Sequential Trade Model for Asymmetrical Information — Part 2

Exercises of Hasbrouck’s Empirical Market Microstructure, Chapter 5

Introduction

In this piece, I’ll extend my exploration into the Sequential Trade Model proposed by Hasbrouck in Chapter 5 of Empirical Market Microstructure (find my previous article here). Moreover, I will solve the suggested exercises, offering my insights and discussing potential further developments.

Exercise 5.1

As a modification to the basic model, take δ = 1/2 and suppose that immediately after V is drawn (as either V̄ or V̱), a broker is randomly drawn. The probability of an informed trader within broker k’s customer set is μₖ. Other brokers have proportion μ, with μₖ < μ. Quote setters post bids and asks. If the order comes into broker k, however, he has option to trade against the customer by matching the bid or ask set by the other dealers.

Compute the broker k’s expected profits on a customer buy order.

For a buy order, the chain of operations between customer and brokers can be visualised as follows:

The customer sends a buy order to Broker K, who is aware that the probability of the order coming from an informed trader is μₖ. He also recognizes that the proportion of informed traders in the customer pool of the Other Broker is μ, with μₖ < μ. Then the Broker K can match the Ask Price of the Other Broker.

Under assumptions of market competitiveness, we computed the Ask Price by imposing the Broker K profit on a Buy trade equal to zero. However, in this scenario, given that the Broker can adopt a different Ask Price, the resulting profit is nonzero. From previous articles, we know that the higher the probability of an informed trader, the higher the Ask and the lower the Bid price(the Bid-Ask spread widens). This means that we expect to compute a higher Ask Price for the Other Broker and hence to have a positive profit for Broker K.

The ask price for Broker K and for Other Broker(remind that δ = 1/2) is:

The profit for Broker K under the new ask is simply the difference between the Ask Price of the Other Broker and the Ask Price of Broker K:

This proves that the profit of Broker K is positive and solves the exercise.

Exercise 5.2

Consider a variant of the model in which there is informed trading only in the low state (V = V̱).

Compute δ₁(Buy), δ₁(Sell), Ask Price, Bid Price.

This exercise assumes that when the actual value of V is in the high state, the trading is always random. Leveraging the same reasonings employed in the former article, we can compute the probability of Buy and Sell:

This leads to the following probabilities:

And hence to the following Ask and Bid Prices:

We notice that the spread tends to be smaller compared to scenarios where informed traders also participate during high states. This is because the broker’s primary focus is on managing trades in the low state, while traders in the high state execute transactions more randomly.

Exercise 5.3

Consider a variant of the model in which informed traders only receive a signal S about V. The signal can be either low or high: S ∈ {S, S}. The accuracy of the signal is γ = Pr[S | V] = Pr[S | V]. Informed traders always trade in the direction of their signal.

Compute δ(Buy).

The model can be described as follows:

In the original model we had a probability of 0 for an informed trader to buy the security when the realised price was in the low state and a probability of 1 to sell it. In this model the buy probability is 1- γ and the sell probability is γ. The same reasoning applies to the high state.

We can compute the probabilities as in the original case and we obtain:

Given that informed traders receive a signal rather than absolute certainty, the Bid-Ask Spread of this model will be smaller compared to the one of the original model. When γ = 1, we return to the conditions of the original model. Conversely, if γ = 0, informed agents always make incorrect trading decisions.

Exercise 5.4

The δₖ following a sequence of orders can be expressed recursively as δₖ(Order₁, . . . , Orderₖ ) = δₖ(Orderₖ ; δₖ₋₁ (Orderₖ₋₁; δₖ₋₂(Orderₖ₋₂; . . .))).

Verify that δ₂(Sell₁, Buy₂) = δ₀, that is, offsetting trades are uninformative.

Using the definition of probability of a low outcome we can compute:

Reflecting on the sequential trade model simulation from my previous article, it’s evident that each pair of opposing trades effectively offset. Such transactions do not add any new information to the broker. Remind that the broker has to determine the fraction of informed traders in the market. This assessment can be conducted by looking at the sequence of buy and sell trades. When the market exhibits a specific trend, i.e. the agents systematically trade in the buy side, it suggests to the broker that the informed agents expect the end of day value of the security to be higher than the initial security value. Given that each agent can visit the market just once, the more the sequence of buys is persistent the higher the fraction of informed traders. Conversely, a noisy sequence of buys and sells lacking any discernible trend, indicates to the broker the absence of informed traders. As a consequence, in such a scenario, the broker finds it challenging to predict the future security’s expected value with any degree of confidence.

Further steps

Even if the sequential trade model we have just examined and developed, is quite simplistic, it was useful to highlight some crucial aspects of the broker’s information asymmetry problem.

In the next article we will deep dive into the Chapter 6 of the book, that describes the order flow and the probability of informed trading, studying the statistical properties of the number of buy and sell orders.

References

Joel Hasbrouck (2007). Empirical Market Microstructure, Chapter 5