Supply emand Symmetry of Market Impact Models · 3 Supply–Demand Symmetry for Forex 4 But Then:...
Transcript of Supply emand Symmetry of Market Impact Models · 3 Supply–Demand Symmetry for Forex 4 But Then:...
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Supply–Demand Symmetry of MarketImpact ModelsNovember 2012
Carlo Acerbi
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Outline
1 A Question Awaiting Formalization
2 Basic FormalismForex: Basic Facts
3 Supply–Demand Symmetry for Forex
4 But Then: Why Only Forex?Geometrical Interpretation of Supply–Demand Symmetry
5 Some Results
6 Conclusions
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A Question Awaiting Formalization
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What is a Bid–Ask Symmetrical Market Impact?
Is it an even function ?
−100 −80 −60 −40 −20 0 20 40 60 80 1000
20
40
60
80
100
120
140
number of contracts
impa
ct (
bps)
even impact function
BUY SELL
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What is a Bid–Ask Symmetrical Market Impact?
Is it an even function ?
−100 −80 −60 −40 −20 0 20 40 60 80 1000
20
40
60
80
100
120
140
number of contracts
impa
ct (
bps)
even impact function
BUY SELL
Wow, nice plot! Great idea!
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What is a Bid–Ask Symmetrical Market Impact?
Is it an even function ?
−100 −80 −60 −40 −20 0 20 40 60 80 1000
20
40
60
80
100
120
140
number of contracts
impa
ct (
bps)
even impact function
BUY SELL
Wow, nice plot! Great idea!
But why even?
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What is a Bid–Ask Symmetrical Market Impact?
Is it an even function ?
−100 −80 −60 −40 −20 0 20 40 60 80 1000
20
40
60
80
100
120
140
number of contracts
impa
ct (
bps)
even impact function
BUY SELL
Wow, nice plot! Great idea!
But why even?
Does it represent any financial symmetry?
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The Large Scale Picture Cannot Be Even
Bid prices are floored at zero. Ask prices are not capped
Bid impact is capped. Ask impact may be unbounded.
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10Marginal and Average Impact
µµ̄
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The Large Scale Picture Cannot Be Even
Bid prices are floored at zero. Ask prices are not capped
Bid impact is capped. Ask impact may be unbounded.
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10Marginal and Average Impact
µµ̄
Parity can’t be a global fundamental symmetry
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How to Formalize the Symmetry?
What is the symmetry behind equivalence of supply and demand ?
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How to Formalize the Symmetry?
What is the symmetry behind equivalence of supply and demand ?
Not so obvious for – say – equities or bonds
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How to Formalize the Symmetry?
What is the symmetry behind equivalence of supply and demand ?
Not so obvious for – say – equities or bonds
But what about forex? Buying one currency is selling another one
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How to Formalize the Symmetry?
What is the symmetry behind equivalence of supply and demand ?
Not so obvious for – say – equities or bonds
But what about forex? Buying one currency is selling another one
Buy/Sell symmetry ⇔ relativity under change of base currency
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How to Formalize the Symmetry?
What is the symmetry behind equivalence of supply and demand ?
Not so obvious for – say – equities or bonds
But what about forex? Buying one currency is selling another one
Buy/Sell symmetry ⇔ relativity under change of base currency
First we need an appropriate formalism
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Basic Formalism
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Price Impact of a General Security
Definition (Average price impact)
Expected price impact µ(s,T ) of an order of s contracts, executed through atime horizon T . Best execution assumed.
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Price Impact of a General Security
Definition (Average price impact)
Expected price impact µ(s,T ) of an order of s contracts, executed through atime horizon T . Best execution assumed.
Definition (Marginal price impact)
Expected price impact µ(s,T ) of additional ds contracts traded.
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Price Impact of a General Security
Definition (Average price impact)
Expected price impact µ(s,T ) of an order of s contracts, executed through atime horizon T . Best execution assumed.
Definition (Marginal price impact)
Expected price impact µ(s,T ) of additional ds contracts traded.
µ(s,T ) =1
s
∫
s
0
µ(z ,T ) dz
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Price Impact of a General Security
Definition (Average price impact)
Expected price impact µ(s,T ) of an order of s contracts, executed through atime horizon T . Best execution assumed.
Definition (Marginal price impact)
Expected price impact µ(s,T ) of additional ds contracts traded.
µ(s,T ) =1
s
∫
s
0
µ(z ,T ) dz
Convention
Buy trade if s < 0; sell trade if s > 0
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Supply–Demand Curves
Let m be the fair price
Definition (Supply–Demand Curve (SDC))
Expected price of the entire order s
m(s,T ) = m − sgn(s)µ(s,T )
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Supply–Demand Curves
Let m be the fair price
Definition (Supply–Demand Curve (SDC))
Expected price of the entire order s
m(s,T ) = m − sgn(s)µ(s,T )
Definition (Marginal Supply–Demand Curve (MSDC))
Expected price of additional ds contracts
m(s,T ) = m − sgn(s)µ(s,T )
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Supply–Demand Curves
Let m be the fair price
Definition (Supply–Demand Curve (SDC))
Expected price of the entire order s
m(s,T ) = m − sgn(s)µ(s,T )
Definition (Marginal Supply–Demand Curve (MSDC))
Expected price of additional ds contracts
m(s,T ) = m − sgn(s)µ(s,T )
‘Effective Order Book’ Interpretation
We interpret a couple {ds,m(s,T )} as a quote available within T
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Supply–Demand Curves
Let m be the fair price
Definition (Supply–Demand Curve (SDC))
Expected price of the entire order s
m(s,T ) = m − sgn(s)µ(s,T )
Definition (Marginal Supply–Demand Curve (MSDC))
Expected price of additional ds contracts
m(s,T ) = m − sgn(s)µ(s,T )
‘Effective Order Book’ Interpretation
We interpret a couple {ds,m(s,T )} as a quote available within T
Definition (Bid and Ask price)
m± ≡ m(0±,∀T ) = m(0±,∀T )
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Example: Piecewise Constant MSDC
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4MSDC and SDC
mm̄
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Marginal and Average Impact
µµ̄
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Example: Piecewise Constant MSDC
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4MSDC and SDC
mm̄
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Marginal and Average Impact
µµ̄
Is it symmetrical?
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Liquidation Operator
Definition (Liquidation operator)
Expected order proceedings
L(s,T ) = m(s,T ) s =
∫
s
0
m(z ,T ) dz
Cash in if L > 0, cash out if L < 0
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Regular Market Hypothesis
The only fundamental hypotheses we make
Regular Market Hypothesis
The MSDC m(s,T ) is non-increasing in s, for all T .
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Regular Market Hypothesis
The only fundamental hypotheses we make
Regular Market Hypothesis
The MSDC m(s,T ) is non-increasing in s, for all T .
It amounts to impose that any further contract sold/bought meets worseand worse prices
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Regular Market Hypothesis
The only fundamental hypotheses we make
Regular Market Hypothesis
The MSDC m(s,T ) is non-increasing in s, for all T .
It amounts to impose that any further contract sold/bought meets worseand worse prices
Or equivalently, that every quote in the market can be filled partially, forarbitrarily small sizes (no block quotes)
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Regular Market Hypothesis
The only fundamental hypotheses we make
Regular Market Hypothesis
The MSDC m(s,T ) is non-increasing in s, for all T .
It amounts to impose that any further contract sold/bought meets worseand worse prices
Or equivalently, that every quote in the market can be filled partially, forarbitrarily small sizes (no block quotes)
Corollary
The liquidation operator L is concave
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Forex: Just a Special Case of Security
One unit of foreign currency is just one particular security
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Forex: Just a Special Case of Security
One unit of foreign currency is just one particular security
Fair exchange rate: X d
f
d: ‘domestic’ currency CCYd
f : ‘foreign’ currency CCYf
Xd
fexpressed in CCYd per unit CCYf traded
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Forex: Just a Special Case of Security
One unit of foreign currency is just one particular security
Fair exchange rate: X d
f
d: ‘domestic’ currency CCYd
f : ‘foreign’ currency CCYf
Xd
fexpressed in CCYd per unit CCYf traded
MSDC: X d
f (s,T )s: number of ’foreign’ currency units traded
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Forex: Just a Special Case of Security
One unit of foreign currency is just one particular security
Fair exchange rate: X d
f
d: ‘domestic’ currency CCYd
f : ‘foreign’ currency CCYf
Xd
fexpressed in CCYd per unit CCYf traded
MSDC: X d
f (s,T )s: number of ’foreign’ currency units traded
Similar convention for all other equivalent functions
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Forex: Just a Special Case of Security
One unit of foreign currency is just one particular security
Fair exchange rate: X d
f
d: ‘domestic’ currency CCYd
f : ‘foreign’ currency CCYf
Xd
fexpressed in CCYd per unit CCYf traded
MSDC: X d
f (s,T )s: number of ’foreign’ currency units traded
Similar convention for all other equivalent functions
All the introduced functions admit a dual representation in the twocurrencies
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Notation: Dropping Time Dependence
In what follows, we fix some horizon T and we stop indicating it
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Forex: Basic Facts
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Forex–Duality of the Framework
The order sizes sa and sb of a trade in the two currencies are related by
sa = − La
b(sb)
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Forex–Duality of the Framework
The order sizes sa and sb of a trade in the two currencies are related by
sa = − La
b(sb) but also sb = −Lb
a(sa)
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Forex–Duality of the Framework
The order sizes sa and sb of a trade in the two currencies are related by
sa = − La
b(sb) but also sb = −Lb
a(sa)
Proposition
Dual liquidation operators are related by
−Lb
a = (−La
b)[−1]
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Forex–Duality of the Framework
The order sizes sa and sb of a trade in the two currencies are related by
sa = − La
b(sb) but also sb = −Lb
a(sa)
Proposition
Dual liquidation operators are related by
−Lb
a = (−La
b)[−1]
Proposition
Dual MSDCs are related by
Xb
a (sa)Xa
b (sb) = 1
Dual SDCs are related by
Xb
a(sa)Xa
b(sb) = 1
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Forex–Duality of the Framework
The order sizes sa and sb of a trade in the two currencies are related by
sa = − La
b(sb) but also sb = −Lb
a(sa)
Proposition
Dual liquidation operators are related by
−Lb
a = (−La
b)[−1]
Proposition
Dual MSDCs are related by
Xb
a (sa)Xa
b (sb) = 1
Dual SDCs are related by
Xb
a(sa)Xa
b(sb) = 1
far less obvious!
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Proofs
Proof.
By definition of inverse[−L
b
a ] ◦ [−La
b](sb) = sb
Differentiating both sides by sb we obtain
Xb
a (−La
b(sb))Xa
b (sb) = 1
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Proofs
Proof.
By definition of inverse[−L
b
a ] ◦ [−La
b](sb) = sb
Differentiating both sides by sb we obtain
Xb
a (−La
b(sb))Xa
b (sb) = 1
Proof.
Applying twice the definition of SDC: L(s) = X (s) s
[−Lb
a ] ◦ [−La
b](sb) = Xb
a(−La
b(sb)) · La
b(sb) = Xb
a(−La
b(sb)) · Xa
b(sb) · sb = sb
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Supply–Demand Symmetry for Forex
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Invariance Under Change of Base Currency
To impose supply–demand symmetry, we require that the two dual foreximpact functions look identical to two investors with opposite basecurrency.
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Invariance Under Change of Base Currency
To impose supply–demand symmetry, we require that the two dual foreximpact functions look identical to two investors with opposite basecurrency.
... up to a constant rescaling to account for notional disparity of the twocurrency units
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Invariance Under Change of Base Currency
To impose supply–demand symmetry, we require that the two dual foreximpact functions look identical to two investors with opposite basecurrency.
... up to a constant rescaling to account for notional disparity of the twocurrency units
Example
Suppose the U/e rate is 100 U/e. In a symmetrical market, we expect thatthe relative impact of liquidating e100 and the relative impact of liquidatingU10’000 should be the same
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Imposing Supply–Demand Symmetry for Small Forex Trades
Impose that the dual relative bid–offer spreads are identical
X− − X+
X=
1/X+ − 1/X−
1/X
Solving for X yields
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Imposing Supply–Demand Symmetry for Small Forex Trades
Impose that the dual relative bid–offer spreads are identical
X− − X+
X=
1/X+ − 1/X−
1/X
Solving for X yields
Proposition
In a symmetrical forex LS, the fair rate is the geometric average of the bid rate
and the offer rate.
Xa
b =√
X a
b
+ X a
b
−
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Imposing Supply–Demand Symmetry for Forex, in General
Definition (Forex Supply–Demand Symmetry)
We say that a forex market is symmetrical, if there exists a constant α > 0such that the mapping sa/α↔ sb
sa
α= −
1
αLa
b(sb)
is an involution
−1
αLa
b =
(
−1
αLa
b
)[−1]
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Imposing Supply–Demand Symmetry for Forex, in General
Definition (Forex Supply–Demand Symmetry)
We say that a forex market is symmetrical, if there exists a constant α > 0such that the mapping sa/α↔ sb
sa
α= −
1
αLa
b(sb)
is an involution
−1
αLa
b =
(
−1
αLa
b
)[−1]
Proposition
If such α exists, it’s the fair rate
α = Xa
b =√
X a
b
+ X a
b
−
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Imposing Supply–Demand Symmetry for Forex, in General
Definition (Forex Supply–Demand Symmetry)
We say that a forex market is symmetrical, if there exists a constant α > 0such that the mapping sa/α↔ sb
sa
α= −
1
αLa
b(sb)
is an involution
−1
αLa
b =
(
−1
αLa
b
)[−1]
Proposition
If such α exists, it’s the fair rate ... as you may have guessed
α = Xa
b =√
X a
b
+ X a
b
−
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Classification of Forex Symmetrical Markets
Theorem
A forex market displays supply–demand symmetry if and only if the liquidation
operator sb 7→ La
b(sb) can be expressed as
La
b(sb) = −Xa
b φ(sb)
where the function φ : Da
b → Da
b
1 is an involution φ = φ[−1]
2 is convex and strictly decreasing
3 φ(0) = 0
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Classification of Forex Symmetrical Markets
Theorem
A forex market displays supply–demand symmetry if and only if the liquidation
operator sb 7→ La
b(sb) can be expressed as
La
b(sb) = −Xa
b φ(sb)
where the function φ : Da
b → Da
b
1 is an involution φ = φ[−1]
2 is convex and strictly decreasing
3 φ(0) = 0
Corollary
In a symmetrical forex market the MSDC and the SDC satisfy
Xa
b (s)Xa
b (s̃) = (X a
b )2
Xa
b(s)Xa
b(s̃) = (X a
b )2
at conjugated points s and s̃ = φ(s).
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But Then: Why Only Forex?
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A Currency is Just One Security Among All Others
If you say
“the euro for a yen based investor is as liquid an asset
as the yen is for a euro based investor”
it might seem you’re speaking of a forex symmetry only
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A Currency is Just One Security Among All Others
If you say
“the euro for a yen based investor is as liquid an asset
as the yen is for a euro based investor”
it might seem you’re speaking of a forex symmetry only
But if you equivalently say
“for a euro based investor selling yens is as liquid as buying yens”
you realize that the yen is just one security among all others
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A Currency is Just One Security Among All Others
If you say
“the euro for a yen based investor is as liquid an asset
as the yen is for a euro based investor”
it might seem you’re speaking of a forex symmetry only
But if you equivalently say
“for a euro based investor selling yens is as liquid as buying yens”
you realize that the yen is just one security among all others
you could have been speaking of a stock, a gold bullion, an oil gallon, ...
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A Currency is Just One Security Among All Others
If you say
“the euro for a yen based investor is as liquid an asset
as the yen is for a euro based investor”
it might seem you’re speaking of a forex symmetry only
But if you equivalently say
“for a euro based investor selling yens is as liquid as buying yens”
you realize that the yen is just one security among all others
you could have been speaking of a stock, a gold bullion, an oil gallon, ...
similarities with “change of numeraire” type of symmetry
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A Currency is Just One Security Among All Others
If you say
“the euro for a yen based investor is as liquid an asset
as the yen is for a euro based investor”
it might seem you’re speaking of a forex symmetry only
But if you equivalently say
“for a euro based investor selling yens is as liquid as buying yens”
you realize that the yen is just one security among all others
you could have been speaking of a stock, a gold bullion, an oil gallon, ...
similarities with “change of numeraire” type of symmetry
Definition (Supply–Demand Symmetry for General Securities)
A security’s market is symmetrical if it has the same properties of a forexsymmetrical market
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Classification of Symmetrical Markets for General Securities
Theorem
A security’s market displays supply–demand symmetry if and only if the
liquidation operator s 7→ L(s) can be expressed as
L(s) = −m φ(s)
where the function φ : D → D
1 is an involution φ = φ[−1]
2 is convex and strictly decreasing
3 φ(0) = 0
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Classification of Symmetrical Markets for General Securities
Theorem
A security’s market displays supply–demand symmetry if and only if the
liquidation operator s 7→ L(s) can be expressed as
L(s) = −m φ(s)
where the function φ : D → D
1 is an involution φ = φ[−1]
2 is convex and strictly decreasing
3 φ(0) = 0
Corollary
In a security’s symmetrical market the MSDC and the SDC satisfy
m(s)m(s̃) = m2
m(s)m(s̃) = m2
at conjugated points s and s̃ = φ(s).
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Geometrical Interpretation
of Supply–Demand Symmetry
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Geometrical Interpretation of Supply–Demand Symmetry
L(x)/m: concave, symmetrical wrt y = −x , increasing, zero in zero
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L(s)/m
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Geometrical Interpretation of Supply–Demand Symmetry
L(x)/m: concave, symmetrical wrt y = −x , increasing, zero in zero
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L(s)/m
the curve is forced to live in the white area of the plane
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Geometrical Interpretation of Supply–Demand Symmetry
L(x)/m: concave, symmetrical wrt y = −x , increasing, zero in zero
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L(s)/m
the curve is forced to live in the white area of the plane
two extremes: perfectly liquid and perfectly illiquid market
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Geometrical Interpretation of Supply–Demand Symmetry
L(x)/m: concave, symmetrical wrt y = −x , increasing, zero in zero
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L(s)/m
the curve is forced to live in the white area of the plane
two extremes: perfectly liquid and perfectly illiquid market
natural notion of partial ordering of liquidity among different L’s
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Supply–Demand Symmetry for Stocks, in Words
s̃ = −L(s)/m and s = −L(s̃)/m
Proposition (A Market is Symmetrical iff)
If |s̃| stocks correspond in fair value to the liquidation of s stocks, then s
stocks correspond in fair value to the cost of buying |s̃| stocks, ∀s
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L/m
s̃
s
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Example: Exponentially Decaying MSDC
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10MSDC and SDC
mm̄
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10Marginal and Average Impact
µµ̄
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Example: Exponentially Decaying MSDC
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Exponentially Decaying MSDC with Spread
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10MSDC and SDC
mm̄
−10 −5 0 5 100
1
2
3
4
5
6
7
8
9
10Marginal and Average Impact
µµ̄
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Example: Exponentially Decaying MSDC with Spread
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Asymptotically Finite MSDC
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2MSDC and SDC
mm̄
−10 −5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7Marginal and Average Impact
µµ̄
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Example: Asymptotically Finite MSDC
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Asymptotically Finite MSDC with Spread
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2MSDC and SDC
mm̄
−10 −5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Marginal and Average Impact
µµ̄
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Example: Asymptotically Finite MSDC with Spread
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Linear Ask MSDC
−10 −5 0 5 100
1
2
3
4
5
6MSDC and SDC
mm̄
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Marginal and Average Impact
µµ̄
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Example: Linear Ask MSDC
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Linear Ask MSDC with Spread
−10 −5 0 5 100
1
2
3
4
5
6
7
MSDC and SDC
mm̄
−10 −5 0 5 100
0.5
1
1.5
2
2.5
Marginal and Average Impact
µµ̄
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Example: Linear Ask MSDC with Spread
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Piecewise Constant MSDC
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4MSDC and SDC
mm̄
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Marginal and Average Impact
µµ̄
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Example: Piecewise Constant MSDC
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4MSDC and SDC
mm̄
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Marginal and Average Impact
µµ̄
It was symmetrical!
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Example: Piecewise Constant MSDC
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Example: Another Piecewise Constant MSDC
−10 −5 0 5 100
0.5
1
1.5MSDC and SDC
mm̄
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25Marginal and Average Impact
µµ̄
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Example: Another Piecewise Constant MSDC
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10L
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Some Results
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An Even Impact Always Corresponds to Excess of Supply
Given any market ‘wing’, there exists one and only one symmetrical wingthat completes a symmetrical market
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An Even Impact Always Corresponds to Excess of Supply
Given any market ‘wing’, there exists one and only one symmetrical wingthat completes a symmetrical market
Therefore, we can build sound notions of‘more liquid market’‘more liquid wing’
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An Even Impact Always Corresponds to Excess of Supply
Given any market ‘wing’, there exists one and only one symmetrical wingthat completes a symmetrical market
Therefore, we can build sound notions of‘more liquid market’‘more liquid wing’
but also a notion of‘excess of demand’ (resp. ‘of supply’): sell side more (resp. less) liquid thanbuy side
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An Even Impact Always Corresponds to Excess of Supply
Given any market ‘wing’, there exists one and only one symmetrical wingthat completes a symmetrical market
Therefore, we can build sound notions of‘more liquid market’‘more liquid wing’
but also a notion of‘excess of demand’ (resp. ‘of supply’): sell side more (resp. less) liquid thanbuy side
Proposition
If the average impact is an even function, the market has an excess of supply
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An Even Impact Always Corresponds to Excess of Supply
Given any market ‘wing’, there exists one and only one symmetrical wingthat completes a symmetrical market
Therefore, we can build sound notions of‘more liquid market’‘more liquid wing’
but also a notion of‘excess of demand’ (resp. ‘of supply’): sell side more (resp. less) liquid thanbuy side
Proposition
If the average impact is an even function, the market has an excess of supply
Proof.
Not so obvious
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Even Impact Always Corresponds to Excess of Supply
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
s
L(s)/m
L+
L̃+
L−
Figure : Illustration of the proposition. L+ represents the bid wing of a LS. The plotcompares the ask wing L− obtained assuming that impact is even and the ask wing
L̃+ assuming a symmetrical market. The former, is always more liquid.
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Even Impact as Small Size Limit of Supply–Demand Symmetry
Proposition
Consider a symmetrical market. Express the bid wing MSDC as
m(s) = m+ −m+ ψ(s) s > 0
with lims→0 ψ(s) = 0. Then, the opposite ask wing MSDC can be
approximated as an expansion in powers of ψ, to give
m(s) = m− +m− ψ(−sm−/m) +O(ψ2) s < 0
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Even Impact as Small Size Limit of Supply–Demand Symmetry
Proposition
Consider a symmetrical market. Express the bid wing MSDC as
m(s) = m+ −m+ ψ(s) s > 0
with lims→0 ψ(s) = 0. Then, the opposite ask wing MSDC can be
approximated as an expansion in powers of ψ, to give
m(s) = m− +m− ψ(−sm−/m) +O(ψ2) s < 0
Proof.
An application of the Lagrange inversion theorem
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Even Impact as Small Size Limit of Supply–Demand Symmetry
Proposition
Consider a symmetrical market. Express the bid wing MSDC as
m(s) = m+ −m+ ψ(s) s > 0
with lims→0 ψ(s) = 0. Then, the opposite ask wing MSDC can be
approximated as an expansion in powers of ψ, to give
m(s) = m− +m− ψ(−sm−/m) +O(ψ2) s < 0
Proof.
An application of the Lagrange inversion theorem
Corollary
At small impact regimes, a symmetrical market can be approximated by an
even impact function iff the bid–offer spread is zero, in which case
µ(s) = mψ(|s|) +O(ψ2) ∀s
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Zooming at Low Impact Scale with No Spread
−10 −5 0 5 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
s
Marginal Impact µ
µ
Figure : A symmetrical (power–law) marginal impact with no bid–ask spread, zoomedat small impact scale. The function is very close to an even one.
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Zooming at Low Impact Scale with Spread
−10 −5 0 5 100.09
0.1
0.11
0.12
0.13
0.14
0.15
s
Marginal Impact µ
µ
Figure : A symmetrical (power–law) marginal impact with finite bid–ask spread. Thefunction can not be approximated by an even one at any scale.
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Zooming at Low Impact Scale with Spread
−10 −5 0 5 100.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
s
Marginal Impact wrt Mid Price: µmid
µmid
Figure : Even if we compute impact from mid price instead of fair price, to offset thecentral gap, the ask wing remains steeper.
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Comparing with the Literature
All models in the literature (with a massive amount of empirical evidence)assume even functions to describe supply–demand equilibrium.
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Comparing with the Literature
All models in the literature (with a massive amount of empirical evidence)assume even functions to describe supply–demand equilibrium. Are theyall wrong ?
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Comparing with the Literature
All models in the literature (with a massive amount of empirical evidence)assume even functions to describe supply–demand equilibrium. Are theyall wrong ?
No. The last proposition tells us that they may be just looking into smallimpact regimes, neglecting bid–offer spread. Which is in fact the typicalassumption in most models
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Comparing with the Literature
All models in the literature (with a massive amount of empirical evidence)assume even functions to describe supply–demand equilibrium. Are theyall wrong ?
No. The last proposition tells us that they may be just looking into smallimpact regimes, neglecting bid–offer spread. Which is in fact the typicalassumption in most models
Our notion of symmetry makes testable predictions at all size scales andimpact regimes, that are supposed to extend previous findings.
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Conclusions
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Summing Up
We have formalized the conditions that describe equivalence of supply anddemand for a general security
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Summing Up
We have formalized the conditions that describe equivalence of supply anddemand for a general security
We have characterized and classified all the possible solutions to the aboveconditions
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Summing Up
We have formalized the conditions that describe equivalence of supply anddemand for a general security
We have characterized and classified all the possible solutions to the aboveconditions
The resulting symmetry generalizes the traditional idea of ‘even’ impactfunctions at all trade size scales
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Summing Up
We have formalized the conditions that describe equivalence of supply anddemand for a general security
We have characterized and classified all the possible solutions to the aboveconditions
The resulting symmetry generalizes the traditional idea of ‘even’ impactfunctions at all trade size scales
Even market impact functions can describe a supply–demand equilibriumonly in absence of bid/ask spread and for small trade sizes. They alwaysexpress an excess of supply.
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Summing Up
We have formalized the conditions that describe equivalence of supply anddemand for a general security
We have characterized and classified all the possible solutions to the aboveconditions
The resulting symmetry generalizes the traditional idea of ‘even’ impactfunctions at all trade size scales
Even market impact functions can describe a supply–demand equilibriumonly in absence of bid/ask spread and for small trade sizes. They alwaysexpress an excess of supply.
Supply–demand symmetry should represent the equilibria points of nomarket imbalance in all market impact models
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Thanks!
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