Defining Long, Short, and Liquidation in a Cointegrated Stock Pair

Summary

The document sets up a spot portfolio containing two stocks whose prices are related by a proposed cointegration expression. It assigns coefficients to the two prices, defines a position indicator that can take long, flat, or short values, and writes the total marked value as the position-scaled stock combination plus cash. The example uses one positive and one negative coefficient to represent the pair exposure.

It then asks for precise mathematical meanings of going long or short the portfolio and liquidating it. No answer or trading rules are provided. In particular, the text does not specify whether the coefficients represent share quantities, dollar weights, or a normalized hedge ratio, nor how cash, transaction costs, borrow, or execution should be handled. The setup is useful for clarifying signed pair positions, but the requested operational definitions remain open.

Key ideas

• The proposed pair portfolio combines two stock prices using signed coefficients.
• The portfolio’s marked value is expressed as its position-scaled stock exposure plus cash.
• The author asks how to define long, short, and liquidation mathematically.
• The document does not specify whether the coefficients are share quantities, dollar weights, or normalized weights.

Tags

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


# 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 ?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.