High Frequency Trading with Speed Hierarchies

Wei LiTopics in Quantitative Finance

Presented by Richard Lin Oct 5th 2015

Agenda

• Introduction• Models and Findings• Benchmark model• General model• Speed competition• Market quality

• Conclusions• Personal Opinions

Introduction

• General idea of HFTsAnticipate incoming orders and trade rapidly with short holding horizons to exploit normal-speed traders' latencies

• Characteristics of HFTsa) High trading volumeb) Very short holding horizonc) Extremely rely on trading speed

• Innovative point in this paperMost existing papers assume all fast traders have homogeneous speeds.In this paper, it allows speed competition among fast traders and analyzes the effect of speed competition.

A typical trading round for HFTMMs post

pricing function that other can trade against

Informed and noise traders

submit market orders

Fast traders rapidly front-

running by trading in the

same direction at better prices

ahead of the orders

Fast traders reverse their

trades and exit their positions at profits when normal-speed traders' orders

arrived

MMs update the final quoted

price

Benchmark model

•Assumptions• Only a monopolistic fast trader (no speed competition

among fast traders)• Based on extended Kyle (1985) framework with trading

and quoting latencies• All traders (include fast trader and normal-speed trader)

are risk neutral

Benchmark model

• Model setup• Two assets:

• A risk free numeraire with zero interest rate• A risky asset with normally distributed fundamental value

• Three types of normal-speed traders:• Strategic informed trader who privately observes the true value of the risky asset• Noise traders who trade randomly for non-informational motives with normally

distribution shares;• Competitive fringe market makers who passively absorb order flow imbalance and set

the pricing function with zero expected profits• A new type of fast trader:

• Anticipate the size of the incoming market orders and rapidly trade twice in one trading round. They do not carry inventory when the trading round ends.

• Timeline and Information Structure

Benchmark model

• Speed, holding horizon and information• Fast trader has speed advantage over all other traders

(both normal-speed traders and market makers)• Fast trader has to liquidate her position by the end of the

trading round• Fast trader has advance information about incoming order

flow

Benchmark model

Benchmark model

• Equilibrium• Define four functions . They denote the fast trader trade size strategy,

the informed trader trade size strategy, market makers pricing function, and the final quote function respectively• In equilibrium, four conditions have to be satisfied:• Informed trader profit maximization• Fast trader profit maximization• Competitive pricing function• Informationally efficient quotes

Benchmark model

• In the paper, the pricing function is assumed to be linear, market makers fill the order at the average price of

denotes the price impact (market depth) factor which is fixed in the trading round.

Benchmark model

• Given above assumption, there is a unique equilibrium where• Fast trading size: • Informed trading size: • Market order pricing: • Initial quote: • Final quote:

• Under the assumption that is linear, are all linear. And the equilibrium is fully characterized by the four parameters and .• is fast trader’s trading intensity.• is informed trader’s trading intensity.• is the temporary price impact per share of an market order on the transaction price .• is permanent price impact per share on the final quote of the aggregated order size .

Benchmark model

• Equilibrium analysis

General Model

• Model setup• Normal-speed traders have the same action timings as in the benchmark

model• Timeline needs to be modified to accommodate multiple fast traders

Suppose there are N strategic fast traders, and each of them can trade twice until time 1 and they are not allowed to carry inventory after time 1. In order to focus on speed differences, the same signal is distributed to all N fast traders. Fast traders’ orders also suffer from latencies but their latencies are much shorter than normal-speed traders’. Between time 0 + and time 1, fast traders’ orders sequentially arrive in J instants . At time , orders arrive simultaneously.

General Model

• Four auxiliary definitions:• Definition 1: The speed profile of fast traders is a vector of numbers where

is the number of fast traders arriving at time .• Definition 2: (Stackelberg-N speed profile) Each of the N fast traders arrives at

a different moment and the speed profile is {1, 1, …, 1}.• Definition 3: (Cournot-N speed profile) All N fast traders arrive at the same

time and the speed profile is {N}.• Definition 4: (Fast traders’ order sizes) For ,

, denotes the total order size from fast traders arriving at time and denotes the total order size from fast traders arriving before time .

General Model

• EquilibriumIn general model, we have to modify the equilibrium conditions:• th Fast trading size: • Informed trading size: • Market order pricing: • Initial quote: • Final quote:

• Speed friction: • Market quality parameter:

General Model

• Equilibrium analysis

General Model

• Findings :• Intuitively, fast traders levy a “speed tax” on the market makers with being the

effective expected tax rate. When market makers receive an order of shares, they mark the price up by and fill the order. The price impact surplus for executing the trade is . Market makers use the surplus to offset the loss to the informed trader and to pay the speed tax to fast traders.• The effective tax rate goes down either because fast traders have the less

accurate signal or fast traders take away a smaller fractionof market makers’ price impact revenue. When the speed tax rate drops , market makers are able to use a larger fraction of the surplus to cover loss to the informed trader.• The temporary price impact increases since the order flow is more informative.

Speed Competition

• Fast traders’ profit and relative speed• The profit of all fast traders arriving at time is

• The profit of all fast traders is

• The aggregate fast trading profit is increasing in the effective speed tax rate.• The fast traders can make more profits when there is more uncertainty about

the fundamental value or there is more noise trading.

Speed Competition

• Speed competition and speed friction

Speed Competition

Market quality

• Information efficiency

Market quality

• Market liquidity• In equilibrium,

Market quality

Conclusions

• HFTs use their speed advantage to extract rents (or levy a speed tax) from normal speed traders and the extracted rents (or tax) are allocated among HFTs according their relative speeds.• Two key factors which contribute to the speed tax rate are information quality

for HFTs and speed friction .• HFTs do not provide market liquidity or information efficiency for normal-

speed traders.• Two policy suggestions:

• lowering the frequency of periodic uniform price auctions reduces the negative impact of HFTs on market quality

• randomizing the sequence of order execution can degrade market quality when the randomizing interval is short.

Personal Opinions

• If the market makers’ pricing function is not a linear one with fixed slope, which is close to real market, the HFTs may behave in a different and complicated way.• The assumption that fast traders have to liquidate their position in

one trading round is restrictive.• Different fast traders may have different information signal rather

than the same one.• Needs more empirical test to improve the model.