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ÿþMAFS5270 (L1) - Mathematical Market Microstructure
Assignment 1  Market Microstructure Variables and Characteristic Time Scale
Due Date: 11:59pm, Jul. 17, 2022
5Ø+5Ø 2 are 5Ø,5Ø ,5Ø , respectively. By definition, 5Ø +5Ø +5Ø =1. Note that all variables 5Ø5Ø5Ø 5Ø5Ø5Ø

Consider a stock that has a bid-ask spread of s = 5Ø ” 5Ø > 0 . For each individual transaction
that happens to this stock, the probabilities of hitting the bid, the ask, and the mid-quote of
here can be functions of time, 5Ø.
Question 1: Derive the formulas for the mean 5Ø 5Ø and the variance 5Ø5Ø5Ø 5Ø of the transaction
price 5Ø with the variables mentioned above.
Question 2: Simplify your results in Question 1 by assuming that 5Ø5Ø ” 5Ø5Ø = 5Ø j” 1 so that higher than 1st order terms of 5Ø5Ø ” 5Ø5Ø can be omitted. In practice, this is the situation when order flow has no significant directional upward or downward movements.
Question 3: Simplify your results in Question 1 by assuming that 5Ø5Ø j” 1 so that higher than 1 order terms of 5Ø5Ø can be omitted. In practice, this is the situation when  off exchange trades (such as those in  dark pools ) are rare.
Question 4: Let 5Ø” be the  market microstructure characteristic time scale for this stock. It can be derived by relating the variance calculated from Questions 1-3 with the variance of a continuous arithmetic Brownian price process with a constant volatility of 5Ø; that is,
5Ø5Ø5Ø 5Ø ~5Ø05Ø 5Ø” 5Ø. (Equation 1)
Here 5Ø0 is a base price so that the dimensions on both sides of Equation 1 become the same. In this assignment, let s assume ³ = 1/3 to reflect potential fat-tail distribution of price returns. Based on the simplified versions of Questions 2 and 3 respectively, give the formulas of 5Ø” as a function of the following variables: 5Ø, 5Ø, 5Ø5Ø, 5Ø5Ø, 5Ø5Ø, 5Ø, 5Ø0 . Comment on how 5Ø” changes as the market becomes more or less  fat-tailed in return distribution (i.e., ³ decreases or increases).
Question 5: Consider five scenarios of 5Ø5Ø = 0, 0.1, 0.2, 0.3, 0.5, respectively. For each scenario, plot 5Ø”(5Ø5Ø)for 5Ø5Ø =0to1.0(for 5Ø5Ø =0)or0.9(for 5Ø5Ø =0.1)or0.8(for 5Ø5Ø =0.2)witheach step of 0.1. Here we assume 5Ø = 264.0, 5Ø = 264.5, Ã = 32% annualized, 5Ø0 = 264.25.
Question 6:  Dark pool trading supporters often argue that market quality can be improved with more usage of dark pools, especially from reducing the level of price volatility. Based on the above calculations, provide your own views on whether you agree with such argument.
End of assignment 1.
Copyright © by Dr. Hongsong Chou, 2012-2022. No part of this material may be: (i) copied, photocopied, or duplicated in any form, by any means, or (ii) redistributed without prior expressed consent from the author. The views expressed here are those of the author himself and himself only.