Equilibrium High Frequency Trading
Equilibrium High Frequency Trading
Bruno Biais
Thierry Foucault
October, 2011
Sophie Moinas
Equilibrium High Frequency Trading
Plan
1. Introduction
2. Model
3. Prices and allocations with high frequency trading
4. Equilibrium level of high frequency trading
5. Is the level of high frequency trading socially optimal?
Equilibrium High Frequency Trading
Introduction
Our research question
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Can we expect market forces to generate the socially
optimal level of investment in high frequency trading? Is
the investment in high frequency technology wasteful?
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Why is this question important?
Equilibrium High Frequency Trading
Introduction
Motivations
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Increasing presence of high frequency traders.
Figure: Source: Celent
Equilibrium High Frequency Trading
Introduction
Motivations
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Increasing presence of high frequency traders.
1. Brogaard (2010): 26 HFTs participate to 68% of the dollar
volume for 120 Nasdaq stocks
2. Kirilenko et al (2010): 34% of the trading volume in the
E.mini S&P500 futures
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High frequency trading is very concentrated: 2% of the
20,000 trading …rms operating in U.S. markets account for
73% of U.S. equity trading volume (Iati (2009), TAAB Group.
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Yet (risk-adjusted) pro…ts are signi…cant (Brogaard
(2011), Kirilenko et al.(2010), Menkveld (2010)) )
Entry costs in this activity must be signi…cant.
Equilibrium High Frequency Trading
Introduction
Motivations
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High frequency trading requires investments:
1. Technology: hardware and connections
2. Fees: co-location, data etc...
3. Human capital.
“Goldman spends tens millions of dollars on this stu¤.
They have more people working in their technology area
than people on the trading desk...The nature of the
market has changed dramatically.” International Herald
Tribune, July 4 2007.
Equilibrium High Frequency Trading
Introduction
Issues
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Can we expect market forces to generate the socially
optimal level of investment in high frequency trading? Is
the investment in high frequency technology wasteful?
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Addressing this question requires a model in which:
1. Trading …rms make investment decisions in high frequency
trading technologies.
2. The aggregate level of high frequency trading is determined in
equilibrium
3. Market outcomes (prices and allocations) and traders’welfare
depend on the aggregate level of high frequency trading
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We provide a model with these three ingredients.
Equilibrium High Frequency Trading
Introduction
Literature
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Growing literature on algorithmic trading (Hendershott et
al.(2011), Hendershott and Riordan (2011), Martinez and
Rosu (2011), Biais et al.(2010), Brogaard (2010), Easley,
Lopez de Prado, and O’Hara (2010), Menkveld and Jovanovic
(2010), Foucault, Kandel, Kadan (2010), Foucault and
Menkveld (2008) etc....
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Yet no analysis of investment decisions in this activity.
1. Key to analyze whether regulatory intervention is needed.
Equilibrium High Frequency Trading
Introduction
Plan
1. Introduction
2. Model
3. Prices and allocations with high frequency trading
4. Equilibrium level of high frequency trading
5. Is the level of high frequency trading socially optimal?
Equilibrium High Frequency Trading
Model
Model
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N securities and a continuum of “…nancial institutions”.
Three dates:
1. Date 0 (Investment): Financial institutions decide to be
“Fast”(invest in the high frequency trading technology) or
“Slow”. Cost of being fast: C . Fraction of fast institutions: α.
2. Date 1 (Trading): Fast and Slow institutions can buy or sell
one share of each security.
3. Date 2: Assets pay o¤
Equilibrium High Frequency Trading
Model
Assets and Institutions
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Final cash-‡ow for a security: v = µ + e or v = µ e with
equal probabilities.
Value of this cash-‡ow for an institution: v + δ or v δ
with equal probabilities
1. Dispersion of valuations for the security can be due to
di¤erences in hedging needs, prudential requirements, or tax
treatments.
2. Generates gains from trade: institutions who value the
security at v + δ should buy it while those who value the
security at v δ should sell it.
3. No noise trading: welfare is well de…ned
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In this presentation, I focus on:
e
< δ < e.
2
Equilibrium High Frequency Trading
Model
Assets and Institutions
Equilibrium High Frequency Trading
Model
Trading Process
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In each asset, institutions are matched with a
“contraparty” (a pool of liquidity providers, e.g., a limit
order book) with some probability.
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Contraparties are risk neutral and competitive =) trades
take place at the asset expected value conditional on the
institution’s desired trade.
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Computerized trading:
1. Facilitate search in highly fragmented markets: fast
traders are more likely to …nd a contraparty.
2. React and use information on asset cash-‡ows faster
than other traders: fast traders are better informed than
their counterparty.
Equilibrium High Frequency Trading
Model
Informed high frequency traders: evidence
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Henderdshott and Riordan (2011): orders (limit and
market) submitted by algorithmic traders contain more
information than orders submitted by human traders.
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Kirilenko et al. (2011): They …nd that HFTs are able to
buy (sell) right before prices increase (decrease), "possibly due
to their speed advantage or superior ability to predict price
changes"
Equilibrium High Frequency Trading
Model
Fast and Slow Institutions
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Implications:
1. Fast traders observe information on v : adverse selection.
2. Fast traders …nd more trading opportunities: e¢ ciency gain.
Equilibrium High Frequency Trading
Model
Timing at date 1
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We …rst solve for the equilibrium at date 1 (price and trades)
taking the fraction of fast institutions, α, as given.
Equilibrium High Frequency Trading
Model
Benchmark: No Fast Institutions
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Prices: All trades take place at p = µ
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Trades:
1. Institutions with a high private value for an asset and who …nd
a counterparty buy it; Welfare: δ
2. Institutions with a low private value for the asset and who …nd
a counterparty sell it; Welfare: δ
3. Institutions who do not …nd a counterparty ("trading
opportunity") do not trade; Welfare: zero.
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Volume: ρ
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Welfare: ρ
δ
Equilibrium High Frequency Trading
Model
Plan
1. Introduction
2. Model
3. Prices and allocations with high frequency trading
4. Equilibrium level of high frequency trading
5. Is the level of high frequency trading socially optimal?
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Equilibria with High Frequency Traders
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At date 1, an institution can have six types:
1.
2.
3.
4.
5.
6.
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Fast with good news and high private valuation (GH)
Fast with good news and low private valuation (GL)
Fast with bad news and high private valuation (BH)
Fast with bad news and low private valuation (GL)
Slow with high private valuation (H)
Slow with low private valuation (L)
Counterparties do not observe the institution’s type =)
Adverse selection: is the institution buying because she has
good news, high private valuation or both?
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Trading Strategies
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βFj : probability that a fast institution of type
j 2 fGH, GL, BH, BL) buys the asset.
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βSj : probability that a slow institution of type j 2 fH, L) buys
the asset.
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We have βFGH = 1 and βFBL = 0 since prices belong to
[µ e, µ + e].
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Prices
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Buy orders execute at: a = E (v jBuy ), i.e.,
α(1 + βFGL
a = µ+
2 (1
βFBH )
α)ρ( βSH + βSL ) + 2α (1 + βFGL + βFBH )
e
µ
) Prices depend on institutions’strategies and institutions’
strategies depend on prices.
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Price impact (aka informational content) of buy orders
become stronger when βFGL increases relative to βFBH , that is
when fast institutions’trades become more driven by their
information
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Five candidates Nash equilibria
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Multiple Equilibria are possible
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Why? Traders’expectations regarding the impact of
their trades on prices can be self-ful…lling.
1. Slow expects price impact to be high
2. They decide not to trade
3. Price impacts are high since only informed institutions trade.
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Pareto-Dominant equilibria
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For each parameter values, there exists a unique
Pareto-Dominant equilibrium.
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Order Flow Toxicity and High Frequency Trading
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Institutions’Welfare and High Frequency Trading
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
High Frequency Trading and Externalities
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Fast institutions obtain greater expected pro…ts than
slow institutions because:
1. They are more likely to …nd a counterparty (good for overall
welfare)
2. They can pro…t from their information at the expense of slow
institutions (bad for overall welfare)
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Entry of a new fast institution exerts a negative
externality on all other institutions because:
1. It leads to greater price impacts ) lower expected pro…t per
trade + states in which even fast institutions decide not to
trade.
2. It triggers exit of slow institutions ("crowding out
equilibrium").
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=) Risk of excessive investment in high frequency
trading technology.
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Volume and High Frequency Trading
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E¤ect of high frequency trading on volume is non
monotonic (as found in Jovanovic and Menkveld (2010))
because:
1. High frequency trading makes it more likely to …nd a
counterparty but
2. Leads some institutions to refrain from trading.
Equilibrium High Frequency Trading
Prices and allocation with high frequency trading
Plan
1. Introduction
2. Model
3. Prices and allocations with high frequency trading
4. Equilibrium level of high frequency trading
5. Is the level of high frequency trading socially optimal?
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Investment in High Frequency Trading
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We now consider institutions’decisions to become high
frequency traders at date 0.
Institutions vary in size. Bigger institutions have access to
more markets.
Assumptions:
1. An institution of "size" t can invest in n (t ) N markets with
n (t ) increasing in size.
2. Institutions’size is distributed over [t, t ] with density:
f (t ) =
N
.
n (t )
3. Note: n (t ) f (t ) = N. Hence, the distribution of institutions’
sizes within a market is uniform in all markets.
4. "Law of the few": we have a few big institutions and many
small.
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Example
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Entry decision 1/2
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Let φ(α) be the expected pro…t of high frequency traders and
ψ(α) be the expected pro…t of slow traders for a …xed level of
high frequency trading.
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It is optimal to invest in high frequency trading if and only if:
φ ( α )n (t )
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C
ψ ( α )n (t )
IMPORTANT: Investment decisions in HFTs are not driven
by the absolute expected pro…t of HFTs relative to the cost
but by the relative expected pro…t of HFTs relative to the
cost.
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Entry decision 2/2
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An institution invests in High Frequency Trading i¤ its
size is greater than t (α) where
t (α) = n
C
1
φ(α)
ψ(α)
.
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Hence, an institution’s decision to invest depends on the level
of high frequency trading, i.e., on other institutions’decisions.
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We consider Nash equilibria of the entry game.
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Let α be the equilibrium level of high frequency trading.
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Equilibrium levels of high frequency trading
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There are 3 possibilities:
1. If 0 < α < 1 then it solves:
α = Pr(t > t (α )) =
t
t
t (α )
,
t
and t < t (α ) < t.
2. If α = 1 then the smallest institution must …nd it optimal to
invest given that all other institutions invest:
δ
t (1) < t () C < Cmin = n (t ) .
2
3. If α = 0 then we must have:
t (0) > t () C > Cmax = n (t )(e
δρ).
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
An arm’s race
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As more institutions become HFTs:
1. HFTs’expected pro…ts decline =) less incentive to
become an HFT
2. BUT slow traders’expected pro…ts decline as well =)
more incentive to become an HFT, especially if slow traders
are sidelined (crowding out equilibrium)
3. The net gain of becoming an HFTs may increase in α.
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=) HFT can be like an arm’s race: if one institution
expects other traders to invest in HFTs, they also …nd it
bene…cial to invest in HFT =) Multiple equilibria.
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This happens when remaining slow is optimal if others
remain slow, i.e., ρ large enough.
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Multiple Equilibria
n (t )
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Result: If ρ > ρ = 2e 21 n (t ) > 21 then Cmax < Cmin . Thus,
if Cmax < C < Cmin , there are at least two possible
equilibrium levels of high frequency trading: α = 0 or α = 1.
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This multiplicity may also happen when
C < Cmax , α > 0.
1
2
< ρ < ρ , but if
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Multiple Equilibria 1
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Parameter values: ρ = 0.9, e = 0.9, δ = 0.9, C = 2, N = 10.
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Multiple Equilibria
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Parameter values: ρ = 0.9,
e = 0.9, δ = 0.9, C = 4.145, N = 10
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Equilibrium levels of high frequency trading
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When ρ < 1/2, the equilibrium is unique. Parameter values:
ρ = 0.9, e = 0.9, δ = 0.9, C = 5, N = 10
Equilibrium High Frequency Trading
Equilibrium level of high frequency trading
Plan
1. Introduction
2. Model
3. Prices and allocations with high frequency trading
4. Equilibrium level of high frequency trading
5. Is the level of high frequency trading socially optimal?
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Welfare
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For a given level of high frequency trading (α), social
welfare is:
W (α) = Slow Institutions + Fast Institutions Aggregate Welfare
= N[
Z t
t
= N (t̄
ψ(α)dt +
t )[(1
Z t̄
t
φ(α)dt ]
C
α)ψ(α) + αφ(α)]
Pr(t > t (α)).
C
Pr(t > t (α)).
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Individual vs. socially optimal investment decisions
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Is the equilibrium level of high frequency trading socially
optimal?
1. Social bene…t: more e¢ cient search: institutions are more
likely to carry out mutually bene…cial trades.
2. Social cost: High frequency trading consumes resources, C +
lower market participation (adverse selection).
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Individual decisions
1. Internalize the bene…t from more e¢ cient search and the cost
of the technology but ignore the negative externality
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=) Excessive investment in high frequency trading.
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Wasteful investment in high frequency trading
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Result: If ρ > 1/2, the socially optimal level of high
frequency trading is 0 but for C < Cmax , the equilibrium level
of high frequency trading is strictly positive and can be as
high as α = 1.
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More generally, the equilibrium level of HFT is higher than the
socially optimal level.
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Numerical Example
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Parameter values: ρ = 0.9,
e = 0.9, δ = 0.9, C = 4.145, N = 10
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Caveat
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The model does not imply that no investment in HFT is
socially optimal. Just that investment in HFT is excessive
relative to the social optimum.
Equilibrium High Frequency Trading
Is the level of high frequency trading socially optimal?
Future Research
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How to avoid excessive investments in high frequency
trading? "Pigouvian Tax" paid by high frequency …rms based
on the overall volume of high frequency trading? How to
design such a tax? How to redistribute it to non high
frequency …rms?
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Endogenous matching probability ρ or ρ dependent on α :
As more …rms become high frequency traders, likelihood of
…nding a counterparty increases ! Positive "liquidity
externality"
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Empirical analysis: How large is the aggregate investment in
high frequency trading technology in reality? Is there evidence
of complementarities/clustering in investment decisions by
prop trading …rms? (e;g., using time series data on number of
algorithmic traders as in Chaboud et al.(2010)).