Posterior Probabilities for Trading Days' States

Due to constant model parameters, the static EHO model do not enable to compute $\pintext$ on a daily basis. Nevertheless, we can harness Bayes' theorem and construct formulas for posterior probabilities (e.g. see @Bayes) of trading days' conditions.

Incorporating the independence of of buys' and sells' Poisson processes, we can write the probability that a trading day $\daysym$ resides in no-news state given that we have observed $B_{\daysym}$ buys and $S_{\daysym}$ sells as

$$ \begin{align} \prsym \left(\nonews \mid (B_{\daysym}, S_{\daysym}) \right) &= \dfrac{\prsym\left( B_{\daysym} \mid \nonews \right) \prsym\left( S_{\daysym} \mid \nonews \right) \prsym\left(\nonews\right)} {\prsym\left(B_{\daysym}, S_{\daysym}\right)} \notag \ &= \dfrac{1 - \probinfevent}{(1 - \probinfevent) + \exp(-\intensinf) \left[\probinfevent(1 - \probbadnews) \left(1 + \dfrac{\intensinf}{\intensuninfbuys}\right)^{B_{\daysym}} + \probinfevent\probbadnews \left(1 + \dfrac{\intensinf}{\intensuninfsells}\right)^{S_{\daysym}}\right]}. \label{eq:postno} \end{align} $$

Likewise, posterior probabilities for a good-news and bad-news trading day are given by

$$ \begin{align} \prsym \left(\goodnews \mid (B_{\daysym}, S_{\daysym}) \right) &= \dfrac{\prsym\left( B_{\daysym} \mid \goodnews \right) \prsym\left( S_{\daysym} \mid \goodnews \right) \prsym\left(\nonews\right)} {\prsym\left(B_{\daysym}, S_{\daysym}\right)} \notag \ &= \dfrac{\probinfevent (1 - \probbadnews) \exp(-\intensinf) \left(1 + \dfrac{\intensinf}{\intensuninfbuys}\right)^{B_{\daysym}}}{(1 - \probinfevent) + \exp(-\intensinf) \left[\probinfevent(1 - \probbadnews) \left(1 + \dfrac{\intensinf}{\intensuninfbuys}\right)^{B_{\daysym}} + \probinfevent\probbadnews \left(1 + \dfrac{\intensinf}{\intensuninfsells}\right)^{S_{\daysym}}\right]} \label{eq:postgood} \end{align} $$

and

$$ \begin{align} \prsym \left(\badnews \mid (B_{\daysym}, S_{\daysym}) \right) &= \dfrac{\prsym\left( B_{\daysym} \mid \badnews \right) \prsym\left( S_{\daysym} \mid \badnews \right) \prsym\left(\nonews\right)} {\prsym\left(B_{\daysym}, S_{\daysym}\right)} \notag \ &= \dfrac{\probinfevent \probbadnews \exp(-\intensinf) \left(1 + \dfrac{\intensinf}{\intensuninfsells}\right)^{S_{\daysym}}}{(1 - \probinfevent) + \exp(-\intensinf) \left[\probinfevent(1 - \probbadnews) \left(1 + \dfrac{\intensinf}{\intensuninfbuys}\right)^{B_{\daysym}} + \probinfevent\probbadnews \left(1 + \dfrac{\intensinf}{\intensuninfsells}\right)^{S_{\daysym}}\right]}. \label{eq:postbad} \end{align} $$

To the best of our knowledge, Bayes' posterior probabilities for conditions of trading days were not used before in the field of the probability of informed trading. While we can calculate the number of trading days static models predict to reside in each of the three possible states, posteriors allow to assign each trading day probabilities of no-news, good-news and bad-news condition.

Exemplary, assuming low probability parameters $\probinfevent$ and $\probbadnews$, we can interpret this as few trading days in the sample period on which insiders triggered by positive private information enter the market. However, we are not able to relate information events to specific trading days in the datasource. Utilizing equations \ref{eq:postno} - \ref{eq:postbad} we can identify good-news days according to the magnitude of $\prsym \left(\goodnews \mid (B_{\daysym}, S_{\daysym}) \right)$ for each trading day. Hence, posterior probabilities deliver useful additional insights in classification of trading days and help to improve analyses of the EHO model.



anre005/pinbasic documentation built on May 6, 2022, 4:40 a.m.