The amazing power of dimensional
analysis: Quantifying market impact∗
arXiv:1702.05434v1 [q-fin.TR] 17 Feb 2017
Mathias Pohl†
Alexander Ristig‡
Walter Schachermayer§
Ludovic Tangpi¶
February 20, 2017
Abstract. This note complements the inspiring work on dimensional analysis
and market microstructure by Kyle and Obizhaeva [15]. Our main theorem shows
by a similar argument as usually applied in physics the following remarkable fact.
If the market impact of a meta-order only depends on four well-defined and financially meaningful variables, then – up to a constant – there is only one possible
form of this dependence. In particular, the market impact is proportional to the
square root of the size of the meta-order.
This theorem can be regarded as a special case of a more general result of
Kyle and Obizhaeva. These authors consider five variables which might have
an influence on the size of the market impact. In this case one finds a richer
variety of functions which we precisely characterize. We also discuss the relation
to classical arguments from physics, such as the period of a pendulum.
1. Introduction
Dimensional analysis is a well known argument in physics. The idea is best explained by
considering the most classical example: the period of a pendulum.
The basic assumption is that the period depends only on the following quantities:
• the length l of the pendulum, measured in meters,
∗ The
authors acknowledge support by the Vienna Science and Technologie Fund (WWTF) through project
MA14-008.
† University of Vienna, Faculty of Mathematics and Faculty of Business, Economics & Statistics, OskarMorgenstern-Platz 1, 1090 Vienna, Austria, [email protected]
‡ University of Vienna, Faculty of Mathematics and Faculty of Business, Economics & Statistics, OskarMorgenstern-Platz 1, 1090 Vienna, Austria, [email protected]
§ University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, [email protected]
¶ University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, [email protected]
1
• the mass m of the pendulum, measured in grams,
• the acceleration g of gravity, measured in meters per second squared.
Hence,
period = f (l, m, g),
(1)
where the period is measured in seconds and f is an – a priori – arbitrary function.
Of course, the relation (1) should not depend on whether we measure length by meters or
inches, time by seconds or minutes, and mass by grams or pounds. Combining these three
requirements with the ansatz
f (l, m, g) = const · ly1 my2 g y3 ,
(2)
these requirements translate into three linear equations in the variables y1 , y2 , y3 . The unique
solution yields the well-known relation (see [13] as well as Appendix A below for the details)
s
l
.
(3)
f (l, m, g) = const ·
g
This result goes back as far as Galileo. The elementary linear algebra used in the above
argument has been formalized in proper generality in the nineteenth century and is known
under the name of “Pi-Theorem” (see Section 3 below).
Recently, Kyle and Obizhaeva [15] have applied this line of argument to analyze the market
impact of a meta-order: think of an investor who wants to buy (or sell) a sizeable amount
of an underlying stock within a limited time (e.g. two days). Of course, when placing this
“meta-order” she will split it into smaller pieces, the actual orders, in some (hopefully) clever
way. Nevertheless, we expect the quoted prices to move to the disadvantage of the agent.
We call the size of this price movement the market impact.
We start by identifying the variables which we expect to have an influence on the size of
the market impact:
• Q
the size of the meta-order, measured in units of shares [S],
• P
the price of the stock, measured in units of money [U],
• V
the traded volume of the stock, measured in the units of shares per time [S/T],
• σ 2 the squared volatility of the stock, measured in percentage of the stock price per
unit of time [T−1 ].
Now we formulate the following basic assumption.
Assumption 1. The market impact G depends only on the above 4 variables, i.e.
G = g(Q, P, V, σ 2 ),
(4)
where the function g : R4+ → R+ as well as the quantity G are invariant under changes of
the units chosen to measure the “dimensions” T, U, S.
2
We note that G is a percentage of the quoted price of the stock; hence it is a dimensionless
quantity. We thus encounter an analogous situation as in the pendulum example. There is,
however, a serious difference to the pleasant situation encountered above: we now have 4
variables, corresponding to the powers of Q, P , V , σ 2 , but only 3 equations resulting from
the scaling invariance for the fundamental dimensions T, U, S. We need one more equation
to obtain such a crisp result as in (3) above. Kyle and Obizhaeva found a remedy: an
additional “no arbitrage” type argument which follows from the findings of the ModiglianiMiller theorem. We consider a stock which is a share of a company. Suppose that the
company changes its capital structure by paying dividends or, passing to the opposite sign,
by raising new capital. The Modigliani-Miller theorem precisely tells us which quantities
remain constant when changing the leverage between equity and debt of the company. This
requirement should furnish one more equation to be satisfied by (4). For the details we refer
to Section 4 below. The subsequent assumption summarizes the additional restriction which
Kyle and Obizhaeva call “leverage neutrality” (see Assumption 5 below for a more formal
definition).
Assumption 2 (Leverage Neutrality). The value of a firm does not depend on its capital
structure, i.e. its debt-equity ratio.
It turns out that this invariance indeed provides one more linear equation analogous to
the equations obtained by the scaling arguments above. We therefore find ourselves in a
perfectly analogous situation as with the pendulum and have the same number of equations
as unknowns, namely four.
Theorem 1. Under Assumptions 1 and 2, the market impact is of the form
r
Q
,
G = const · σ
V
(5)
for some constant const > 0.
In particular, we find the square-root dependence on the order size Q in accordance with
several theoretical as well as empirical findings (see the review of the literature below).
In fact, the above line of arguments does not correspond exactly to what Kyle and
Obizhaeva have done in [15] (compare also [16]). They were more ambitious and considered one more variable which may have influence on the market impact. These authors
suppose that the agent faces a fixed “inventory cost” C when preparing the placement of a
meta-order. It is assumed to be independent of the order size Q as well as of the quantities
P , V , σ 2 discussed above. Hence, they consider an additional quantity which might have
an influence on the market impact:
• C
the fixed inventory cost, measured in units of money [U].
In other words, Kyle and Obizhaeva only use the subsequent hypothesis which is weaker
than Assumption 1 above.
Assumption 3. The market impact G depends only on the above 5 variables, i.e.
G = g(Q, P, V, σ 2 , C),
(6)
where the function g : R5+ → R+ as well as the quantity G are invariant under changes of
the units chosen to measure the “dimensions” T, U and S.
3
Starting from this weaker assumption Kyle and Obizhaeva apply a similar reasoning as
above, in particular the argument of leverage neutrality. This leads to four linear equations
in five unknowns. The solution is not unique anymore, but rather a 1-dimensional affine
space in R5 . In other words, we still have one degree of freedom left which leads to the
following theorem.
Theorem 2 (Kyle and Obizhaeva). Under Assumptions 2 and 3, the market impact is of
the form
1
(7)
G = f (Z)
L
where f : R+ → R+ is an arbitrary function and the quantities L and Z are given by
L=
PV
σ2 C
1/3
and
Z=
Q3 P 2 σ 2
V C2
1/3
.
(8)
A priori, the generality of the function f : R+ → R+ is not restricted by Assumption 3. Specializing further as in the ansatz (2) above, one may consider f to be of the
form f (z) = z p , for some p ≥ 0, so that G = const · Z p /L. In this case, the choice p = 1/2
leads precisely to the relation (5) obtained in Theorem 1 above. Moreover, we would like to
emphasize that the quantities L and Z have an intuitive and financially meaningful interpretation in terms of measuring liquidity and the size of meta-orders (see [15]).
The roadmap of this note is as follows. In Section 2, we provide a brief review of the
existing literature. Section 3 introduces some notation as well as the so-called Pi-Theorem
from dimensional analysis, which is the key to rigorously proving Theorems 1 and 2 in
Section 4. Section 5 concludes. Appendix A applies dimensional analysis to the pendulum
in more details and some proofs are moved to the Appendix B.
2. Literature review
As pointed out in recent reviews [5, 10], market impact can arise from different sources.
For instance, Kyle derives from an agent-based model in his seminal paper [14] that market
impact is linear in the order size and permanent in time. The majority of studies, however,
does not support this conclusion of Kyle’s model. Instead, a body of literature finds market
impact being non-linear in the order size and fading in time, e.g. [5]. In particular, the market impact is frequently found to be concave in the size of the meta-order and in particular
close to the square-root function, which causes the name square-root law for market impact,
see [2, 3, 6, 9, 12, 17, 19, 21]. Among other results, a market microstructure foundation
in favor of the square-root law is provided in [11]. The broad evidence for the square-root
law relies on studies having data from different venues, maturities, historical periods and
geographical areas and thus provides the square-root law with universality. On the other
hand, it deserves to be mentioned that some studies reveal empirically substantial deviations
from the square-root law, e.g. [1, 22].
Let us try to elaborate on the relation between dimensional analysis and general theory
by alluding once more to the analogy with the period of the pendulum. Complementary
to the introductory example, relation (3) from physics can also be derived from solving
differential equations. Analogously, the square-root law for market impact can also be
derived via solving partial differential equations, see Donier et al. [8, Section 5]. These
4
authors formulate the dynamics of the average buy and sell volume density of the latent
order book in terms of partial differential equations under minimal model requirements.
While the latent order book is a theoretical concept which records the trading intentions
of market participants, traders typically do not display their true supply and demand, so
that the fictitious, non-public latent order book differs from the observed limit order book.
As we derive the square-root law for market impact via dimensional analysis, Theorem 1
complements the existing literature.
3. Some linear algebra
To review the basic results of dimensional analysis we follow Chapter 1 of the book by
Bluman and Kumei [4]. Additionally, the interested reader is referred to [20] for a historical
perspective and to [7] for a purely mathematical treatment of dimensional analysis. We
formalize the assumptions behind dimensional analysis.
Assumption 4 (Dimensional analysis).
(i) Let the quantity of interest U ∈ R+ depend on n quantities W1 , . . . , Wn ∈ R+ , i.e.
U = H(W1 , W2 , . . . , Wn )
(9)
for some function H : Rn+ → R+ .
(ii) The quantities U, W1 , . . . , Wn are measured in terms of m fundamental dimensions
labelled L1 , . . . , Lm . For any positive quantity X, its dimension [X] satisfies [X] =
Lx1 1 · · · Lxmm for some x1 , . . . , xm ∈ R. If [X] = 1, the quantity X is called dimensionless.
The dimensions of the quantities U, W1 , W2 , . . . , Wn are known and given in the form
of the vectors a and b(i) ∈ Rm , i = 1, . . . , n, satisfying [U ] = La1 1 · · · Lamm and [Wi ] =
Lb11i · · · Lbmmi , i = 1, . . . , n. Denote by B = (b(1) , b(2) , . . . , b(n) ) the m × n matrix with
column vectors b(i) = (b1i , . . . , bmi )⊤ , i = 1, . . . , n.
(iii) For the given set of fundamental dimensions L1 , . . . , Lm , a system of units is chosen
in order to measure the value of a quantity. A change from one system of units to
another amounts to rescaling all considered quantities. In particular, dimensionless
quantities remain unchanged and formula (9) is invariant under arbitrary scaling of
the fundamental dimensions.
We can now state the main result from dimensional analysis (see [4]).
Theorem 3 (Pi-Theorem). Under Assumption 4, let x(i) := (x1i , . . . , xni )⊤ , i = 1, . . . , k :=
n − rank(B), and y := (y1 , . . . , yn )⊤ be solutions to the homogeneous and inhomogenous
systems Bx = 0 and By = a respectively. Then, there is a function F : Rk+ → R+ such that
U = W1y1 · · · Wnyn F (π1 , . . . , πk ),
(10)
where πi := W1x1i · · · Wnxni are dimensionless quantities, for i = 1, . . . , k.
We shall only need the special cases k = 0 and k = 1, which are spelled out in the two
subsequent corollaries.
5
Corollary 4. Under Assumption 4, suppose that rank(B) = n and let y := (y1 , . . . , yn )⊤ be
the unique solution to the linear system By = a. Then there is a constant const > 0 such
that
U = const ·W1y1 · · · Wnyn .
Corollary 5. Under Assumption 4, suppose that rank(B) = n − 1 and let x := (x1 , . . . , xn )
and y := (y1 , . . . , yn )⊤ be solutions to the homogeneous and inhomogeneous systems Bx = 0
and By = a respectively. Then there is a function F : R+ → R+ such that
U = W1y1 · · · Wnyn F (W1x1 · · · Wnxn ).
4. Market impact
The aim of this section is to formalize and prove Theorems 1 and 2 stated in the introduction
by applying Corollaries 4 and 5. In order to derive the price impact function from these
corollaries, we need to formalize Assumption 2 in the framework of Section 3. Therefore, we
take a closer look at this assumption and hence, the behavior of the quantities G, Q, P, V, σ 2
and C in case of leveraging the firm’s debt-equity ratio. From a conceptual point of view,
this gives an additional constraint on their behavior. This constraint can be understood
as an additional “dimension” in our analysis, which we refer to as the “Modigliani-Millerdimension” M. Scaling M by a factor A 6= 0 is equivalent to leveraging with a factor A.
Following [15], Assumption 2 can be reformulated in the following way:
Assumption 5. Scaling M by a factor A implies that
• Q, V and C remain constant,
• P changes by a factor A−1 ,
• σ 2 changes by a factor A2 ,
• G changes by a factor A.
To recapitulate in prose: setting A = 2 corresponds to paying out half of the equity as
dividends in the sense that each share yields a dividend of (1 − A−1 )P = P/2. The stock
price thus is multiplied by 1/2 while the volatility σ and the percentage price impact G are
multiplied by A = 2. The remaining quantities are not affected by changing the leverage, in
accordance with the insight of Modigliani and Miller [18] and the recent work by Kyle and
Obizheava [15]. As indicated in the introduction, this assumption is referred to as leverage
neutrality.
We now formulate Theorem 1 from the introduction in terms of Assumption 5 and prove
it:
Theorem 6. Under Assumptions 1 and 5, the market impact is of the form
r
Q
,
G = const · σ
V
for some constant const > 0.
6
(11)
S
U
T
M
Q
1
0
0
0
P
-1
1
0
-1
V
1
0
-1
0
σ2
0
0
-1
2
C
0
1
0
0
G
0
0
0
1
Table 1: A labelled overview of the matrix B related to the dimensions of the quantities (Q, P, V, σ 2 ) and the matrix K related to the dimensions of the quantities
(Q, P, V, σ 2 , C), respectively, as well as the vector a related to the dimensions of G.
Proof. Combining Assumptions 1 and 5 with the dimensions of Q, P, V and σ 2 introduced
in Section 1, we obtain that the matrix B and the vector a are given by

1
 0
B=
 0
0

−1 1
0
1
0
0 

0 −1 −1 
−1 0
2

0
 0 

and a = 
 0 .
1

(12)
Table 1 illustrates how B and a relate to the considered quantities and their dimensions. As
the matrix B has full rank, i.e. rank(B) = 4 = n, applying Corollary 4 yields
G = const · Qy1 P y2 V y3 σ 2y4
(13)
for some constant const> 0, where y = (y1 , y2 , y3 , y4 )⊤ is the unique solution of the linear
system By = a which is given by y = (0.5, 0, −0.5, 0.5)⊤.
Theorem 6 implies the well known square-root law for market impact. We would like
to highlight that the present derivation of the square-root law does not rely on economic,
empirical or theoretical assumptions except the dependence of G on Q, P, V and σ 2 as well
as leverage neutrality. Donier et al. [8] present an alternative derivation of the square-root
law relying on partial differential equations.
As discussed above, Kyle and Obizheava [15] consider yet another variable to influence
the market impact, namely the fixed inventory cost C leading to the weaker Assumption 3.
An economic motivation to include C in the analysis is provided also in [16]. Based on Assumption 3, Kyle and Obizheava [15] derive a more general result summarized in Theorem 2.
The methodology of Section 3 can be employed to prove a similar result stated below.
First of all, the matrix B used in the proof of Theorem 6 is extended by one column,
which corresponds to C, to obtain the matrix

1 −1 1
 0 1
0
K=
 0 0 −1
0 −1 0
0
0
−1
2

0
1 
.
0 
0
(14)
The vector a = (0, 0, 0, 1)⊤ defined in (12) remains unchanged. Table 1 summarizes how K
and a can be derived and should be read as follows: Assume the measurement of dimension S
referring to the unit of shares changes by a factor S, then Q changes by a factor S, P changes
7
by a factor S −1 , V changes by a factor S, σ 2 and C do not change as S 0 = 1. Likewise, the
last row labelled by M indicates that if the stock’s leverage changes by a factor A, Q does
not change, P changes by a factor A−1 , and so on.
Let us now compute the solution space H of the homogeneous system Kx = 0, which is
given by the kernel of the linear map induced by K, as well as the solution space I of the
inhomogeneous system Ky = a:
  
 


 




3
−1
3









 2

−1

 2





  
 

 

 
 
 
H = λ  −1 , λ ∈ R
and I =  0 + λ −1 , λ ∈ R .
(15)
 

 


 





  1
 0 


 1












−2
1
−2
Theorem 7. Suppose Assumptions 3 and 5 hold. Fix x = (x1 , . . . , x5 )⊤ ∈ H and y =
(y1 , . . . , y5 )⊤ ∈ I. There is a function f : R+ → R+ such that
(16)
G = Qy1 P y2 V y3 σ 2y4 C y5 f Qx1 P x2 V x3 σ 2x4 C x5 .
For example, setting
⊤
2 1 1 2
∈H
x = 1, , − , , −
3 3 3 3
⊤
1 1 1 1
y = 0, − , − , ,
∈ I,
3 3 3 3
and
we obtain
G=
=
σ2 C
PV
1/3
f
1
f (Z) ,
L
Q3 P 2 σ 2
V C2
1/3 !
(17)
where L and Z are defined in (8).
Proof. Combining Assumptions 3 and 5 with the dimensions of Q, P, V , σ 2 and C introduced
in Section 1, we recover the matrix K and the vector a given in (14). Since Assumption 4
is satisfied and rank(K) = 4, applying Corollary 5 completes the proof.
Remark 1. It is rather obvious from a financial point of view that the function f : R+ → R+
should be monotonically non-decreasing, by considering the variable Q while keeping the
others constant. Likewise, from a financial perspective it seems reasonable to require f to be
continuous.
Fixing x and y as in Theorem 7, different choices of f in Equation (17) lead to some
particularly relevant market impact models studied in the literature.
(a) The proportional price impact: f ≡ const leads to
G = const ·
σ2 C
PV
1/3
(b) The square-root impact: f (z) = const ·z 1/2 leads to
r
Q
.
G = const · σ
V
8
.
(c) The linear price impact: f (z) = const ·z leads to
G = const · Q
σ4 P
CV 2
1/3
.
Kyle and Obizhaeva [15] also point out that these models are coherent with different functional forms for f in (17), namely f (z) = z p , so that (a), (b) and (c) arise from setting (a)
p = 0, (b) p = 1/2 and (c) p = 1. It should be noticed that restricting the functional form
of f to f (z) = z p coincides with replacing the general function g in Assumption 3 with a
function taking the form of the ansatz (2) mentioned in the introduction in the context of
the pendulum.
Alternatively, the price impact models (a), (b) and (c) can also be recovered by choosing
the vectors x ∈ H and y ∈ I from Equation (16) appropriately. In more detail, setting
x = (0, 0, 0, 0, 0)⊤ implies that f Qx1 P x2 V x3 σ 2x4 C x5 from (16) reduces to a constant, so
that choosing
⊤
1 1 1 1
∈ I,
(a) y = 0, − , − , ,
3 3 3 3
⊤
1
1 1
(c) y =
, 0, − , , 0
∈ I,
2
2 2
⊤
1 2 2 1
∈ I,
(b) y = 1, , − , , −
3 3 3 3
results in the respective models (a), (b) and (c).
5. Conclusion
The main contribution of this manuscript is a novel derivation of the square-root law for
market impact. The strong empirical support in favor of this law provides it with a universal
character. Inspiring for our work, Kyle and Obizhaeva [15] derive a general form for the
market impact function relying on dimensional analysis as well as the concepts of leverage
neutrality and market microstructure invariance, where the square-root law turns out to be
a special case of their result. Complementary to their approach, we present a direct and
simple derivation of the square-root law requiring only two assumptions: Firstly, the market
impact of a given meta-order only depends on its size, the corresponding stock price, the
trading volume in the stock as well as its volatility. Secondly, we employ the concept of
leverage neutrality as in [15]. This idea can be traced back to the Modigliani-Miller theorem
[18] and explains how the considered quantities behave when leveraging a firm’s debt-equity
ratio. Relying on these plausible assumptions, we apply dimensional analysis in a rigorous
way to show that the market impact of a meta-order is proportional to the volatility as well
as to the square-root of this order’s size and inversely proportional to the trading volume.
In the setting of Theorem 1, somehow surprisingly, the market impact does not depend
on the stock price, although – a priori – the price is included in our analysis. There is a
simple explanation: the stock price is the only quantity in our analysis measured in units
of money. Hence it cannot play a role in the market impact as this dimensionless quantity
does not involve the “dimension” money. In Appendix A we give a more detailed discussion
of this argument in the form of an analogy to the case of the pendulum.
9
If we follow [15] and include an “inventory cost” into the set of quantities under investigation, we cannot uniquely determine a functional form for the market impact anymore (see
Theorem 7). Rather we can precisely characterize its general form.
A. The pendulum
We want to illustrate the difference between Theorem 1 and 2 by presenting its analogues
in the context of the pendulum mentioned at the very beginning of this paper.
Consider a pendulum with length l (measured in meters), mass m (measured in grams)
and period (measured in seconds). Assume that the period depends only on l, m and the
acceleration g of gravity (measured in meters per seconds squared). That is, there is a
function f : R3+ → R+ such that
p = period = f (l, m, g).
From Table 2, we get the matrices

1 0
D= 0 1
0 0

1
0 
−2


0
and c =  0 
1
which represent the dimensions of the quantities (l, m, g) and the period p respectively in
the unit system meter, gram and seconds. As D is of full rank, it follows from Corollary 4
that
(18)
period = const · ly1 my2 g y3
for some const > 0, where the unique solution of the linear system Dy = c is given by
y = (0.5, 0, −0.5)⊤. Thus,
s
l
.
(19)
period = const ·
g
The crucial assumption in the reasoning above is that the period of the pendulum is
completely determined by its length, its mass and the acceleration due to gravity. However,
it is conceivable that the period also depends on other variables, e.g. the amplitude a
(measured in meters) of the observed swing. In other words, we might also start from the
weaker assumption
period = h(l, m, g, a).
(20)
The matrix containing the dimensions of the observed quantities is now given by


1 0 1 1
e =  0 1 0 0 .
D
0 0 −2 0
It follows from Corollary 5 that the general form of the relation (20) is given by
period = ly1 my2 g y3 ay4 h̃(lx1 mx2 g x3 ax4 )
e =0
for some function h̃ : R+ → R+ , where x = (x1 , x2 , x3 , x4 ) is a solution of the system Dx
e = c. Thus,
and y = (y1 , y2 , y3 , y4 ) a solution of the system Dy
period = ly1 g −1/2 a1/2−y1 h̃(lx1 ax1 )
10
(21)
lenght
mass
time
l
1
0
0
m
0
1
0
g
1
0
-2
a
1
0
0
period
0
0
1
Table 2: A labelled overview of the matrix summarizing the dimension of the quantities
considered to determine the period of a pendulum.
for some y1 , x1 ∈ R corresponding to the solutions x, y of the above linear systems. The
solution depends on one degree of freedom involved in the choice of x1 and y1 . This is in
contrast to the solution (18) above, where we simply had x1 = 0 and y1 = 12 . In the setting
(20), dimensional analysis is not able to determine how the degree of freedom should be
used to fix the choice of x1 and y1 . We need some additional information. In physics we
have the possibility of experiments. Already Galileo noticed the – at first glance surprising
– experimental result that the period of the pendulum does not depend on the amplitude
at least, in first order approximation. Using this insight from experimental physics, we
conclude that we must have x1 = 0 and y1 = 0.5, i.e. the only physically relevant solution
of the general solution (21) is given by
f (l, m, g) = const · l1/2 g −1/2 ,
(22)
so that we are back to (19).
Let us make the analogy to the case of Theorems 1 and 2 above. Dimensional analysis
alone cannot decide whether the special solution given by Theorem 1 is the “true” relation
between the market impact and the relevant variables, or whether some other solutions
provided by Theorem 2 yields the “true” relation. To answer this question one has to take
recourse either to economic theory or to empirical analysis. This is analogous to the above
discussed situation of the pendulum where physical experiments yields that the special case
(22) of the more general solution (21) is in fact the“true” relation (as long as the amplitude
remains within reasonable bounds).
We conclude this discussion by observing that the question whether the solution (11)
is the “true” relation among the more general candidates given by (16) is tantamount to
saying whether the market impact actually depends – ceteris paribus – on the price of the
stock in (6). In the setting of Theorem 2, fix the quantities Q, V σ 2 and C. Varying the
price P of the stock then corresponds to varying the size, i.e. the market capitalization of
the company, while leaving everything else unchanged. The above question then boils down
to the question whether this variation of size of the company should have an effect on the
relative market impact of the meta-order or not.
B. Proof of Pi-Theorem
Proof of Theorem 3: We pass to logarithmic coordinates by using the following notation:
e for Z
e = log(Z). On the logarithm scale U
e satisfies
given Z ∈ R+ we shall write Z
e=g W
f1 , . . . , W
fn
(23)
U
for some function g : Rn → R.
11
If the left and the right hand sides of (23) do not depend on L1 , then it is sufficient to
work with the units (L2 , . . . , Lm ). On the other hand, if a1 6= 0 and b11 = · · · = b1n = 0 then
g ≡ 0. If there exists i ∈ {1, . . . , n} such that b1i 6= 0, we assume without loss of generality
f1 + U
e and X
ei−1 := − b1i W
f1 + W
fi , i = 2, . . . , n we have
that b11 6= 0. Putting Ve := − ba111 W
b11
a1 f
b1n f
b1i f
e
e
e
f
V =−
W1 + g W1 ,
W1 + X1 , . . . ,
W1 + Xn−1
b11
b11
b11
f1 , X
e1 , . . . , X
en−1 ,
=f W
ei−1 ,
for some function f . Let λ ∈ R and put L∗1 := eλ L1 . Since the dimensions of Ve and X
e
e
en−1
i = 2, . . . , n are given in terms of the units (L1 , . . . , Lm ), the quantities V , X1 , . . . , X
∗
remain unchanged upon passing to the system of units (L1 , L2 , . . . Lm ). On the other hand,
P
e i so that in the system of units (L∗1 , L2 , . . . , Lm ) it
e ∗1 + m bi1 L
log ([W1 ]) = −λb11 + b11 L
i=2
holds
f1 , X
e1 , . . . , X
en−1 .
Ve = f λb11 + W
Since λ was taken arbitrary, f does not depend on the first component, that is,
e1 , . . . , X
en−1 .
Ve = f X
(24)
By repeating the argument rank(B) − 1 times, we obtain


n
n
n
X
X
X
fj  ,
fj , . . . ,
fj = h 
e−
xkj W
x1j W
yj W
U
j=1
j=1
j=1
where k := n − rank(B) and x(i) := (xi1 , . . . , xin )⊤ satisfies xi(k+i) = 1 and xi(k+j) = 0 for
j ∈ {1, . . . , k} \ {i} for all i = 1, . . . , k. Thus, x(i) are linearly independent. Notice that
P
fj are dimensionless.
fj for i = 1, . . . , k as well as U
e − Pn yj W
by construction nj=1 xij W
j=1
(i)
Therefore, x ∈ ker(B) and y is the solution of the inhomogeneous system By = a. Hence,
there is a function F : Rk → R+ such that
U W1−y1 · · · Wn−yn = F (π1 , . . . , πn )
x
x
with πj := W1 1j · · · Wn nj , j = 1, . . . , k.
(25)
Proof of Corollary 4: First notice that if k = 0 then n = rank(B). Repeating the
argument leading to (24) rank(B) − 2 times, we can find (z1 , . . . , zn−1 ) ∈ Rn−1 , a quantity
m
X with dimension [X] = Lα
m and a function f : R → R such that
e−
Ye := U
n−1
X
j=1
fj = f (X).
e
zj W
Let us denote by Lcmm the dimension of Y . We can assume without loss of generality that
αm 6= 0. As in the proof of Theorem 3, we have
e−
Ve := U
n−1
X
fj − cm X = − cm X + f (X)
e = g(X).
e
zj W
αm
αm
j=1
12
for some function g. Let λ ∈ R and put L∗m = eλ Lm . Since Ve is dimensionless, its value does
not change when passing to the unit L∗m . On the other hand, log([X]) = −αm λ + αm L∗m .
Hence, with respect to the unit L∗m we have
Ve = g(αm λ + X).
Since λ was taken arbitrary, the function g must be a constant. Thus, there is const > 0
and y = (y1 , . . . , yn ) ∈ Rn such that U = const · W1y1 · · · Wnyn . Since the right hand side of
the latter equation has the dimension of U , we have By = a.
Proof of Corollary 5: The result follows from a direct application of Theorem 3.
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