Informed traders’ arrival in foreign exchange markets: Does geography matter?

Abstract

This article critically investigates the possibility that private information offering systematic profit opportunities exists in the spot foreign exchange market. Using a unique dataset with trader-specific limit and market order histories for more than 10,000 traders, we detect transaction behavior consistent with the informed trading hypothesis, where traders consistently make money. We then work within the theoretical framework of a high-frequency version of a structural microstructure trade model, which directly measures the market maker’s beliefs. Both the estimates of the trade model parameters and our model-free analysis of the data suggest that the time-varying pattern of the probability of informed trading is rooted in the strategic arrival of informed traders on a particular day-of-week, hour-of-day, or geographic location (market).

Appendices

Appendix 1: independent arrival model (Easley et al. 1996b)

The model consists of informed and uninformed traders and a risk-neutral competitive market maker. The traded asset is a foreign currency for the domestic currency. Similar to the portfolio shifts model (Evans and Lyons 2002), the trades and the governing price process are generated by the quotes of the market maker over a 24-h trading day. Within any trading hour, the market maker is expected to buy and sell currencies from his posted bid and ask prices. The price process is the expected value of the currency based on the market maker’s information set at the time of the trade.

The hourly arrival of news occurs with the probability $\alpha$. This represents bad news with probability $\delta$ and good news with $1-\delta$ probability. Let $\{p_{i}\}$ be the hourly price process over $i=1,2,\ldots ,24$ h. $p_{i}$ is assumed to be correlated across hours and will reveal the intraday time dependence and intraday persistence of the price behavior across these two classes of traders. The lower and upper bounds for the price process should satisfy $p_{i}^{b} < p_{i}^{n} < {p_{i}^{g}}$ where $p_{i}^{b}$, $p_{i}^{n}$, and $p_{i}^{g}$ are the prices conditional on bad, no news, and good news, respectively. Within each hour, time is continuous and indexed by $t \in [0,T]$.

In any trading hour, the arrivals of informed and uninformed traders are determined by independent Poisson processes. At each instant within an hour, uninformed buyers and sellers each arrive at a rate of $\varepsilon$. Informed traders only trade when there is news, and arrive at a rate of $\mu$. All informed traders are assumed to be risk neutral and competitive and are therefore expected to maximize profits by buying when there is good news and selling otherwise.²⁵ For good news hours, the arrival rates are $\varepsilon + \mu$ for buy orders and $\varepsilon$ for sell orders. For bad news hours, the arrival rates are $\varepsilon$ for buy orders and $\varepsilon + \mu$ for sell orders. When no news exists, the buy and sell orders arrive at a rate of $\varepsilon$ per hour.

The market maker is assumed to be a Bayesian who uses the arrival of trades and their intensity to determine whether a particular trading hour belongs to a no news, good news, or bad news category. Since the arrival of hourly news is assumed to be independent, the market maker’s hourly decisions are analyzed independently from 1 h to the next. Let $P(t) = (P_n(t), P_b(t), P_g(t))$ be the market maker’s prior beliefs with no news, bad news, and good news at time $t$. Accordingly, his or her prior beliefs before trading starts each day are $P(0) = (1-\alpha , \alpha \delta , \alpha (1-\delta ))$.

Let $S_t$ and $B_t$ denote sell and buy orders at time $t$. The market maker updates the prior conditional on the arrival of an order of the relevant type. Let $P(t|S_{t})$ be the market maker’s updated belief conditional on a sell order arriving at $t$. $P_n(t|S_{t})$ is the market maker’s belief about no news conditional on a sell order arriving at $t$. Similarly, $P_{b}(t|S_{t})$ is the market maker’s belief about the occurrence of bad news events conditional on a sell order arriving at $t$, and $P_{g}(t|S_{t})$ is the market maker’s belief about the occurrence of good news conditional on a sell order arriving at $t$.

The probability that any trade occurring at time $t$ is information based is (please see Appendix 2)

$$\begin{aligned} i(t)=\frac{\mu ( 1-P_{n}( t) ) }{2\varepsilon +\mu ( 1-P_{n}( t) ) } \end{aligned}$$

(4)

Since each buy and sell order follows a Poisson Process at each trading hour and orders are independent, the likelihood of observing a sequence of orders containing $B$ buys and $S$ sells in a bad news hour of total time $T$ is given by

$$\begin{aligned} L_{b}( (B,S)|\theta ) =L_{b}( B|\theta ) L_{b}( S|\theta ) =e^{-( \mu +2\varepsilon ) T}\frac{\varepsilon ^{B}( \mu +\varepsilon ) ^{S}T^{B+S}}{B!S!}, \end{aligned}$$

(5)

where $\theta =( \alpha ,\delta ,\varepsilon ,\mu )$.

Similarly, in a no-event hour, the likelihood of observing any sequence of orders that contains B buys and S sells is

$$\begin{aligned} L_{n}( (B,S)|\theta ) =L_{n}( B|\theta )L_{n}( S|\theta ) =e^{-2\varepsilon T}\frac{\varepsilon ^{B+S}T^{B+S}}{B!S!} \end{aligned}$$

(6)

In a good-event hour, this likelihood is

$$\begin{aligned} L_{g}( (B,S)|\theta ) =L_{g}( B|\theta )L_{g}( S|\theta ) =e^{-( \mu +2\varepsilon ) T}\frac{\varepsilon ^{S}( \mu +\varepsilon ) ^{B}T^{B+S}}{B!S!} \end{aligned}$$

(7)

The likelihood of observing $B$ buys and $S$ sells in an hour of unknown type is the weighted average of Eqs. (2), (3), and (4) using the probabilities of each type of hour occurring.

$$\begin{aligned} L( (B,S)|\theta )&= ( 1-\alpha ) L_{n}( (B,S)|\theta ) +\alpha \delta L_{b}( (B,S)|\theta ) +\alpha ( 1-\delta ) L_{g}( (B,S)|\theta ) \nonumber \\&= ( 1-\alpha ) e^{-2\varepsilon T}\frac{\varepsilon ^{B+S}T^{B+S} }{B!S!}+\alpha \delta e^{-( \mu +2\varepsilon ) T}\frac{ \varepsilon ^{B}( \mu +\varepsilon ) ^{S}T^{B+S}}{B!S!} \nonumber \\&+\,\alpha ( 1-\delta ) e^{-( \mu +2\varepsilon ) T} \frac{\varepsilon ^{S}( \mu +\varepsilon ) ^{B}T^{B+S}}{B!S!} \end{aligned}$$

(8)

Because hours are independent, the likelihood of observing the data $M=( B_{i},S_{i}) _{i=1}^{I}$ over 24 h $(I=24)$ is the product of the hourly likelihoods,

$$\begin{aligned} L ( M|\theta )&= \prod _{i=1}^{I}L( \theta |B_{i},S_{i}) =\prod _{i=1}^{I}\frac{e^{-2\varepsilon T}T^{B_{i}+S_{i}}}{ B_{i}!S_{i}!} \times \nonumber \\&\left[ ( 1-\alpha ) \varepsilon ^{B_{i}+S_{i}}+\alpha \delta e^{-\mu T}\varepsilon ^{B_{i}}( \mu +\varepsilon ) ^{S_{i}}+\alpha ( 1-\delta ) e^{-\mu T}\varepsilon ^{S_{i}}( \mu +\varepsilon ) ^{B_{i}}\right] \nonumber \\ \end{aligned}$$

(9)

The log likelihood function is

$$\begin{aligned}&\ell ( M|\theta ) =\sum _{i=1}^{I}\ell ( \theta |B_{i},S_{i}) \nonumber \\&\quad =\sum _{i=1}^{I}[-2\varepsilon T+( B_{i}+S_{i}) \ln T] \nonumber \\&\qquad +\sum _{i=1}^{I}\ln \left[ ( 1\!-\!\alpha ) \varepsilon ^{B_{i}+S_{i}}\!+\!\alpha \delta e^{-\mu T}\varepsilon ^{B_{i}}( \mu +\varepsilon ) ^{S_{i}}\!+\!\alpha ( 1-\delta ) e^{-\mu T}\varepsilon ^{S_{i}}( \mu +\varepsilon ) ^{B_{i}}\right] \nonumber \\&\qquad -\sum _{i=1}^{I}(\ln B_{i}!+\ln S_{i}!) \end{aligned}$$

(10)

As in Easley et al. (2008), the log likelihood function, after dropping the constant and rearranging,²⁶ is given by

$$\begin{aligned} \ell ( M|\theta )&= \sum _{i=1}^{I}\left[ -2\varepsilon + M_i\ln x + (B_i +S_i)\ln (\mu + \varepsilon ) \right] \nonumber \\&+ \sum _{i=1}^{I}\ln \left[ \alpha (1-\delta )e^{-\mu }x^{S_i-M_i} + \alpha \delta e^{-\mu } x^{B_i-M_i} + (1+\alpha )x^{B_i+S_i-M_i} \right] ,\nonumber \\ \end{aligned}$$

(11)

where $M_i\equiv \min (B_i,S_i) + \max (B_i,S_i)/2$, and $x =\frac{\varepsilon }{\varepsilon +\mu } \in [0,1]$.

Appendix 2: derivation of the PIN

By Bayes’ rule, the market maker’s posterior probability with no news at time $t$, if an order to sell arrives at $t$, is

$$\begin{aligned} P_n(t|S_{t})&= \frac{P_{n}(S_{t}|t) P_{n}( t) }{P( S_{t}) } \\&= \frac{P_{n}( S_{t}|t ) P_{n}( t) }{P_{n}( S_{t}|t ) P_{n}( t) +P_{g}( S_{t}|t ) P_{g}( t) +P_{b}( S_{t}|t ) P_{b}(t) } \\&= \frac{\varepsilon P_{n}( t) }{\varepsilon ( 1-P_{g}( t) -P_{b}(t) ) +\varepsilon P_{g}( t) +( \varepsilon +\mu ) P_{b}( t) } \\&= \frac{\varepsilon P_{n}( t) }{\varepsilon +\mu P_{b}( t) }, \end{aligned}$$

where $P_{n}( S_{t}|t )$ is the probability of the arrival of a sell order conditional on no news at time $t$, $P_{g}( S_{t}|t )$ is the probability of the arrival of a sell order conditional on good news at time $t$, and $P_{b}(S_{t}|t )$ is the probability of the arrival of a sell order conditional on bad news at time $t$.

Similarly, the posterior probability on bad news is

$$\begin{aligned} P_{b}( t|S_{t})&= \frac{P_{b}( S_{t}|t ) P_{b}( t) }{P( S_{t}) } \\&= \frac{( \varepsilon +\mu ) P_{b}( t) }{\varepsilon +\mu P_{b}( t)}, \end{aligned}$$

and the posterior probability on good news is

$$\begin{aligned} P_{g}( t|S_{t})&= \frac{P_{g}( S_{t} | t ) P_{g}( t) }{P( S_{t}) } \\&= \frac{\varepsilon P_{g}( t) }{\varepsilon +\mu P_{b}( t) } \end{aligned}$$

The bid price, $b(t)$, conditional on $S_{t}$ at time $t$ at hour $i$ is

$$\begin{aligned} b( t)&= P_n(t|S_{t}) p^n_{i}+P_{b}(t|S_{t}) p^b_{i}+P_{g}(t|S_{t}) p^g_i\\&= \frac{\varepsilon P_{n}(t) p_{i}^n+( \varepsilon +\mu ) P_{b}( t) p^b_{i}+\varepsilon P_{g}(t) p^g_{i}}{\varepsilon +\mu P_{b}(t) } \end{aligned}$$

Similarly, the ask price $a(t)$ is the market maker’s expected value of the asset conditional on the history prior to $t$ and on $B_{t}$.

Thus, the ask at time $t$ at hour $i$ is

$$\begin{aligned} a( t)&= P_{n}( t|B_{t}) p_{i}^n+P_{b}( t|B_{t}) p^b_{i}+P_{g}( t|B_{t}) p^g_{i}\\&= \frac{\varepsilon P_{n}( t) p_{i}^n+\varepsilon P_{b}( t) p^b_{i}+( \varepsilon +\mu ) P_{g}( t) p^g_{i}}{\varepsilon +\mu P_{g}( t) } \end{aligned}$$

The expected price conditional on $t$ is

$$\begin{aligned} E\left[ p_{i}|t\right] =P_{n}( t) p_{i}^n+P_{b}( t) p^b_{i}+P_{g}( t) p^g_{i}, \end{aligned}$$

where $P_{n}( t)$, $P_{b}( t)$ and $P_{g}( t)$ are the prior beliefs of the market maker for no news, bad news, and good news at time $t$.

Substituting the expected price equation into the equations for bid and ask prices yields

$$\begin{aligned} b(t)&= \frac{\varepsilon P_{n}( t) p_i^n+\varepsilon P_{b}( t) p_i^b+\varepsilon P_{g}( t) p_i^g+\mu P_{b}( t) p_i^b}{ \varepsilon +\mu P_{b}( t) } \\&= \frac{\varepsilon E\left[ p_{i}|t\right] +\mu P_{b}(t) p_i^b}{\varepsilon +\mu P_{b}(t) } \\&= \left[ 1-\frac{\mu P_{b}( t) }{\varepsilon +\mu P_{b}( t) }\right] E\left[ p_{i}|t\right] +\frac{\mu P_{b}( t) p_i^b}{\varepsilon +\mu P_{b}( t) } \\&= E\left[ p_{i}|t\right] -\frac{\mu P_{b}( t) }{\varepsilon +\mu P_{b}( t) }\left( E\left[ p_{i}|t\right] -p_i^b\right) , \end{aligned}$$

and

$$\begin{aligned} a(t)&= \frac{\varepsilon P_{n}(t) p_{i}^n+\varepsilon P_{b}(t) p_i^b+ \varepsilon P_{g}(t) p_i^g+\mu P_{g}(t) p_i^g}{\varepsilon +\mu P_{g}(t) } \\&= \frac{\varepsilon E\left[ p_{i}|t\right] +\mu P_{g}( t) p_i^g}{\varepsilon +\mu P_{g}( t) } \\&= \left[ 1-\frac{\mu P_{g}( t) }{\varepsilon +\mu P_{g}( t) }\right] E\left[ p_{i}|t\right] +\frac{\mu P_{g}( t) p_i^g}{\varepsilon +\mu P_{g}( t) } \\&= E\left[ p_{i}|t\right] +\frac{\mu P_{g}( t) }{\varepsilon +\mu P_{g}( t) }\left( p_i^g-E\left[ p_{i}|t\right] \right) \end{aligned}$$

Let $d(t) =a(t)-b(t)$ be the spread at time $t$.

$$\begin{aligned} d(t)&= E\left[ p_i|t\right] +\frac{\mu P_{g}( t) }{\varepsilon +\mu P_{g}( t) }\left( p_i^g-E \left[ p_i|t\right] \right) \\&-\left[ E\left[ p_i|t\right] -\frac{\mu P_{b}( t) }{\varepsilon +\mu P_{b}( t) }\left( E\left[ p_i|t\right] -p_i^b\right) \right] \\&= \frac{\mu P_{g}( t) }{\varepsilon +\mu P_{g}( t) } \left( p_i^g-E\left[ p_i|t\right] \right) +\frac{\mu P_{b}( t) }{\varepsilon +\mu P_{b}( t) }\left( E\left[ p_i|t\right] -p_i^b\right) \end{aligned}$$

The spread for the opening quotes is

$$\begin{aligned} d(0)&= \frac{\mu P_{g}( 0) }{\varepsilon +\mu P_{g}( 0) }\left( p_i^g-E\left[ V_{i}\right] \right) + \frac{\mu P_{b}( 0) }{\varepsilon +\mu P_{b}( 0) } \left( E\left[ p_{i}\right] -p_i^b\right) \\&= \frac{\mu \alpha ( 1-\delta ) }{\varepsilon +\mu \alpha ( 1-\delta ) }\left( p_i^g-E\left[ p_{i}|t\right] \right) + \frac{\mu \alpha \delta }{\varepsilon +\mu \alpha \delta }\left( E\left[ p_{i}|t\right] -p_i^b\right) \end{aligned}$$

If good and bad events are equally likely, that is, if $\delta =1-\delta$ , $\delta =0.5$. Thus

$$\begin{aligned} d(0)&= \frac{\mu \alpha }{2\varepsilon +\mu \alpha }( p_i^g-p_i^b) \end{aligned}$$

The probability that any trade occurring at time $t$ is information based is

$$\begin{aligned} i(t)&= \frac{P_{b}( t) \mu +P_{g}( t) \mu }{P( B_{t},S_{t}) } \\&= \frac{\mu ( 1-P_{n}( t) ) }{P_{n}( t) P_{n}( B_{t},S_{t}|t) +P_{g}( t) P_{g}( B_{t},S_{t}|t) +P_{b}( t) P_{b}( B_{t},S_{t}|t) }\\&= \frac{\mu ( 1-P_{n}( t) ) }{2\varepsilon +\mu ( 1-P_{n}( t) ) } \end{aligned}$$