sign, one positive, the other negative. HTH. – Alex C Mar 17 '19 at 2:41
• @AlexC I have modified the question. Could you provide an answer including the "scale factor" you mention ? – Arnold Mar 17 '19 at 14:33