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

Grid Search Algorithm

@YanZhang present a methodology to generate initial values by grid search technique. We need five starting values for the EHO model. The two probability parameters $\probinfevent$ and $\probbadnews$ can take on any values in the range of 0 to 1. The remaining parameters $\left(\intensuninfbuys, \intensuninfsells, \intensinf \right)$ have no upper bound and can be any positive real number.

To determine initial values for the non-probability parameters @YanZhang make use of the marginal distributions of buys and sells.^[ More details are given in the appendix of @YanZhang.] The marginal distributions for buys and sells in the EHO model can be written as $$ \begin{align} \label{eq:margdensehobuys} \prsym\left(# \text{Buys} = B\right) &= \left(1-\probinfevent\right)\left(\exp\left(-\intensuninfbuys \right)\dfrac{\intensuninfbuys^{B}}{B!} \right) \notag \ & + \probinfevent \left(1-\probbadnews\right) \left(\exp\left(-(\intensuninfbuys + \intensinf) \right)\dfrac{\left(\intensuninfbuys + \intensinf \right)^{B}}{B!} \right) \notag \ & + \probinfevent \probbadnews \left(\exp\left(-\intensuninfbuys \right)\dfrac{\intensuninfbuys^{B}}{B!} \right) \end{align} $$ and $$ \begin{align} \label{eq:margdensehosells} \prsym\left(# \text{Sells} = S\right) &= \left(1-\probinfevent\right)\left(\exp\left(-\intensuninfsells \right) \dfrac{\intensuninfsells^{S}}{S!} \right) \notag \ & + \probinfevent \left(1-\probbadnews\right) \left(\exp\left(-\intensuninfsells \right) \dfrac{\intensuninfsells^{S}}{S!} \right) \notag \ & + \probinfevent \probbadnews \left(\exp\left(-(\intensuninfsells + \intensinf) \right) \dfrac{\left(\intensuninfsells + \intensinf \right)^{S}}{S!} \right), \end{align} $$ with $B, S \in \mathbb{N}_0$. It is obvious that both marginal densities are weighted sums of Poisson distributed random variables. We can utilize the linearity of the expectation operator and write the expected values for the marginal distributions of the EHO model as $$ \begin{align} \label{eq:expectedbuyseho} \expect\left(B\right) = \probinfevent (1 - \probbadnews) \intensinf + \intensuninfbuys \ \label{eq:expectedsellseho} \expect\left(S\right) = \probinfevent \probbadnews \intensinf + \intensuninfsells, \end{align} $$ These moment conditions for the expected values enable us to set initial values for all five parameters.

First step is to get initial values for the probabilities $\probinfevent$ and $\probbadnews$. To prevent the initial guesses for these two parameters to lie on the boundaries we go with @YanZhang and take a sub-interval of $\left[0, 1\right]$. Starting values for $\probinfevent$ and $\probbadnews$ are limited to belong to a series of equidistant real-valued numbers in the range of $0.1$ to $0.9$, always beginning and ending with the minimum and maximum , respectively.^[ These bounds are chosen according to the work by @YanZhang. In principle, any value reasonably greater than 0 can be chosen as lower bound and any value reasonably smaller than 1 as upper bound.] In the next step, the sample averages $\overline{B}$ and $\overline{S}$ of the series of daily buys and sells replace the expectations $\expect(B)$ and $\expect(S)$. Since the term $\probinfevent (1 - \probbadnews) \intensinf$ in equation \ref{eq:expectedbuyseho} is always positive, $\intensuninfbuys$ needs to be smaller than $\overline{B}$. The average daily number of buys $\overline{B}$ is then multiplied by the same equally spaced series of values which are chosen as initial values for $\probinfevent$ and $\probbadnews$ to generate starting values for the intensity of uninformed buyers $\intensuninfbuys$. The last step is to receive initial values for the intensity of sells initiated by noise traders $\intensuninfsells$ and the intensity of transaction fulfilled by informed traders $\intensinf$. Therefore, equations \ref{eq:expectedbuyseho} and \ref{eq:expectedsellseho} are simultaneously solved.

A set of initial values for the EHO model, $\thetaehoinishort = \left(\probinfevent^0, \probbadnews^0, \intensuninfbuys^0, \intensuninfsells^0, \intensinf^0 \right)$, can then be calculated as, $$ \begin{align} \probinfevent^0 &= \probinfevent_i, \notag \ \probbadnews^0 &= \probbadnews_j, \notag \ \intensuninfbuys^0 &= \gamma_k \overline{B}, \notag \ \intensinf^0 &= \dfrac{\overline{B} - \intensuninfbuys^0}{\probinfevent^0 \left(1- \probbadnews^0\right)}, \notag \ \intensuninfsells^0 &= \overline{S} - \probinfevent^0 \probbadnews^0 \intensinf^0, \notag \end{align} $$ where each of the three parameters $\probinfevent_i$, $\probbadnews_j$ and $\gamma_k$ take on equally distanced values between 0.1 and 0.9, one at a time. @YanZhang choose the length of the series of initial values for $\probinfevent_i$, $\probbadnews_j$ and $\gamma_k$ to be five. Hence, the starting values for each of the three parameters are 0.1, 0.3, 0.5, 0.7 and 0.9. This results in a total of $5^3 = 125$ potential sets of initial values.

However, not all combinations are feasible due to negative values for the intensity of uninformed sells $\intensuninfsells^0$. In addition, @Ersan recommend to exclude sets of starting values with irrelevant values for $\intensinf^0$ which is the case if $\intensinf^0$ exceeds the maximum number of daily buys or sells ($\intensinf^0 > \max(B_{\daysym}, S_{\daysym}), \:\: d = 1, \dots, \totaldays$).

pinbasic documentation built on May 2, 2019, 2:07 a.m.