Estimating the proportion of informed and speculative traders in financial markets: evidence from exchange rate

Abstract

We study the Glosten–Milgrom model and estimate the proportion of informed traders or speculators using bid–ask spread and price range. The GM model is generalized in terms of a key parameter $\theta$—the probability of making a correct decision by an agent. Informed traders have $\theta = 1$, and uninformed traders have $\theta = 1/2$ in the GM model. Speculators are defined to be agents with $1/2 < \bar{\theta } < 1$. We show that bid–ask spread can be generated when speculators and uninformed traders are in the market—the presence of informed traders is unnecessary. We estimate the proportion of informed traders or speculators using the spread-to-range ratio as a proxy, which entails a new estimation method. Using three exchange rate data, we obtain the conditional mean of the proportion of informed traders and speculators over a seven-year period. Speculators can achieve probability $\bar{\theta } > 1/2$ using simple trading rules within short trading horizons and net of transaction cost.

Appendix A: Bid and ask from the GM model with speculators

We introduce a third type of trader—speculator, denoted by $S$—into the GM model. We define speculators in terms of their ability in making a correct trading decision:

$$ P\left( {{\text{buy}}\left| {U, \bar{V}} \right.} \right) = P\left( {{\text{buy}}\left| {U, \underline{V} } \right.} \right) = P\left( {{\text{sell}}\left| {U, \bar{V}} \right.} \right) = P\left( {{\text{sell}}\left| {U,\underline{V} } \right.} \right) = 1/2 $$

(A1)

$$ P\left( {{\text{buy}}\left| {I, \bar{V}} \right.} \right) = P\left( {{\text{sell}}\left| {I,\underline{V} } \right.} \right) = 1 $$

(A2)

$$ P\left( {{\text{buy}}\left| {S, \bar{V}} \right.} \right) = P\left( {{\text{sell}}\left| {S,\underline{V} } \right.} \right) = \theta > P\left( {{\text{buy}}\left| {S,\underline{V} } \right.} \right) = P\left( {{\text{sell}}\left| {S, \bar{V}} \right.} \right) = 1 - \theta $$

(A3)

In the GM model, uninformed traders $U$ have 50% probability in making a correct decision and informed traders $I$ always make a correct decision. A speculator $S$ has a probability $\frac{1}{2} < \bar{\theta } < 1$ in making a correct trading decision.

The probability tree of the GM model with speculators now has 10 scenarios as shown by the additional rows in Table 1. We denote the proportion of informed traders and speculators by $\mu_{i}$ and $\mu_{s}$. The bid-and-ask prices can be derived by a similar approach in the GM model in three steps:

Step 1 Find $E\left[ {V\left| {S, {\text{buy}}} \right.} \right]$ and $E\left[ {V\left| {S, {\text{sell}}} \right.} \right]$

First, we have:

$$ E\left[ {V\left| {S, {\text{buy}}} \right.} \right] = P\left( {\bar{V}\left| {S, {\text{buy}}} \right.} \right)\bar{V} + P\left( {\underline{V} \left| {S,{\text{buy}}} \right.} \right)\underline{V} $$

(A4)

where

$$ P\left( {\bar{V}\left| {S, {\text{buy}}} \right.} \right) = \frac{\pi \theta }{{\pi \theta + \left( {1 - \pi } \right)\left( {1 - \theta } \right)}} $$

(A5)

and $P\left( {\underline{V} \left| {S,{\text{buy}}} \right.} \right) = 1 - P\left( {\bar{V}\left| {S, {\text{buy}}} \right.} \right)$. For $E\left[ {V\left| {S, {\text{sell}}} \right.} \right]$ we have:

$$ E\left[ {V\left| {S, {\text{sell}}} \right.} \right] = P\left( {\bar{V}\left| {S, {\text{sell}}} \right.} \right)\bar{V} + P\left( {\underline{V} \left| {S,{\text{sell}}} \right.} \right)\underline{V} $$

(A6)

where

$$ P\left( {\bar{V}\left| {S, {\text{sell}}} \right.} \right) = \frac{{\pi \left( {1 - \theta } \right)}}{{\pi \left( {1 - \theta } \right) + \left( {1 - \pi } \right)\theta }} $$

(A7)

and $P\left( {\underline{V} \left| {S,{\text{sell}}} \right.} \right) = 1 - P\left( {\bar{V}\left| {S, {\text{sell}}} \right.} \right)$.

Step 2 Calculate $P\left( {\text{buy}} \right)$ and $P\left( {\text{sell}} \right)$

The unconditional probability of seeing a buy order $P\left( {\text{buy}} \right)$ can be calculated by summing up the probabilities of relevant scenarios in Table 1:

$$ \begin{aligned} P\left( {\text{buy}} \right) & = \pi \mu_{i} + \frac{{\pi \left( {1 - \mu_{i} - \mu_{s} } \right)}}{2} + \pi \mu_{s} \theta \\ &\quad + \frac{{\left( {1 - \pi } \right)\left( {1 - \mu_{i} - \mu_{s} } \right)}}{2} + \left( {1 - \pi } \right)\mu_{s} \left( {1 - \theta } \right) \end{aligned} $$

In the same way, the unconditional probability $P\left( {\text{sell}} \right)$ is given by:

$$ \begin{aligned} P\left( {\text{sell}} \right) & = \frac{{\pi \left( {1 - \mu_{i} - \mu_{s} } \right)}}{2} + \pi \mu_{s} \left( {1 - \theta } \right) + \left( {1 - \pi } \right)\mu_{i} \\ &\quad + \frac{{\left( {1 - \pi } \right)\left( {1 - \mu_{i} - \mu_{s} } \right)}}{2} + \left( {1 - \pi } \right)\mu_{s} \theta \end{aligned} $$

Step 3 Obtain the bid-and-ask formula

Under the assumption that $E\left[ {{\text{Profit}}\left| {\text{Buy}} \right.} \right] = E\left[ {{\text{Profit}}\left| {\text{Sell}} \right.} \right] = 0$, the formula for bid-and-ask prices in the GM model with speculators is:

$$ \left\{ {\begin{array}{*{20}l} {B_{\text{GM}}^{S} = P\left( {I\left| {\text{sell}} \right.} \right)\underline{V} + P\left( {U\left| {\text{sell}} \right.} \right)E\left[ V \right] + P\left( {S\left| {\text{sell}} \right.} \right)E\left[ {V\left| {S, {\text{sell}}} \right.} \right]} \hfill \\ {A_{\text{GM}}^{S} = P\left( {I\left| {\text{buy}} \right.} \right)\bar{V} + P\left( {U\left| {\text{buy}} \right.} \right)E\left[ V \right] + P\left( {S\left| {\text{buy}} \right.} \right)E\left[ {V\left| {S, {\text{buy}}} \right.} \right]} \hfill \\ \end{array} } \right. $$

(A8)

where

$$ \begin{aligned} P\left( {I\left| {\text{sell}} \right.} \right) & = \frac{{\left( {1 - \pi } \right)\mu_{i} }}{{P\left( {\text{sell}} \right)}},\quad P\left( {U\left| {\text{sell}} \right.} \right) = \frac{{\left( {1 - \mu_{i} - \mu_{s} } \right)/2}}{{P\left( {\text{sell}} \right)}},\\ P\left( {S\left| {\text{sell}} \right.} \right) & = \frac{{\mu_{s} \left( {\pi + \theta - 2\pi \theta } \right)}}{{P\left( {\text{sell}} \right)}}, \end{aligned} $$

and

$$ \begin{aligned} P\left( {I\left| {\text{buy}} \right.} \right) & = \frac{{\pi \mu_{i} }}{{P\left( {\text{buy}} \right)}},\quad P\left( {U\left| {\text{buy}} \right.} \right) = \frac{{\left( {1 - \mu_{i} - \mu_{s} } \right)/2}}{{P\left( {\text{buy}} \right)}},\\ P\left( {S\left| {\text{buy}} \right.} \right) & = \frac{{\pi \mu_{s} \theta + \left( {1 - \pi } \right)\mu_{s} \left( {1 - \theta } \right)}}{{P\left( {\text{buy}} \right)}}. \end{aligned} $$