Hueristic
Legendary
Activity: 4550
Merit: 7169
Doomed to see the future and unable to prevent it
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I want to argue this, because it is such a fatalistic take:
1. CME started derivatives in late 2017 and was clearly responsible for "taming" (as they said) bitcoin from 20K down to 3k.
2. However, CME/derivatives did not prevent bitcoin subsequent rise from 3K to 69K with a great intensity in 2018-2021.
3. The entry of Microstrategy represents a time period where bitcoin slowed down significantly.
4. That said, it could be some other players that slow it down. I have some (that are major WS players) in mind.
5. It is still possible that the internal properties of bitcoin (possibly represented by the powerlaw) would allow it to persevere.
TL;DR: saying that it is ALL derivatives makes it a fatalistic determination, as they are not going away, imho.
1. CME's rollout in 2017 is correct, but it was gradual. 2. There were no options and many other things, limiting derivatives and hedging. 3. It is exactly derivatives that enable massive buying without creating a parabolic rise. I will just give examples of 2 mechanisms. a. Massive hedging options. Buy BTC in spot, short futures being one. b. More participants and derivatives means that they can dissipate big buys through futures and options instead of forcing the spot price up. If there were no derivatives of any kind and everything else that happened afterwards stayed the same, the price would be who knows where. Several fold higher at least. There are other factors and impacts of course, that is always the case. But primarily derivatives are what tamed Bitcoin and continue to suppress the price despite of massive buying. I don't have the time for this but here is the the proof they are mathematically linked if that's what your trying to question. https://www.researchgate.net/publication/368842542_Price_Discovery_for_Derivatives8 Conclusion
We have considered a basic framework of general price discovery in contingent claim markets and
characterized the economic mechanism through which contingent claim prices jointly incorporate in-
formation. Price discovery takes place not only within markets but also between markets. Our setting
extends the theory of informed trading to derivatives and higher-order information. In doing so, our
results explain empirical practice and stylized facts, bridge the gap with the empirical literature, and
yield new empirical questions. Within the general framework, special cases of interest and possible
extensions can be analyzed further. These are avenues for future research.
A Appendix
A.1 Mathematical Assumptions
The rigorous formulations of both the market maker’s and insider’s problems are obtained under
the follow assumptions. These assumptions are maintained for all mathematical statements made in
the paper.
For δ > 0, recall the H¨older space Cδ([x, ¯x],R) consists of δ-H¨older continuous elements of C([x, ¯x],R),
equipped with the H¨older seminorm [f]δ= sup
x6=y
|f(x)−f(y)|
|x−y|δ.
Assumption A.1.
Let Fbe the Borel σ-field generated by the uniform norm k · k∞on Ω,µbe the probability measure
on the measurable space (Ω,F)such that the canonical process x7→ ωxhas the same probability law as
(σ(x)Bx).
Fix γ∈(1
3,1
2)and δ∈(0,1] with δ+γ > 1. Let Cδ([x, ¯x],R)and Cγ([x, ¯x],R)be the corresponding
H¨older spaces, and Ωγ={ω∈Ω : ω(0) = 0 and ω∈Cγ([x, ¯x],R)}.
(i) For every s∈S, the insider’s portfolio of contingent claims W(·, s): [x, ¯x]→Rlies in
Cδ([x, ¯x],R).
(ii) Possible realization ωof total order flow across states lies in the measurable space (Ωγ,Fγ),
where Fγ=F ∩ Ωγ, and µγdenotes the measure µrestricted to Ωγ.
45
(iii) The insider’s strategy s7→ W(·, s), S →(Cδ[x, ¯x],R), is continuous.68
(iv) The (squared) noise trading intensity σ2(·)of Arrow-Debreu markets lies in Cδ([x, ¯x],R)and
σ2(x)>0for all x.
The condition δ+γ > 1 in Assumption A.1 ensures that the integrand W(·, s) and integrator
ωtogether have sufficient pathwise (H¨older) regularity to define a pathwise integral. This condition
means that W(·, s) and ωjointly satisfy a regularity condition that is infinitesimally stronger than
the Lipschitz condition.69 Assumption A.1(i) imposes H¨older δ-continuity in xon W(x, s). It is
(much) weaker than differentiability and imposes no constraint on the insider’s demand from a practical
perspective. Assumption A.1(ii) replaces the underlying probability space (Ω,F, µ) by (Ωγ,Fγ, µγ),
where H¨older γ-continuity holds for the sample paths. This is without loss of generality because
Ωγ∈ F and µ(Ωγ) = 1.70
Assumption A.1(iii) assumes continuity of insider’s strategy with respect to signal s. This imposes
no constraint on the insider’s strategy when η(·, s) is continuous in s. Assumption A.1(iv) assumes
that noise traders are present in all AD markets. If noise traders are absent in a neighborhood of states,
there would be no trade in that neighborhood of states.
Lemma A.1.
Under Assumption A.1, the following holds.
(i) (Pathwise Integral) For all ω∈Ωγ,W∈Cδ([x, ¯x],R), and x∈[x, ¯x], the limit of Riemann
sums
Zx
x
Wy
σ2(y)dωy≡lim
max
k|xk+1−xk|→0
x=x0<···<xn=¯x
n−1
X
k=0
Wxk
σ2(xk)·[ωxk+1∧x−ωxk∧x]
exists and therefore defines an ω-by-ωRiemann integral.
(ii) (Joint Measurability of Data and Parameter) The map
(ω, W )
| {z }
(data, parameter)
7→ Z¯x
x
Wx
σ2(x)dωx,(Ωγ,[·]γ)×Cδ([x, ¯x],R)→R,
is continuous—in particular, measurable.
(iii) (Conditional Likelihood of Data) For all x∈[x, ¯x]and W∈Cδ([x, ¯x],R), there exists a µγ-null
68The topology on Sis the same topology that generates the Borel σ-field on S.
69Having a H¨older exponent strictly larger than one can be interpreted as infinitesimal stronger than being Lipschitz.
70µ(Ωγ) = 1 because Brownian paths lie in C1
2−([x, ¯x],R) for any ∈(0,1
2).
46
set N, which may depend on xand W, such that, for all ω∈Ωγ\N,
Zx
x
Wy
σ2(y)dωy=Zx
x
Wy
σ2(y)dBy(ω),
where the integral on the left-hand side is the pathwise integral defined in (i) and the integral on the
right-hand side is a version of the Itˆo integral.
Proof. (i) We note first that H¨older continuity is preserved by taking quotients when the denominator
is bounded away from zero. Let W,σ∈Cδ([x,¯x],R) with min
x∈[x,¯x]σ(x)> α > 0 for some α > 0. Then
W(y)
σ(y)−W(x)
σ(x)≤|σ(x)| |W(y)−W(x)|+|W(x)| |σ(y)−W(x)|
|σ(y)| |σ(x)|
≤kWk∞[W]γ+kσk∞[σ]γ
α2· |y−x|δ,
which implies W
σ∈Cδ([x, ¯x],R). Therefore it suffices to prove the claim with Win place of W
σ2.
Given α,β > 0, we shall consider the space Cα,β
2([x, ¯x],R) of all functions Ξ from {(y, x) : x≤y≤
x≤¯x}to Rwith seminorm
[Ξ]α,β ≡[Ξ]α
|{z}
≡sup
y<x
|Ξy,x|
(x−y)α
+ sup
y<r<x
|Ξy,x −Ξy,r −Ξr,x|
|x−y|β<∞.
For ω∈Ωγand W∈Cδ([x, ¯x],R), define Ξω,W by
Ξω,W
y,x ≡Wy·(ωx−ωy).
Since Ξω,W
y,x ≤[ω]γ|x−y|γand, for x≤y≤r≤x≤¯x,
Ξω,W
y,x −Ξω,W
y,r −Ξω,W
r,x =|(Wy−Wr)·(ωx−ωr)| ≤ [W]δ[ωγ|x−y|γ+δ,
we have Ξω,W ∈Cγ,γ+δ
2([x, ¯x],R). Thus, because γ≤1< γ +δ, it follows from the Sewing Lemma of
Friz and Hairer (2020) that the limit of Riemann sums
lim
max
k|xk+1−xk|→0
x=x0<···<xn=¯x
n−1
X
k=0
Wxk·[ωxk+1∧x−ωxk∧x]
exists. This proves the claim.
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(ii) We will prove the continuity of the map71
(ω, W )7→ Z•
0
Wtdωt,(Ωγ,[·]γ)×Cδ([x, ¯x]R)→Cγ([x, ¯x],R).
By the Sewing Lemma quoted in (i), it suffices to check the continuity of
(ω, W )7→ Ξω,W ,(Ωγ,[·]γ)×Cδ([x, ¯x],R)→Cγ ,γ+δ
2([x, ¯x],R).
Given ω, ˜ω∈Ωγ,W,˜
W∈Cδ([x, ¯x],R),
(Ξω,W
y,x −Ξω,W
y,r −Ξω,W
r,x )−(Ξ˜ω, ˜
W
y,x −Ξ˜ω, ˜
W
y,r −Ξ˜ω, ˜
W
r,x )
=(Wy−Wr)·(ωx−ωr)−(˜
Wy−˜
Wr)·(˜ωx−˜ωr)
≤[W−˜
W]δ[ω]γ|x−y|γ+δ+ [ ˜
W]δ[ω−˜ω]γ|x−y|γ+δ.
Therefore (below we use the H¨older norm k · kWinstead of the seminorm [ ·]W)
Ξω,W
y,x −Ξ˜ω, ˜
W
y,x =Wy·(ωx−ωy)−˜
Wy·(˜ωx−˜ωy)
≤ kWk∞[ω]γ|x−y|γ+kW−˜
Wkδ·(bδ∨1) [˜ωγ|x−y|γ.
This proves the claim.
(iii) Let Rx
xWydBybe a given version of the Itˆo integral. By the continuity of W, the chosen version
is a probability limit72
Zx
x
WydBy= lim
max
k|xk+1−xk|→0
a=x0<···<xn=x
n−1
X
k=0
Wxk·[Bxk+1∧x−Bxk∧x] in µγ-probability.
Then one can pass to a subsequence of the (implicitly given) sequence of partitions such that, for
µγ-a.a. ω∈Ωγ(where we use the same notation for the subsequence),
Zx
x
WydBy(ω)≡lim
max
k|xk+1−xk|→0
x=x0<···<xn=x
n−1
X
k=0
Wxk·[ωxk+1∧x−ωxk∧x] = Zx
x
Wydωy.
This proves the claim.
71This is a stronger property than that stated in Lemma A.1(ii).
72See Revuz and Yor (2013).
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A.2 Proof of Theorem 3.1
Claim A.1. Under Assumption A.1, for each W∈Cδ([x, ¯x],R), define a probability measure PWon
the measurable space (Ωγ,Fγ)via the Radon-Nikodym density
dPW
dµγ
=eR¯x
x
Wx
σ2(x)dωx−1
2R¯x
x
W2
x
σ2(x)dx,(A.1)
where µγis the canonical Wiener measure µrestricted to Ωγ, and R¯x
x
Wx
σ2(x)dωxis the pathwise integral
defined in Lemma A.1(i). Then the canonical process on (Ωγ,Fγ,PW)has the same law as
Wxdx +σ(x)dBx
where (Bx)is a standard Brownian motion.
Proof. This follows immediately from Lemma A.1(iii).
Proof of Theorem It suffices to prove (i), after which (ii) follows trivially. Consider the probability
space (Cγ([x, ¯x],R),G, µγ),where the σ-field Gis the Borel σ-field given by the uniform norm, and µγ
is such that the canonical process has the same law as (σ(x)Bx) where (Bx) is a standard Brownian
motion.
Under Assumption A.1(i), (iii), and (iv), we have by Lemma A.1(i) that the expression π1(ds, ω;f
W)
is well-defined for each ω∈Cγ([x, ¯x],R). By Claim A.1, under the probability measure Pf
W(·,s)defined
by the Radon-Nikodym density
dPf
W(·,s)
dµγ
=eR¯x
xf
W(x,s)
σ2(x)dωx−1
2R¯x
xf
W(x,s)2
σ2(x)dx,
the canonical process on Cγ([x, ¯x],R) has the same law as f
W(x, s)dx+σ(x)dBxwhere (Bx) is a standard
Brownian motion. Therefore the probability measure on Cγ([x, ¯x],R)×Sdefined by
eR¯x
xf
W(x,s)
σ2(x)dωx−1
2R¯x
xf
W(x,s)2
σ2(x)dx ·µγ(dω)⊗π0(ds) = eR¯x
xf
W(x,s)
σ2(x)dωx−1
2R¯x
xf
W(x,s)2
σ2(x)dx ·µγ(dω)⊗π0(ds)
correctly specifies the joint probability law of (ω, s) according to the market maker’s conjecture f
W.
Under Assumption A.1(ii), we have by Lemma A.1(ii) that eR¯x
xf
W(x,s)
σ2(x)dωx−1
2R¯x
xf
W(x,s)2
σ2(x)dx is jointly mea-
surable in (ω, s). Therefore, an application of Fubini-Tonelli Theorem shows that π1(ds, ω;f
W) specifies
the probability measure on Sconditional on ω, i.e. {π1(ds, ω;f
W)}ω∈Cγ([x,¯x],R)is the ω-disintegration
of the family {Pf
W(·,s)}s∈S, where Pf
W(·,s)is the probability law of aggregate order flow conditional on
s, according to market maker belief f
W(·,·) regarding the insider’s trading strategy. This proves the
49
theorem.
A.3 Proof of Theorem 4.2
To make the dependence of expected AD price P(x, W ;f
W) (18) on Wmore explicit, we write
P(x, W ;f
W) = EPW[P(x, ω ;f
W)]
=EPW[ZS
η(x, s)π1(ds, ω;f
W)]
=EPW[ZS
η(x, s)eI(ω,s;f
W)π0(ds)
/C(ω;f
W)]
=Eµγ[ZS
η(x, s)eI(ω,s;f
W)eR¯x
xf
W(x0,s)W(x0)
σ2(x0)dx0
·π0(ds)
/C0(ω;f
W)],(A.2)
where
I(ω, s;f
W) = Z¯x
xf
W(x, s)
σ2(x)dωx−1
2Z¯x
xf
W(x, s)2
σ2(x)dx
and C0(ω, W ;f
W) = RSeI(ω,s0;f
W)eR¯x
xf
W(x0,s0)W(x0)
σ2(x0)dx0
π0(ds0). The equality (A.2) follows from the fact that
the law of the canonical process x7→ ωxunder PWis the same as the law of x7→ Rx
0W(x0)dx0+ωx
under µγ.
The Gˆateaux derivative of the payoff functional W7→ R¯x
xW(x)η(x, s)ds is trivially identified with
η(·, s), and it suffices to consider the cost functional
Jc(W) = Z¯x
x
W(x)·P(x, W ;f
W)dx.
Let v∈Cδ([x, ¯x],R) and define
f(ε) = Jc(W+εv).
The Gˆateaux derivative dJc(W) evaluated at vcan be computed by invoking the Dominated Conver-
gence Theorem and differentiating under the integral signs:
dJc(W) = f0(0)
=Z¯x
x
v(x)·P(x, W ;f
W)dx +Z¯x
x
W(x)g(x)dx (A.3)
for some function g(x). The first integral of (A.3) verifies the AD term NAD(W)(v) of Equation (20).
It remains to show the price impact term NK(W)(v) of Equation (21) is the second integral of (A.3).
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Have fun disproving it if you want to. |
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