def get_instrument_replica(prices, returns, m, alphas=None):
assert len(prices) == len(returns[0])
if alphas is None:
alphas = np.array([i / m for i in range(1, m + 1)])
t = len(prices)
mu = returns[:, -1]
print('number of nans in returns:', np.isnan(returns).sum())
drawdown = ut.drawdown(prices)
rhos = np.array([ut.cvar(drawdown, alpha) for alpha in alphas])
# create variables
lambdas = cp.Variable(m)
v = cp.Variable()
us = cp.Variable((m, t))
aux = cp.Variable((m, t)) # aux = max(us, 0)
objective = cp.Minimize(lambdas @ rhos - v)
constraints = [
lambdas >= 0.,
cp.sum(lambdas) == 1., v * mu == returns @ cp.sum(us, axis=0),
aux >= 0., aux >= us
]
# add constraints for u and aux
for i in range(m):
constraints.append(cp.sum(us[i, :]) == 0.)
constraints.append(cp.sum(aux[i, :]) <= lambdas[i])
constraints.append(cp.sum(us[i, :]) >= 0.)
for k in range(t):
if k >= 1:
constraints.append(cp.sum(us[i, :k]) >= 0.)
constraints.append(us[i, k] >= -1. * lambdas[i] / (alphas[i] * t))
# optimization
problem = cp.Problem(objective, constraints)
problem.solve(solver=cp.GLPK)
return lambdas.value, v.value, us.value, problem.value
def get_constraint_violation(optimal_returns, all_returns, alpha):
n, t = np.shape(all_returns)
# calculate CDaR
drawdowns = ut.drawdown(optimal_returns)
q_var = cp.Variable(t)
objective = cp.Maximize(q_var @ drawdowns)
constraints = [q_var >= 0, q_var <= 1. / (alpha * t), cp.sum(q_var) == 1]
problem = cp.Problem(objective, constraints)
problem.solve(solver=cp.GLPK)
cdar = problem.value
q_opt = q_var.value
betas = np.zeros(n)
print('cdar1=', cdar, 'cdar2=', ut.cvar(drawdowns, alpha))
# construct list of indices of max elements
k = []
for j in range(1, t + 1):
k.append(np.argmax(optimal_returns[:j]))
# construct betas
for i in range(n):
for j in range(t):
betas[i] += q_opt[j] * (all_returns[i, k[j]] -
all_returns[i, j]) / cdar
violation = all_returns[:, -1] - optimal_returns[-1] * betas
print('violations are', violation)
return np.linalg.norm(violation, ord=2), np.linalg.norm(violation, ord=1)
def f(y):
y_ext = np.zeros(len(y) + 1)
y_ext[1:] = y
y_ext[0] = (1. - returns[1:, -1] @ y) / returns[0, -1]
ret = 0.
for i in range(m):
if lambdas[i] > 0:
ret += lambdas[i] * ut.cvar(ut.drawdown(y_ext @ returns),
alphas[i])
return ret
def f(y):
ret = 0.
for i in range(m):
ret += lambdas[i] * ut.cvar(ut.drawdown(y @ returns), alphas[i])
return ret
def check_rhos(prices, alphas):
for alpha in alphas:
print(ut.cvar(ut.drawdown(prices), alpha))
def f(y):
ret = 0.
for i in range(m):
ret += lambdas[i]*ut.cvar(ut.drawdown(y*prices1+(1-y)*prices2), alphas[i])
return ret