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angerisangeris
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‎.gitignore‎

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__pycache__
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*.pdf
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.DS_Store

‎arbitrage.py‎

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import numpy as np
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import cvxpy as cp
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# Problem data
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global_indices = list(range(4))
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local_indices = [
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[0, 1, 2, 3],
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[0, 1],
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[1, 2],
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[2, 3],
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[2, 3]
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]
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reserves = list(map(np.array, [
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[4, 4, 4, 4],
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[10, 1],
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[1, 5],
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[40, 50],
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[10, 10]
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]))
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fees = [
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.998,
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.997,
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.997,
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.997,
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.999
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]
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# "Market value" of tokens (say, in a centralized exchange)
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market_value = [
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1.5,
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10,
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2,
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3
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]
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# Build local-global matrices
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n = len(global_indices)
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m = len(local_indices)
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A = []
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for l in local_indices:
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n_i = len(l)
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A_i = np.zeros((n, n_i))
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for i, idx in enumerate(l):
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A_i[idx, i] = 1
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A.append(A_i)
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# Build variables
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deltas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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lambdas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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psi = cp.sum([A_i @ (L - D) for A_i, D, L in zip(A, deltas, lambdas)])
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# Objective is to maximize "total market value" of coins out
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obj = cp.Maximize(market_value @ psi)
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# Reserves after trade
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new_reserves = [R + gamma_i*D - L for R, gamma_i, D, L in zip(reserves, fees, deltas, lambdas)]
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# Trading function constraints
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cons = [
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# Balancer pool with weights 4, 3, 2, 1
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cp.geo_mean(new_reserves[0], p=np.array([4, 3, 2, 1])) >= cp.geo_mean(reserves[0]),
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# Uniswap v2 pools
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cp.geo_mean(new_reserves[1]) >= cp.geo_mean(reserves[1]),
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cp.geo_mean(new_reserves[2]) >= cp.geo_mean(reserves[2]),
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cp.geo_mean(new_reserves[3]) >= cp.geo_mean(reserves[3]),
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# Constant sum pool
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cp.sum(new_reserves[4]) >= cp.sum(reserves[4]),
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new_reserves[4] >= 0,
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# Arbitrage constraint
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psi >= 0
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]
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# Set up and solve problem
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prob = cp.Problem(obj, cons)
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prob.solve()
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print(f"Total output value: {prob.value}")

‎latexify.py‎

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#Original Author: Prof. Nipun Batra
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# nipunbatra.github.io
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from math import sqrt
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import matplotlib
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SPINE_COLOR = 'gray'
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def latexify(fig_width=None, fig_height=None, font_size=12, columns=2):
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"""Set up matplotlib's RC params for LaTeX plotting.
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Call this before plotting a figure.
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Parameters
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----------
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fig_width : float, optional, inches
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fig_height : float, optional, inches
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columns : {1, 2}
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"""
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# code adapted from http://www.scipy.org/Cookbook/Matplotlib/LaTeX_Examples
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# Width and max height in inches for IEEE journals taken from
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# computer.org/cms/Computer.org/Journal%20templates/transactions_art_guide.pdf
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assert(columns in [1,2])
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if fig_width is None:
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fig_width = 3.39 if columns==1 else 6.9 # width in inches
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if fig_height is None:
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golden_mean = (sqrt(5)-1.0)/2.0 # Aesthetic ratio
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fig_height = fig_width*golden_mean # height in inches
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MAX_HEIGHT_INCHES = 8.0
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if fig_height > MAX_HEIGHT_INCHES:
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print("WARNING: fig_height too large:" + fig_height +
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"so will reduce to" + MAX_HEIGHT_INCHES + "inches.")
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fig_height = MAX_HEIGHT_INCHES
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# NB (bart): default font-size in latex is 11. This should exactly match
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# the font size in the text if the figwidth is set appropriately.
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# Note that this does not hold if you put two figures next to each other using
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# minipage. You need to use subplots.
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params = {'backend': 'ps',
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'text.latex.preamble': ['\\usepackage{gensymb}'],
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'axes.labelsize': font_size, # fontsize for x and y labels (was 12 and before 10)
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'axes.titlesize': font_size,
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'font.size': font_size, # was 12 and before 10
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'legend.fontsize': font_size, # was 12 and before 10
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'xtick.labelsize': font_size,
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'ytick.labelsize': font_size,
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'text.usetex': True,
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'figure.figsize': [fig_width,fig_height],
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'font.family': 'serif'
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}
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matplotlib.rcParams.update(params)
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def format_axes(ax):
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for spine in ['top', 'right']:
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ax.spines[spine].set_visible(False)
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for spine in ['left', 'bottom']:
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ax.spines[spine].set_color(SPINE_COLOR)
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ax.spines[spine].set_linewidth(0.5)
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ax.xaxis.set_ticks_position('bottom')
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ax.yaxis.set_ticks_position('left')
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for axis in [ax.xaxis, ax.yaxis]:
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axis.set_tick_params(direction='out', color=SPINE_COLOR)
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return ax

‎liquidation.py‎

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import numpy as np
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import cvxpy as cp
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# Problem data
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global_indices = list(range(5))
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local_indices = [
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[0, 1, 2, 3, 4],
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[0, 1],
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[2, 3],
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[3, 4],
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[3, 4]
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]
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reserves = list(map(np.array, [
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[4, 4, 4, 4, 4],
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[10, 1],
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[1, 5],
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[40, 50],
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[10, 10]
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]))
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fees = [
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.998,
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.997,
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.997,
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.997,
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.999
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]
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current_assets = [
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2,
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1,
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3,
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5,
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10
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]
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# Build local-global matrices
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n = len(global_indices)
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m = len(local_indices)
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A = []
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for l in local_indices:
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n_i = len(l)
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A_i = np.zeros((n, n_i))
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for i, idx in enumerate(l):
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A_i[idx, i] = 1
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A.append(A_i)
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# Build variables
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deltas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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lambdas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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psi = cp.sum([A_i @ (L - D) for A_i, D, L in zip(A, deltas, lambdas)])
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# Objective is to liquidate everything into token 4
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obj = cp.Maximize(psi[4])
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# Reserves after trade
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new_reserves = [R + gamma_i*D - L for R, gamma_i, D, L in zip(reserves, fees, deltas, lambdas)]
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# Trading function constraints
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cons = [
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# Balancer pool with weights 5, 4, 3, 2, 1
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cp.geo_mean(new_reserves[0], p=np.array([5, 4, 3, 2, 1])) >= cp.geo_mean(reserves[0]),
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# Uniswap v2 pools
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cp.geo_mean(new_reserves[1]) >= cp.geo_mean(reserves[1]),
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cp.geo_mean(new_reserves[2]) >= cp.geo_mean(reserves[2]),
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cp.geo_mean(new_reserves[3]) >= cp.geo_mean(reserves[3]),
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# Constant sum pool
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cp.sum(new_reserves[4]) >= cp.sum(reserves[4]),
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new_reserves[4] >= 0,
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# Liquidate all assets, except 4
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psi[0] + current_assets[0] == 0,
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psi[1] + current_assets[1] == 0,
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psi[2] + current_assets[2] == 0,
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psi[3] + current_assets[3] == 0
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]
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# Set up and solve problem
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prob = cp.Problem(obj, cons)
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prob.solve()
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print(f"Total liquidated value: {psi.value[4]}")

‎output/empty‎

Whitespace-only changes.

‎two-asset.py‎

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import numpy as np
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import cvxpy as cp
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import matplotlib.pyplot as plt
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from latexify import latexify
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# Problem data
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global_indices = list(range(3))
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local_indices = [
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[0, 1, 2],
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[0, 1],
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[1, 2],
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[0, 2],
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[0, 2]
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]
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reserves = list(map(np.array, [
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[3, .2, 1],
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[10, 1],
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[1, 10],
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# Note that there is arbitrage in the next two pools:
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[20, 50],
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[10, 10]
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]))
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fees = np.array([
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.98,
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.99,
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.96,
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.97,
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.99
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])
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amounts = np.linspace(0, 50)
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u_t = np.zeros(len(amounts))
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all_values = [np.zeros((len(l), len(amounts))) for l in local_indices]
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for j, t in enumerate(amounts):
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current_assets = np.array([
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t,
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0,
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0,
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])
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# Build local-global matrices
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n = len(global_indices)
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m = len(local_indices)
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A = []
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for l in local_indices:
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n_i = len(l)
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A_i = np.zeros((n, n_i))
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for i, idx in enumerate(l):
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A_i[idx, i] = 1
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A.append(A_i)
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# Build variables
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deltas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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lambdas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
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psi = cp.sum([A_i @ (L - D) for A_i, D, L in zip(A, deltas, lambdas)])
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# Objective is to trade t of asset 1 for a maximum amount of asset 3
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obj = cp.Maximize(psi[2])
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# Reserves after trade
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new_reserves = [R + gamma_i*D - L for R, gamma_i, D, L in zip(reserves, fees, deltas, lambdas)]
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# Trading function constraints
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cons = [
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# Balancer pool with weights 3, 2, 1
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cp.geo_mean(new_reserves[0], p=np.array([3, 2, 1])) >= cp.geo_mean(reserves[0], p=np.array([3, 2, 1])),
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# Uniswap v2 pools
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cp.geo_mean(new_reserves[1]) >= cp.geo_mean(reserves[1]),
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cp.geo_mean(new_reserves[2]) >= cp.geo_mean(reserves[2]),
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cp.geo_mean(new_reserves[3]) >= cp.geo_mean(reserves[3]),
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# Constant sum pool
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cp.sum(new_reserves[4]) >= cp.sum(reserves[4]),
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new_reserves[4] >= 0,
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# Allow all assets at hand to be traded
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psi + current_assets >= 0
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]
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# Set up and solve problem
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prob = cp.Problem(obj, cons)
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prob.solve()
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for k in range(len(local_indices)):
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all_values[k][:, j] = lambdas[k].value - deltas[k].value
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print(f"Total liquidated value: {psi.value[2]}")
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for i in range(5):
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print(f"Market {i}, delta: {deltas[i].value}, lambda: {lambdas[i].value}")
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u_t[j] = obj.value
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latexify(fig_width=6, fig_height=3.5)
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for k in range(len(local_indices)):
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curr_value = all_values[k]
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for i in range(curr_value.shape[0]):
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coin_out = curr_value[i, :]
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plt.plot(amounts, coin_out, label=f"$(\\Lambda_{{{k+1}}} - \\Delta_{{{k+1}}})_{{{i+1}}}$")
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plt.legend(loc='center left', bbox_to_anchor=(1, .5))
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plt.xlabel("$t$")
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plt.savefig("output/all_plot.pdf", bbox_inches="tight")
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latexify(fig_width=4, fig_height=3)
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plt.plot(amounts, u_t, "k")
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plt.ylim([0, np.max(u_t)+2])
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plt.ylabel("$u(t)$")
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plt.xlabel("$t$")
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plt.savefig("output/u_plot.pdf", bbox_inches="tight")

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