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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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  • $\begingroup$ It's interesting but there's no stochastic process in your second example so I don't think one can define cointegration in that case. This is because $y_t$ is deterministic. $\endgroup$
    – mark leeds
    Commented Oct 20, 2019 at 14:01
  • $\begingroup$ Thanks @markleeds, excellent point. I tried to edit my question a bit, such that you can basically think of $y_t$ being a jump process with random latency. In other words, $y_t$ does not need to be deterministic at all..does this change something from your perspective? $\endgroup$
    – Stefan Voigt
    Commented Oct 21, 2019 at 1:59
  • $\begingroup$ Glad to try to help. I didn't see an edit when I read it but a jump process sounds non-stationary to me. Is it possible to obtain a stationary process from a jump process and an I(1). I'm not sure. Maybe google for jump processes and cointegration. Jump processes are DEFINITELY not my field so maybe someone else can chime in. Sorry but maybe if you edit it again, I or someone else can understand more clearly. $\endgroup$
    – mark leeds
    Commented Oct 21, 2019 at 14:45

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