axis=1)
C = self.X.ix[c.index[c >= self.rho] + 1]
if C.shape[0] == 0:
b = np.ones(m) / float(m)
else:
b = self.optimal_weights(C)
return b
def optimal_weights(self, X):
X = np.mat(X)
n, m = X.shape
P = 2 * matrix(X.T * X)
q = -3 * matrix(np.ones((1, n)) * X).T
G = matrix(-np.eye(m))
h = matrix(np.zeros(m))
A = matrix(np.ones(m)).T
b = matrix(1.)
sol = solvers.qp(P, q, G, h, A, b)
return np.squeeze(sol['x'])
# use case
if __name__ == '__main__':
tools.quickrun(CORN())
# -*- coding: utf-8 -*-
from universal.algo import Algo
from universal.algos import CRP
import universal.tools as tools
import numpy as np
class BCRP(CRP):
""" Best Constant Rebalanced Portfolio = Constant Rebalanced Portfolio constructed
with hindsight. It is often used as benchmark.
Reference:
T. Cover. Universal Portfolios, 1991.
http://www-isl.stanford.edu/~cover/papers/paper93.pdf
"""
def weights(self, X):
""" Find weights which maximize return on X in hindsight! """
self.b = tools.bcrp_weights(X)
return super(BCRP, self).weights(X)
if __name__ == '__main__':
tools.quickrun(BCRP())
forever. """
PRICE_TYPE = 'raw'
def __init__(self, b=None):
"""
:params b: Portfolio weights at start. Default are uniform.
"""
super(BAH, self).__init__()
self.b = b
def weights(self, S):
""" Weights function optimized for performance. """
b = np.ones(S.shape[1]) / S.shape[1] if self.b is None else self.b
# weights are proportional to price times initial weights
w = S * b
# normalize
w = w.div(w.sum(axis=1), axis=0)
# shift
w = w.shift(1)
w.ix[0] = 1. / S.shape[1]
return w
if __name__ == '__main__':
tools.quickrun(BAH())
leverage = max(self.leverage, 1. / m)
stretch = (leverage - 1. / m) / (1. - 1. / m)
self.W = (self.W - 1. / m) * stretch + 1. / m
def step(self, x, last_b):
# calculate new wealth of all CRPs
self.S = np.multiply(self.S, self.W * np.matrix(x).T)
b = self.W.T * self.S
return b / sum(b)
def plot_leverage(self, S, leverage=np.linspace(1, 10, 10), **kwargs):
""" Plot graph with leverages on x-axis and total wealth on y-axis.
:param S: Stock prices.
:param leverage: List of parameters for leverage.
"""
wealths = []
for lev in leverage:
self.leverage = lev
wealths.append(self.run(S).total_wealth)
ax = pd.Series(wealths, index=leverage, **kwargs).plot(**kwargs)
ax.set_xlabel('Leverage')
ax.set_ylabel('Total Wealth')
return ax
if __name__ == '__main__':
data = tools.dataset('sp500')
tools.quickrun(UP(), data)
def step(self, x, last_b, history):
# calculate return prediction
x_pred = self.predict(x, history.iloc[-self.window:])
b = self.update(last_b, x_pred, self.eps)
return b
def predict(self, x, history):
""" Predict returns on next day. """
return (history / x).mean()
def update(self, b, x, eps):
""" Update portfolio weights to satisfy constraint b * x >= eps
and minimize distance to previous weights. """
x_mean = np.mean(x)
lam = max(0., (eps - np.dot(b, x)) / np.linalg.norm(x - x_mean)**2)
# limit lambda to avoid numerical problems
lam = min(100000, lam)
# update portfolio
b = b + lam * (x - x_mean)
# project it onto simplex
return tools.simplex_proj(b)
if __name__ == '__main__':
tools.quickrun(OLMAR())
# lambda from equation 7
foo = (V - x_upper * x.T * np.sum(sigma, axis=1)) / M**2
a = 2 * theta * V * foo
b = foo + 2 * theta * V * (eps - log(M))
c = eps - log(M) - theta * V
a,b,c = a[0,0], b[0,0], c[0,0]
lam = max(0,
(-b + sqrt(b**2 - 4 * a * c)) / (2. * a),
(-b - sqrt(b**2 - 4 * a * c)) / (2. * a))
# bound it due to numerical problems
lam = min(lam, 1E+7)
# update mu and sigma
mu = mu - lam * sigma * (x - x_upper) / M
sigma = inv(inv(sigma) + 2 * lam * theta * diag(x)**2)
"""
tmp_sigma = inv(inv(sigma) + theta*lam/U_sqroot*diag(xt)^2);
% Don't update sigma if results are badly scaled.
if all(~isnan(tmp_sigma(:)) & ~isinf(tmp_sigma(:)))
sigma = tmp_sigma;
end
"""
return mu, sigma
# use case
if __name__ == '__main__':
tools.quickrun(CWMR())
PRICE_TYPE = 'raw'
REPLACE_MISSING = True
def __init__(self, window=5, eps=10., tau=0.001):
"""
:param window: Lookback window.
:param eps: Constraint on return for new weights on last price (average of prices).
x * w >= eps for new weights w.
:param tau: Precision for finding median. Recommended value is around 0.001. Strongly
affects algo speed.
"""
super(RMR, self).__init__(window, eps)
self.tau = tau
def predict(self, x, history):
""" find L1 median to historical prices """
y = history.mean()
y_last = None
while y_last is None or norm(y - y_last) / norm(y_last) > self.tau:
y_last = y
d = norm(history - y)
y = history.div(d, axis=0).sum() / (1. / d).sum()
return y / x
if __name__ == '__main__':
tools.quickrun(RMR())
def update(self, b, x, eps, C):
""" Update portfolio weights to satisfy constraint b * x <= eps
and minimize distance to previous weights. """
x_mean = np.mean(x)
le = max(0., np.dot(b, x) - eps)
if self.variant == 0:
lam = le / np.linalg.norm(x - x_mean)**2
elif self.variant == 1:
lam = min(C, le / np.linalg.norm(x - x_mean)**2)
elif self.variant == 2:
lam = le / (np.linalg.norm(x - x_mean)**2 + 0.5 / C)
# limit lambda to avoid numerical problems
lam = min(100000, lam)
# update portfolio
b = b - lam * (x - x_mean)
# project it onto simplex
return tools.simplex_proj(b)
if __name__ == '__main__':
tools.quickrun(PAMR())
def step(self, x, last_b, history):
# find indices of nearest neighbors throughout history
ixs = self.find_nn(history, self.k, self.l)
# get returns from the days following NNs
J = history.iloc[[history.index.get_loc(i) + 1 for i in ixs]]
# get best weights
return tools.bcrp_weights(J)
def find_nn(self, H, k, l):
""" Note that nearest neighbors are calculated in a different (more efficient) way than shown
in the article.
param H: history
"""
# calculate distance from current sequence to every other point
D = H * 0
for i in range(1, k+1):
D += (H.shift(i-1) - H.iloc[-i])**2
D = D.sum(1).iloc[:-1]
# sort and find nearest neighbors
D.sort()
return D.index[:l]
if __name__ == '__main__':
tools.quickrun(BNN())
Reference:
Li Gao, Weiguo Zhang
Weighted Moving Averag Passive Aggressive Algorithm for Online Portfolio Selection, 2013.
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6643896
"""
PRICE_TYPE = 'ratio'
def __init__(self, window=5, **kwargs):
"""
:param w: Windows length for moving average.
:param kwargs: Additional arguments for PAMR.
"""
super(WMAMR, self).__init__(**kwargs)
if window < 1:
raise ValueError('window parameter must be >=1')
self.window = window
def step(self, x, last_b, history):
xx = history[-self.window:].mean()
# calculate return prediction
b = self.update(last_b, xx, self.eps, self.C)
return b
# use case
if __name__ == '__main__':
tools.quickrun(WMAMR())
for (i=0; i<Nc[1]; i++) {
float total_claim=0.;
for (j=0; j<Nc[1]; j++) {
total_claim += claim[i][j];
}
if(total_claim != 0){
for (j=0; j<Nc[1]; j++) {
transfer[i][j] = W2(t,i) * claim[i][j] / total_claim;
}
}
}
for (i=0; i<Nc[1]; i++) {
W2(t+1,i) = W2(t,i);
for (j=0; j<Nc[1]; j++) {
W2(t+1,i) += transfer[j][i] - transfer[i][j];
}
}
}
"""
return weave.inline(code, ['c', 'mu', 'w'])
get_weights_c(CORR, EX, weights)
return weights
if __name__ == '__main__':
tools.quickrun(Anticor())
PRICE_TYPE = 'raw'
def __init__(self, b=None):
"""
:params b: Portfolio weights at start. Default are uniform.
"""
super(BAH, self).__init__()
self.b = b
def weights(self, S):
""" Weights function optimized for performance. """
b = np.ones(S.shape[1]) / S.shape[1] if self.b is None else self.b
# weights are proportional to price times initial weights
w = S * b
# normalize
w = w.div(w.sum(axis=1), axis=0)
# shift
w = w.shift(1)
w.ix[0] = 1./S.shape[1]
return w
if __name__ == '__main__':
tools.quickrun(BAH())
C = self.X.ix[c.index[c >= self.rho] + 1]
if C.shape[0] == 0:
b = np.ones(m) / float(m)
else:
b = self.optimal_weights(C)
return b
def optimal_weights(self, X):
X = np.mat(X)
n,m = X.shape
P = 2 * matrix(X.T * X)
q = -3 * matrix(np.ones((1,n)) * X).T
G = matrix(-np.eye(m))
h = matrix(np.zeros(m))
A = matrix(np.ones(m)).T
b = matrix(1.)
sol = solvers.qp(P, q, G, h, A, b)
return np.squeeze(sol['x'])
# use case
if __name__ == '__main__':
tools.quickrun(CORN())
else:
data.plot(logy=False)
if show_3d:
assert dim == 3, '3D plot works for exactly 3 assets.'
plt.figure()
fun = lambda b: CRP(b).run(data).total_wealth
ternary.plot_heatmap(fun, steps=20, boundary=True)
elif dim == 2:
x, y = _crp(data)
s = pd.Series(y, index=x)
s.plot(ax=axes[1], logy=True)
plt.title('CRP performance')
plt.xlabel('weight of {}'.format(data.columns[0]))
elif dim > 2:
fig, axes = plt.subplots(ncols=dim - 1, nrows=dim - 1)
for i in range(dim - 1):
for j in range(i + 1, dim):
x, y = _crp(data[[i, j]])
ax = axes[i][j - 1]
ax.plot(x, y)
ax.set_title('{} & {}'.format(data.columns[i],
data.columns[j]))
ax.set_xlabel('weights of {}'.format(data.columns[i]))
if __name__ == '__main__':
result = tools.quickrun(CRP())
print(result.information)
Reference:
Li Gao, Weiguo Zhang
Weighted Moving Averag Passive Aggressive Algorithm for Online Portfolio Selection, 2013.
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6643896
"""
PRICE_TYPE = 'ratio'
def __init__(self, window=5, **kwargs):
"""
:param w: Windows length for moving average.
:param kwargs: Additional arguments for PAMR.
"""
super(WMAMR, self).__init__(**kwargs)
if window < 1:
raise ValueError('window parameter must be >=1')
self.window = window
def step(self, x, last_b, history):
xx = history[-self.window:].mean()
# calculate return prediction
b = self.update(last_b, xx, self.eps, self.C)
return b
# use case
if __name__ == '__main__':
tools.quickrun(WMAMR())
class EG(Algo):
""" Exponentiated Gradient (EG) algorithm by Helmbold et al.
Reference:
Helmbold, David P., et al.
"On‐Line Portfolio Selection Using Multiplicative Updates."
Mathematical Finance 8.4 (1998): 325-347.
"""
def __init__(self, eta=0.05):
"""
:params eta: Learning rate. Controls volatility of weights.
"""
super(EG, self).__init__()
self.eta = eta
def init_weights(self, m):
return np.ones(m) / m
def step(self, x, last_b):
b = last_b * np.exp(self.eta * x / sum(x * last_b))
return b / sum(b)
if __name__ == '__main__':
data = tools.dataset('nyse_o')
tools.quickrun(EG(eta=0.5), data)
data.plot(ax=axes[0], logy=True)
else:
data.plot(logy=False)
if show_3d:
assert dim == 3, '3D plot works for exactly 3 assets.'
plt.figure()
fun = lambda b: CRP(b).run(data).total_wealth
ternary.plot_heatmap(fun, steps=20, boundary=True)
elif dim == 2:
x,y = _crp(data)
s = pd.Series(y, index=x)
s.plot(ax=axes[1], logy=True)
plt.title('CRP performance')
plt.xlabel('weight of {}'.format(data.columns[0]))
elif dim > 2:
fig, axes = plt.subplots(ncols=dim-1, nrows=dim-1)
for i in range(dim-1):
for j in range(i + 1, dim):
x,y = _crp(data[[i,j]])
ax = axes[i][j-1]
ax.plot(x,y)
ax.set_title('{} & {}'.format(data.columns[i], data.columns[j]))
ax.set_xlabel('weights of {}'.format(data.columns[i]))
if __name__ == '__main__':
result = tools.quickrun(CRP())
print(result.information)
for (i=0; i<Nc[1]; i++) {
float total_claim=0.;
for (j=0; j<Nc[1]; j++) {
total_claim += claim[i][j];
}
if(total_claim != 0){
for (j=0; j<Nc[1]; j++) {
transfer[i][j] = W2(t,i) * claim[i][j] / total_claim;
}
}
}
for (i=0; i<Nc[1]; i++) {
W2(t+1,i) = W2(t,i);
for (j=0; j<Nc[1]; j++) {
W2(t+1,i) += transfer[j][i] - transfer[i][j];
}
}
}
"""
return weave.inline(code, ['c', 'mu', 'w'])
get_weights_c(CORR, EX, weights)
return weights
if __name__ == '__main__':
tools.quickrun(Anticor())
P = matrix((sigma - r*mu*mu.T + (n*r)**2) / (1+r))
q = matrix(-mu)
G = matrix(-np.eye(n))
h = matrix(np.zeros(n))
sol = solvers.qp(P, q, G, h)
return np.squeeze(sol['x'])
def plot_fraction(self, S, fractions=np.linspace(0.,2.,10), **kwargs):
""" Plot graph with Kelly fraction on x-axis and total wealth on y-axis.
:param S: Stock prices.
:param fractions: List (ndarray) of fractions used.
"""
wealths = []
for fraction in fractions:
self.fraction = fraction
wealths.append(self.run(S).total_wealth)
ax = pd.Series(wealths, index=fractions, **kwargs).plot(**kwargs)
ax.set_xlabel('Kelly Fraction')
ax.set_ylabel('Total Wealth')
return ax
# use case
if __name__ == '__main__':
tools.quickrun(Kelly())
# -*- coding: utf-8 -*-
from universal.algo import Algo
from universal.algos import CRP
import universal.tools as tools
import numpy as np
class BCRP(CRP):
""" Best Constant Rebalanced Portfolio = Constant Rebalanced Portfolio constructed
with hindsight. It is often used as benchmark.
Reference:
T. Cover. Universal Portfolios, 1991.
http://www-isl.stanford.edu/~cover/papers/paper93.pdf
"""
def weights(self, X):
""" Find weights which maximize return on X in hindsight! """
self.b = tools.bcrp_weights(X)
return super(BCRP, self).weights(X)
if __name__ == '__main__':
tools.quickrun(BCRP())
return np.ones(m) / m
def step(self, x, last_b, history):
# calculate return prediction
x_pred = self.predict(x, history.iloc[-self.window:])
b = self.update(last_b, x_pred, self.eps)
return b
def predict(self, x, history):
""" Predict returns on next day. """
return (history / x).mean()
def update(self, b, x, eps):
""" Update portfolio weights to satisfy constraint b * x >= eps
and minimize distance to previous weights. """
x_mean = np.mean(x)
lam = max(0., (eps - np.dot(b, x)) / np.linalg.norm(x - x_mean)**2)
# limit lambda to avoid numerical problems
lam = min(100000, lam)
# update portfolio
b = b + lam * (x - x_mean)
# project it onto simplex
return tools.simplex_proj(b)
if __name__ == '__main__':
tools.quickrun(OLMAR())
def step(self, r, p):
# calculate gradient
grad = np.mat(r / np.dot(p, r)).T
# update A
self.A += grad * grad.T
# update b
self.b += (1 + 1. / self.beta) * grad
# projection of p induced by norm A
pp = self.projection_in_norm(self.delta * self.A.I * self.b, self.A)
return pp * (1 - self.eta) + np.ones(len(r)) / float(len(r)) * self.eta
def projection_in_norm(self, x, M):
""" Projection of x to simplex indiced by matrix M. Uses quadratic programming.
"""
m = M.shape[0]
P = matrix(2 * M)
q = matrix(-2 * M * x)
G = matrix(-np.eye(m))
h = matrix(np.zeros((m, 1)))
A = matrix(np.ones((1, m)))
b = matrix(1.)
sol = solvers.qp(P, q, G, h, A, b)
return np.squeeze(sol['x'])
if __name__ == '__main__':
tools.quickrun(ONS())
def step(self, r, p):
# calculate gradient
grad = np.mat(r / np.dot(p, r)).T
# update A
self.A += grad * grad.T
# update b
self.b += (1 + 1./self.beta) * grad
# projection of p induced by norm A
pp = self.projection_in_norm(self.delta * self.A.I * self.b, self.A)
return pp * (1 - self.eta) + np.ones(len(r)) / float(len(r)) * self.eta
def projection_in_norm(self, x, M):
""" Projection of x to simplex indiced by matrix M. Uses quadratic programming.
"""
m = M.shape[0]
P = matrix(2*M)
q = matrix(-2 * M * x)
G = matrix(-np.eye(m))
h = matrix(np.zeros((m,1)))
A = matrix(np.ones((1,m)))
b = matrix(1.)
sol = solvers.qp(P, q, G, h, A, b)
return np.squeeze(sol['x'])
if __name__ == '__main__':
tools.quickrun(ONS())
def step(self, x, last_b):
# calculate return prediction
b = self.update(last_b, x, self.eps, self.C)
return b
def update(self, b, x, eps, C):
""" Update portfolio weights to satisfy constraint b * x <= eps
and minimize distance to previous weights. """
x_mean = np.mean(x)
le = max(0., np.dot(b, x) - eps)
if self.variant == 0:
lam = le / np.linalg.norm(x - x_mean)**2
elif self.variant == 1:
lam = min(C, le / np.linalg.norm(x - x_mean)**2)
elif self.variant == 2:
lam = le / (np.linalg.norm(x - x_mean)**2 + 0.5 / C)
# limit lambda to avoid numerical problems
lam = min(100000, lam)
# update portfolio
b = b - lam * (x - x_mean)
# project it onto simplex
return tools.simplex_proj(b)
if __name__ == '__main__':
tools.quickrun(PAMR())
