In anre005/pinbasic: Fast and Stable Estimation of the Probability of Informed Trading (PIN)

EHO Factorization

\label{sec:ehofactr} @EHO2010 reformulated the likelihood function in the static model with different intensities for uninformed buys and uninformed sells. The authors rearranged the likelihood function and dropped the constant term $-\log\left(B_{\daysym}!S_{\daysym}!\right)$ so we can maximize the algebraically equivalent but more stable and robust factorization

$$ \begin{align} \log \likelihood\left( \thetaehoshort \mid \datasymbs \right) = &\sum\limits_{\daysym=1}^{\totaldays} \Biggl( -\intensuninfbuys - \intensuninfsells + M_{\daysym} \left(\log x_b + \log x_s\right) + B_{\daysym} \log \left(\intensinf + \intensuninfbuys\right) + S_{\daysym} \log \left(\intensinf + \intensuninfsells\right) \Biggr) \notag \ & + \sum\limits_{\daysym=1}^{\totaldays} \log \Biggl( \left(1-\probinfevent\right) x_s^{S_{\daysym} - M_{\daysym}} x_b^{B_{\daysym} - M_{\daysym}} + \probinfevent \left(1-\probbadnews\right) \exp\left(-\intensinf \right) x_s^{S_{\daysym} - M_{\daysym}} x_b^{-M_{\daysym}} \notag \ & + \probinfevent\probbadnews \exp\left(-\intensinf \right) x_b^{B_{\daysym} - M_{\daysym}} x_s^{-M_{\daysym}} \Biggr), \label{eq:ehofactr5par} \end{align} $$ where $M_{\daysym} = \min \left(B_{\daysym}, S_{\daysym} \right) + \dfrac{\max \left(B_{\daysym}, S_{\daysym} \right)}{2}$, $x_s = \dfrac{\intensuninfsells}{\intensuninfsells + \intensinf}$ and $x_b = \dfrac{\intensuninfbuys}{\intensuninfbuys + \intensinf}$.

According to @EHO2010 the computation of the probability of informed trading benefits from the reformulation due to two facts. The computing efficiency is increased and the truncation errors (over- and underflow) are reduced. No evaluation of factorials is needed, additionally $x_b$ and $x_s$ are always weakly smaller than 1 which leads to more stable calculations of the terms involving power operations. However, if the number of buyer- or seller-initiated transactions is very high for a trading day, evaluations of the terms $x_b^{-M_d}$ and $x_s^{-M_d}$ can be problematic and may result in infinite values. Hence, diminishing the frequency of over- and underflow errors is essential in calculating $\pintext$ for (very) frequently traded stocks.

@LinKe state that the $\pintext$ computation is downward-biased if the EHO likelihood formulation is used for stocks with a large transaction number. In the same work an accurate likelihood factorization is presented which we will discuss in the next section.

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