Price Equilibrium and Arbitrage
Given a price range
[F,C][F,C]
within which the price of the specific cryptocurrency varies, where
FF
is the strike price for call and
CC
is the strike price for put. Let
c[F,C]c\in [F,C]
be the market price of the cryptocurrency on the platform. Thus
cFc-F
is the intrinsic value of a call token, while
CcC-c
is the intrinsic value of a put token. Let
x,yx,y
be the amount of call and put token respectively, and
zz
be the total reserved value. The total reserved value is all the asset collected that are used to generate Antimatter tokens. If call and put tokens increase at the same ratio, the price should not change. We aim to build a model that
zz
is a function of
xx
and
yy
,
z=f(x,y)z=f(x,y)
, such that
kf(x,y)=f(kx,ky)kf(x,y)=f(kx,ky)
With this equation, we are able to define the price of each token:
zx,zy\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}
When the volumes of both tokens are equal, it is expected that the prices of both tokens are equal(In reality, the price of call tokens might be higher, because it has more upside potential). When the volume of call tokens far exceeds the volume of put tokens, it is expected that the price of put token approaches
00
. When the volume of put tokens far exceeds the volume of call tokens, it is expected that the price of put token approaches
CFC-F
. The behavior of call token is wilder but in the same way. One note is that the price of both tokens depends on the ratio of volume of both tokens, rather than the difference between them.
For example, we work with ETH and USDT. The price of ETH in terms of USDT varies within the interval
[1000,4000][1000,4000]
. Here
10001000
is the strike price of call, and
40004000
is the strike price of put. If the ratio between call token and put token generated is $6:4$, it is expected that the market price is
35003500
In this case, the cost to generate a call token is about
60006000
and the cost to generate a put token is about
240240
. When there is a difference between the market price and market price of ETH, one can buy and sell call or put token in two market to make profit.
The actual model goes as follows. Let
z=(CF)y2x2+y2+eCFCx2x2+y2z=\frac{(C-F)y^2}{\sqrt{x^2+y^2}}+e\cdot\frac{C-F}{C}\frac{x^2}{\sqrt{x^2+y^2}}
where
ee
is the price of ETH. This expression ensures that
zx,zy0\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\geq0
. This model works well with prices of tokens because of positive definiteness. Because the lowest possible price of ETH is
00
and the highest price of ETH is infinity, the price of call token will be appreciated to a greater extend, if the price moves in the right direction. The prices of both tokens remain in the interval
[0,CF][0,C-F]
for current version.
Starting from
z=f(x,y)z=f(x,y)
, such that
kf(x,y)=f(kx,ky)kf(x,y)=f(kx,ky)
, we have the equivalent following
xfx+yfy=fxf_x+yf_y=f
. Let
r=x2+y2,θ=tan1(yx)(0,π2)r=\sqrt{x^2+y^2},\theta=\tan^{-1}(\frac{y}{x})\in (0,\frac{\pi}{2})
. Also,
x=rcosθ,y=rsinθx=r\cos\theta,y=r\sin\theta
. We further transform:
xfx+yfy=fxrfx+yrfy=frxf_x+yf_y=f\equiv\frac{x}{r}f_x+\frac{y}{r}f_y=\frac{f}{r}
. By chain rule, we have
fr=fr\frac{\partial f}{\partial r}=\frac{f}{r}
Therefore
f=A(θ)rf=A(\theta)r
for some function
AA
(This is true for any function of
θ\theta
). Now we put these aside and consider the second question. The idea is that because
(CF)f=U+Ee(C-F)f=U+Ee
, we may find
U,EU,E
first and then sum them up because of homogeneity. We need
Ux<0,Uy>0,Ex>0,Ey<0\frac{\partial U}{\partial x}<0,\frac{\partial U}{\partial y}>0,\frac{\partial E}{\partial x}>0,\frac{\partial E}{\partial y}<0
. It worth checking that
U=y(yx2+y2)i,E=x(xx2+y2)iU=y\cdot(\frac{y}{\sqrt{x^2+y^2}})^i,E=x\cdot(\frac{x}{\sqrt{x^2+y^2}})^i
satisfy the above inequalities. By homogeneity conditions, it turns out summing through index
ii
does no harm anything (although it turns out that the summation is redundant). We relate
UU
and
EE
with
θ\theta
by
sinθ=yx2+y2,cosθxx2+y2\sin\theta=\frac{y}{\sqrt{x^2+y^2}},\cos\theta\frac{x}{\sqrt{x^2+y^2}}
. Then by choice of
AA
, we can obtain that
z=f(x,y)=xei=1ei(xx2+y2)i+yi=1ui(yx2+y2)iz=f(x,y)=x\cdot e\cdot\sum_{i=1}^\infty e_i(\frac{x}{\sqrt{x^2+y^2}})^i+y\cdot\sum_{i=1}^\infty u_i(\frac{y}{\sqrt{x^2+y^2}})^i
Next, we consider why
II
must be equal to
11
. For simplicity, we let those constants be
11
Suppose
I=2I=2
, then
E=x3x2+y2,Ex=x2(x2+3y3)(x2+y2)2E=\frac{x^3}{x^2+y^2},E_x=\frac{x^2(x^2+3y^3)}{(x^2+y^2)^2}
. Similarly,
Ux=2y3x(x2+y2)2U_x=\frac{-2y^3x}{(x^2+y^2)^2}
. If we consider
zx=Ux+eEx=ex4+y2x(3ex2y)(x2+y2)2z_x=U_x+eE_x=\frac{ex^4+y^2x(3ex-2y)}{(x^2+y^2)^2}
and
zy=Uy+eUy=y4+3x2y22ex3y(x2+y2)2z_y=U_y+eU_y=\frac{y^4+3x^2y^2-2ex^3y}{(x^2+y^2)^2}
That their numerators are different means that we need more relations on
xx
and
yy
, which is implausible since
xx
and
yy
are the token volume. Thus
k=1k=1
. The final thing is to take some constants that have good numerical performance and it is
E=CFCx2x2+y2,U=(CF)y2x2+y2E=\frac{C-F}{C}\cdot\frac{x^2}{\sqrt{x^2+y^2}},U=(C-F)\cdot\frac{y^2}{\sqrt{x^2+y^2}}
.
zz
follows directly.
Last modified 3mo ago
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