High Frequency Market Making Yacine A t-Sahalia Mehmet Saglam LFTs are randomly arriving noise...

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  • High Frequency Market Making

    Yacine Äıt-Sahalia Mehmet Saglam

    Princeton University and NBER Princeton University

    The Evolving Structure of the U.S. Treasury Market Federal Reserve Bank of New York

    October 20-21, 2015

    1

  • 1 A MODEL OF HFMM

    1. A Model of HFMM

    • Premise: Compared to traditional market makers

    – HFMMs are better informed than their counterparties: able to ex-

    tract signals about the direction of the order flow

    – And are faster

    • What can we expect when HFMMs become the primary providers of liquidity?

    2

  • 1 A MODEL OF HFMM

    • Inventory discipline is the primary means of risk control by the HFMM who is risk-neutral, but penalized for holding inventory

    • LFTs are randomly arriving noise traders submitting market orders.

    • The HFMM posts quotes and aims to capture the spread as often as possible.

    • The HFMM receives a signal that is informative, but not perfect, about the sign of the incoming market order from LFTs.

    • Optimal Policy: HFMM always quote unless inventory thresholds are exceeded.

    • When deciding whether to quote or not, the HFMM is constantly weighing the potential of capturing the spread vs. the cost of increas-

    ing his inventory.

    3

  • 2 PREDICTIONS OF THE MODEL

    2. Predictions of the Model

    • Objective value and optimal inventory limits as a function of model parameters

    – the arrival rate of the LFTs, λ

    – the arrival rate of the HFMM’s signal, µ

    – the accuracy of the signal, p

    – the bid-offer spread, c

    – the coefficient of inventory aversion, γ

    4

  • 2.1 LFTs’ Market Orders Arrival Rate 2 PREDICTIONS OF THE MODEL

    2.1. LFTs’ Market Orders Arrival Rate

    Optimal value and inventory trading limits

    50 100 150 200

    2

    4

    6

    8

    10

    12

    14

    16

    18

    Arrival rate of market orders (λ)

    O b

    je ct

    iv e

    V al

    u e

    0 50 100 150 200

    -15

    -10

    -5

    0

    5

    10

    Arrival rate of market orders (λ) C

    ri ti

    ca l L

    im it

    s

    U L

    5

  • 2.2 HFMM’s Signal Arrival Rate (or Latency) 2 PREDICTIONS OF THE MODEL

    2.2. HFMM’s Signal Arrival Rate (or Latency)

    Optimal value and inventory trading limits

    100 200 300 400 500 600 700 4

    4.5

    5

    5.5

    6

    6.5

    7

    7.5

    8

    8.5

    Arrival rate of signals (μ)

    O b

    je ct

    iv e

    V al

    u e

    0 100 200 300 400 500 600 700 -15

    -10

    -5

    0

    5

    Arrival rate of signals (μ) C

    ri ti

    ca l L

    im it

    s

    U L

    6

  • 2.3 Signal Accuracy 2 PREDICTIONS OF THE MODEL

    2.3. Signal Accuracy

    Optimal value and inventory trading limits

    0.5 0.6 0.7 0.8 0.9 1 0

    0.005

    0.01

    0.015

    0.02

    0.025

    Signal accuracy (p)

    v( -1

    ,1 )-

    v( -1

    ,- 1)

    0.5 0.6 0.7 0.8 0.9 1 -10

    -8

    -6

    -4

    -2

    0

    2

    4

    6

    8

    Signal accuracy (p) C

    ri ti

    ca l L

    im it

    s

    U L

    7

  • 2.4 Bis-Ask Spread 2 PREDICTIONS OF THE MODEL

    2.4. Bis-Ask Spread

    Optimal value and inventory trading limits

    0.05 0.1 0.15 0.2 0.25

    4

    6

    8

    10

    12

    14

    16

    18

    20

    22

    Bid-Offer Spread (C)

    O b

    je ct

    iv e

    V al

    u e

    0.05 0.1 0.15 0.2 0.25

    -10

    -5

    0

    5

    10

    Bid-Offer Spread (C) C

    ri ti

    ca l L

    im it

    s

    U L

    8

  • 2.5 Inventory Aversion 2 PREDICTIONS OF THE MODEL

    2.5. Inventory Aversion

    Optimal value and inventory trading limits

    0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

    6

    6.5

    7

    7.5

    8

    Inventory Aversion (Γ)

    O b

    je ct

    iv e

    V al

    u e

    0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

    -10

    -5

    0

    5

    10

    Inventory Aversion (Γ) C

    ri ti

    ca l L

    im it

    s

    U L

    9

  • 2.6 Provision of Liquidity by the HFMM 2 PREDICTIONS OF THE MODEL

    2.6. Provision of Liquidity by the HFMM

    Long-run Probability of LFTs’ Orders Being Filled by the HFMM

    0 200 400 600 800 1000 0.62

    0.63

    0.64

    0.65

    0.66

    0.67

    0.68

    0.69

    0.7

    0.71

    Average Time between HFT Decisions (ms)

    F ill

    R at

    e o

    f M

    ar ke

    t O

    rd er

    s

    10

  • 2.7 Endogenous Cancellations by the HFMM 2 PREDICTIONS OF THE MODEL

    2.7. Endogenous Cancellations by the HFMM

    Long-run probability of an existing quote being canceled by the HFMM

    0 200 400 600 800 1000 0.12

    0.14

    0.16

    0.18

    0.2

    0.22

    0.24

    0.26

    0.28

    0.3

    Average Time between HFT Decisions (ms)

    L o

    n g

    R u

    n C

    an ce

    lla ti

    o n

    R at

    es

    11

  • 3 PRICE VOLATILITY

    3. Price Volatility

    • Add price variability in the form of jumps in the asset’s fundamental value.

    • The HFMM has no informational advantage regarding these price movements; his only signal is about the likely direction of the order

    flow.

    • Volatility introduces adverse selection: the HFMM may get stuck with stale quotes that can be sniped by another HFT

    12

  • 3 PRICE VOLATILITY

    Example: A Simulated Path with Volatility

    0 5 9.95

    10

    10.05

    10.1

    τµ,s1 τ ζ,up 2 τ

    λ,s 3 τ

    µ,b 4

    Loss

    Time (s)

    P ri ce

    Bid Quote Ask Quote Price Trade Signal Jump

    13

  • 3 PRICE VOLATILITY

    Long-run probability of quoting as a function of the price jump arrival

    rates

    0 20 40 60 80 100 0.6

    0.62

    0.64

    0.66

    0.68

    0.7

    0.72

    Rate of jumps (/min)

    F ill

    R at

    e o

    f M

    ar ke

    t O

    rd er

    s

    14

  • 3 PRICE VOLATILITY

    • When the price is more volatile, the likelihood that the HFMM will provide liquidity decreases.

    • This is because this volatility introduces a new source of risk for the HFMM (excess inventory) that is not compensated for and for which

    he holds no advantage (no signal).

    • So while the HFMM provides plenty liquidity in normal times, it is op- timal for the HFMM to withdraw when the market needs that liquidity

    the most...

    15

  • 4 COMPETITION AMONG HFMMS

    4. Competition Among HFMMs

    • Duopoly: Splitting the Rent

    Optimal value achieved by the HFMM: Monopoly vs. Duopoly

    0 100 200 300 400 500 600 700 800 4

    4.5

    5

    5.5

    6

    6.5

    7

    7.5

    Average Time between HFT Decisions (ms)

    O b

    je ct

    iv e

    V al

    u e

    Monopoly Duopoly

    16

  • 4 COMPETITION AMONG HFMMS

    • The rent extracted from LFTs gets split between the two market mak- ers.

    • The faster the HFMM, the more of the rent he is able to capture: there are benefits to becoming faster among HFMMs.

    • LFTs are better off when market makers compete compared to the monopolistic HFMM situation.

    17

  • 5 COMPARING DIFFERENT HFT REGULATIONS

    5. Comparing Different HFT Regulations

    • Three policies in the context of the model: imposing a transaction tax on each trade, setting minimum rest times on limit orders and taxing

    cancellations of limit orders.

    • Objective: induce the HFMM to provide liquidity that is more resilient to increases in volatility = procyclical with respect to volatility

    – We find that none of the three policies result in an improvement compared to doing nothing.

    – Transaction taxes result in less liquidity both in low and high volatil- ity environments.

    – Both minimum rest times and a cancellation tax result in more liquidity in good (low volatility) environments but less in bad (high

    volatility) environments = countercyclical.

    18

  • 5.1 Tobin Tax: Taxing Transactions 5 COMPARING DIFFERENT HFT REGULATIONS

    5.1. Tobin Tax: Taxing Transactions

    • Equivalent to a reduction in the spread. Transaction taxes reduce the incentive to quote.

    0 5 10 15 20 25 30

    0.35

    0.4

    0.45

    0.5

    0.55

    0.6

    0.65

    Transaction Tax (bps)

    L o

    n g

    -r u

    n P

    ro b

    ab ili

    ty o

    f Q

    u o

    ti n

    g

    0 5 10 15 20 0.63

    0.64

    0.65

    0.66

    0.67

    0.68

    0.69

    0.7

    Rate of jumps (/m