DISTRITO (Cuotas) POBLACIÓN

A special case that leads to a useful approximation of the model is when the market maker’s thresholds for which he revises his quotes become large in magnitude. We defined the upper and lower thresholds for the market maker’s prior as ¯θ and θ. Recall that in Section 2.3 the boundaries around the market maker’s likelihood ratio wereHand _H1. LetHand _H1 correspond to the values of the likelihood ratio pi

1−pi when pi =

¯

θand pi = θrespectively. Note that since ¯

θandθtake values in (0,1), H and _H1 take values in (0,∞). In the limit as H goes to infinity, the informed trader makes infinite profits because there are infinitely many trades before the market maker prefers to revise his quotes. However, it is interesting to derive the informed trader’s optimal strategy in the limit as H goes to infinity as an approximation to the case of finiteH. The advantage of consideringHlarge is that we are able to show the optimal strategy is a constant ρ∗ which is independent of the state variable, the market maker’s prior pi. We will showρ∗ _{is given by the solution to D}

ρ[_E[log(E[f(_ξρ₁)]₎|ρ]] = 0 where E[f(ρ)] represents expected

per-period profits, E[log(ξ1)|ρ] is the expected “first step” taken by the market maker’s log-

likelihood ratio andDρis the differential operator with respect toρ.

It is intuitive that the optimal strategy is independent of piwhenH is large. In general, the informed trader balances immediate trading profits with price impact in order to preserve the potential for future profits. When the log-likelihood ratio is close to the market maker’s thresh- old and the potential for future profits is not high relative to immediate profits, the informed trader puts relatively less weight on minimizing price impact. Because the relative contribu- tions of immediate and future profits change with the distance to the boundary, the informed trader will adjust his strategy accordingly. However, when H is large, the log-likelihood ratio is always “far” from the boundary. The balance between immediate and future profits is stable; each trade results in a move toward a boundary that is negligible and therefore no adjustment to the strategy is needed after a trade. We state the result in the following Proposition.

Proposition 2.4.2 The informed trader’s optimal trading strategy, in the sense that it satisfies the first-order condition of the Bellman equation, is given by the solution to Dρ[_E[log(E[f(_ξρ₁)]₎|ρ]] = 0

The denominator of the formula in Proposition 2.4.2 will be nonzero, intuitively, because every trading round will be informative. The informed trader will always submit either a fast limit or market order with positive probability α so the market maker’s Bayesian update will be nontrivial.

Figure 2.9:Optimal strategy.The figure shows the optimal probability of the informed trader submitting a market on the y-axis and the degree of segmentation on the x-axis. The prices used to generate the figure werexl = 1₃ andxh = 2₃. The blue line was created using a value of

1 10 forα.

Degree of Segmentation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Optimal Probability of Market Order

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 H large

H large and , small

We can use software to differentiate symbolically and then numerically find the root of the condition in Proposition 2.4.2 that the optimal strategy satisfies. Results for certain parameter values are shown in Figure 2.9. The figure shows that the informed trader submits a higher proportion of limit orders than market orders as the degree of retail segmentation increases. When certain market orders are identified as retail, the relative informativeness of a non-retail market order increases, all else equal. The informed trader mitigates this effect by decreasing the probability he submits a market order as the proportion of retail tradersβincreases.

Observation 2.4.3 The informed trader submits a greater proportion limit orders as the mea- sure of retail traders increases;ρ∗is decreasing inβin the limit as H goes to infinity.

We will prove the result stated in Observation 2.4.3 for a limiting case in next subsection with Proposition 2.4.5.

We are also interested in the effect of segmentation on price efficiency. Recall that in Section 2.3 we defined price efficiency using the expected number of trades before the market maker’s prior hit arbitrary upper or lower boundaries. The resulting formula in Proposition

2.3.2 had the interpretation of expected number of trades (time) being equal to expected total change in the market maker’s prior before it hits a boundary (distance) over expected “first step” (speed). We will need to change our notion of price efficiency here since the expected total change in the market maker’s prior goes to infinity asHgoes to infinity. We will focus on the “speed” at which the market maker’s prior moves, that is, the expected size of the revision to the market maker’s prior after a trade,E[log(ξ1)|ρ]. In Section 2.4.11, we will show that this

definition of price efficiency is a good proxy the expected number of trades before the maker’s prior hits a boundary.

Figure 2.10 shows how price inefficiency, defined as the reciprocal of the expected change in the market maker’s prior evolves following a trade, changes with the degree of segmentation. The graph shows that the importance of strategic order submission is low for low degrees of segmentation and high for high degrees of segmentation.

Observation 2.4.4 Price efficiency, as measured by the rate at which the market maker’s prior approaches the asset’s true value,1or0, increases with the measure of retail tradersβwhen H is large.

Observation 2.4.4 shows that despite the ability of the informed trader to mitigate the effect of identifying retail traders there is still an improvement in price efficiency in equilibrium.

The equation solved by the optimal strategy, Dρ[_E[log(E[f(_ξρ)]

1)|ρ]]= 0, has a natural interpretation.

Total profits earned by the informed trader can be loosely described as the expected number of trades he executes n, times the expected profit earned per trade E[f(ρ)]. Wald’s lemma

tells us that under certain conditions, the expected number of trades before a hitting time is the expected distance travelled by the process divided by the expected move at each step. The expected move at each step isE[log(ξ1)|ρ]. Let the expected distance travelled by the market

maker’s prior be a constantc. The derivative of total profits with respect to the probability of a market order are:

Dρ[nE[f(ρ)]] =Dρ[ cE[f(ρ)] E[log(ξ1)|ρ] ] =cDρ[ E[f(ρ)] E[log(ξ1)|ρ] ] =⇒Dρ[ E[f(ρ)] E[log(ξ1)|ρ] ]=0,

whenc is finite. The condition Dρ[_E[log(E[f(_ξρ₁)]₎|ρ]] = 0 can be seen as a optimizing a factor that is

proportional to total profits. The proof of Proposition 2.4.2 is mostly devoted to showing that

cis indeed independent ofρfor largeH. Althoughcwill be infinite whenH goes to infinity, we will show the first order condition is still meaningful.

Approximation: Large Switching Threshold and Small Probability of Informed Trading Since the condition in Proposition 2.4.2 is not directly solvable, we pursue a further simplifi- cation of the formula; the limit asαgoes to zero. Whenαgoes to zero, the result is an analytic

Figure 2.10: Price efficiency. They-axis is price inefficiency, defined as the reciprocal of the expected change in the market maker’s prior evolves following a trade. On the x-axis is the degree of segmentation. The solid blue line represents price inefficiency when the informed trader acts strategically, while the dashed green line serves as a benchmark and represents the case when the informed trader submits limit orders and market orders with probabilities 1₂. The figure shows that despite the informed trader’s strategic order choice, price inefficiency decreases increases with the degree of segmentation. The figure was created using values of

α= 1 10,xl = 1 3, and xh= 2 3 Degree of Segmentation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Price Inefficiency

With Strategic Order Submission No Strategic Order Submission

formula for a constant optimal strategy with the same qualitative properties pointed out in Ob- servation 2.4.3. Taking the limit of Dρ[_E[log(E[f(_ξρ)]

1)|ρ]], the first order condition whenH is large, as

αgoes to zero and solving forρ∗_gives:

ρ∗₌ (xl−1)β−2x 2 l +2−(2(xl− x 2 l)β 2_−(7x2 l +1)β+10x 2 l −4xl+2) 1 2 (3xl−1)(β−2) (2.17)

We use symbolic computing software (Matlab) to take the limit since the condition in Proposi- tion 2.4.2 is difficult to work with by hand when expanded. The resulting strategy in Equation 2.17 is shown in Figure 2.9. It is qualitatively similar to the plot of the more general strategy. We can now state the analogue of Observation 2.4.3 as a provable proposition.

Proposition 2.4.5 In the limit as both H goes to infinity and αgoes to 0, the probability the informed trader submits a limit order strictly decreases as the degree of retail segmentationβ increases. A proof is found in Appendix A.8.

In the next section we will use numerical techniques to show that the limiting results derived above are qualitatively similar to a more general set of model parameters.