Cointegration Between Venue Prices with Random Delays

Summary

The document explores when prices from two trading venues can be cointegrated. In its first setup, both prices consist of a shared efficient price that follows Brownian motion plus separate stationary noise. Although each observed price has a unit root, subtracting one from the other removes the common stochastic trend, leaving a stationary spread. This illustrates how shared price movements can produce a cointegrating relationship.

It then considers a different setup in which one venue reports the efficient price without noise and the second mirrors it after a potentially random delay. The resulting price difference is the efficient price’s movement over the delay interval, and the question asks when that spread is stationary, including whether the delay itself must be stationary. The document poses this theoretical question but gives no answer or empirical evidence. Its formulas motivate analysis of delay behavior; they do not establish sufficient conditions for cointegration.

Key ideas

• A shared stochastic trend plus separate stationary price noise can yield a stationary difference between venue prices.
• Each venue’s price may be nonstationary even when the price spread is stationary.
• With a delayed mirror price, the spread reflects efficient-price changes over the delay interval.
• The document asks how delay dynamics affect stationarity but does not resolve the question.

Tags

# Law of one Price and Cointegration relationship


# Law of one Price and Cointegration relationship












I have a question on the relationship between the law of one price and cointegration of (financial) time series.

To set things clear I start with something simple:

- Suppose there is an unobserved "efficient" price $v_t$ that follows a standard Brownian motion such that $$v_t = v_0 + \int\limits_{0}^t \sigma dW_s$$.

- Two venues trade the asset and quote prices ($x_t$ and $y_t$) which reflect the efficient price and some stationary noise component: $$x_t = v_t + u_t \text{ and } y_t = v_t + \tilde u_t$$

- Then, both $x_t$ and $y_t$ exhibit a unit root but $z_t := x_t - y_t$ resembles a valid cointegration relationship because $z_t = u_t + \tilde u_t$ which is stationary.

However, I have trouble understanding the concept if things are different:

- Suppose, there is no noise at one market such that $x_t = v_t$.

- The other market simply mirrors the prices but with a possibly random lag such that $$y_t = x_{t-\tau} = v_{t-\tau}$$

- Then, under which circumstances is $z_t = x_t - y_t = \int\limits_{t-\tau}^t \sigma dW_s \sim N(0, \sigma^2\tau)$ again a stationary variable and resembles a reasonable cointegration relationship? Is it necessary that $\tau$ is stationary or are there other sufficient conditions?

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