Overall Statistics Total Trades 429 Average Win 0.49% Average Loss -1.57% Compounding Annual Return 19.398% Drawdown 18.600% Expectancy 0.270 Net Profit 351.187% Sharpe Ratio 1.097 Loss Rate 3% Win Rate 97% Profit-Loss Ratio 0.31 Alpha 0.095 Beta 4.785 Annual Standard Deviation 0.171 Annual Variance 0.029 Information Ratio 0.983 Tracking Error 0.171 Treynor Ratio 0.039 Total Fees \$650.11
```import numpy as np
from scipy import stats
from statsmodels.distributions.empirical_distribution import ECDF
from scipy.stats import kendalltau, pearsonr, spearmanr
from scipy.optimize import minimize
import sys

def Initialize(self):

## Initialise the data and resolution required, as well as the cash and start-end dates for your algorithm. All algorithms must initialized.

self.SetStartDate(2010,1,1)
self.SetEndDate(2018,7,1)
self.SetCash(100000)
self.numdays = 1000  # set the length of formation period which determine the copula we use
self.lookbackdays = 250 # set the length of history data in trading period
self.cap_CL = 0.95  # set the cap confidence level
self.floor_CL = 0.05 # set the floor confidence level

self._pair_selection()

## Pull history daily close data of past one year
self.syl = []
for i in self.ticker:
self.syl.append(equity.Symbol.Value)

self.price_list = {}
self.price_list[self.syl[0]] = []
self.price_list[self.syl[1]] = []

history = self.History(self.syl, self.numdays,Resolution.Daily)
close = history['close'].unstack(level=0)
logreturn = (np.log(close) - np.log(close.shift(1))).dropna()
x, y = logreturn[self.ticker[0]], logreturn[self.ticker[1]]
logreturn={} # A dictionary to store the history log return series of paired stocks

''' Convert the two returns series to two uniform values u and v using the empirical distribution functions '''

ecdf_x, ecdf_y  = ECDF(x), ECDF(y)
u, v = [ecdf_x(a) for a in x], [ecdf_y(a) for a in y]

# compute the Akaike Information Criterion (AIC) for different copulas and choose copula with minimum AIC

self.family = ['clayton', 'frank', 'gumbel']
tau = kendalltau(x, y)[0]  # estimate Kendall'rank correlation
AIC ={}  # generate a dict with key being the copula family, value = [theta, AIC]
for i in self.family:
lpdf = [self._lpdf_copula(i, float(self._parameter(i,tau)), x, y) for (x, y) in zip(u, v)]
# Replace nan with zero and inf with finite numbers in lpdf list
lpdf = np.nan_to_num(lpdf)
loglikelihood = sum(lpdf)
AIC[i] = [self._parameter(i,tau), -2*loglikelihood + 2]
# choose the copula with the minimum AIC
self.copula = min(AIC.items(), key = lambda x: x[1][1])[0]
self.Log(str(self.copula))
self.Log(str(AIC))

# schedule an event to fire on first day each month for a security
# the action compute the mispricing indices to generate the trading signals
self.Schedule.On(self.DateRules.MonthStart(self.syl[0]), self.TimeRules.AfterMarketOpen(self.syl[0]), Action(self._set_signal))

def OnData(self,data):

for i in self.syl:
self.price_list[i].append(self.Portfolio[i].Price)

# compute today's log return of 2 stocks
if len(self.price_list[self.syl[0]]) < 2 or len(self.price_list[self.syl[1]]) < 2: return
else:
return_x = np.log(float(self.price_list[self.syl[0]][-1]/self.price_list[self.syl[0]][-2]))
return_y = np.log(float(self.price_list[self.syl[1]][-1]/self.price_list[self.syl[1]][-2]))

# Convert the two returns to uniform values u and v using the empirical distribution functions
u_value = self.ecdf_x(return_x)
v_value = self.ecdf_y(return_y)

# Compute the mispricing indices for u and v by using estimated copula
self._misprice_index(self.copula, self.theta, u_value, v_value)
quantity = float(self.CalculateOrderQuantity(self.syl[1],0.4))
if self.MI_u_v < self.floor_CL and self.MI_v_u > self.cap_CL:
if self.Portfolio[self.syl[0]].Quantity < 0 and self.Portfolio[self.syl[1]].Quantity > 0:
self.Liquidate()
quantity = float(self.CalculateOrderQuantity(self.syl[1],0.4))
self.Sell(self.syl[1], 1 * quantity )
else:
self.Sell(self.syl[1], 1 * quantity )

elif self.MI_u_v > self.cap_CL and self.MI_v_u < self.floor_CL:
if self.Portfolio[self.syl[0]].Quantity > 0 and self.Portfolio[self.syl[1]].Quantity < 0:
self.Liquidate()
quantity = float(self.CalculateOrderQuantity(self.syl[1],0.4))
self.Sell(self.syl[0], self.coef * quantity)
else:
self.Sell(self.syl[0], self.coef * quantity)

def _set_signal(self):

history = self.History(self.ticker, self.lookbackdays,Resolution.Daily)

# generate the log return series of paired stocks
close = history['close'].unstack(level=0)
logreturn = (np.log(close) - np.log(close.shift(1))).dropna()
x, y = logreturn[self.ticker[0]], logreturn[self.ticker[1]]

# estimate Kendall'rank correlation each trading day
tau = kendalltau(x, y)[0]

# etstimate the copula parameter: theta
self.theta = self._parameter(self.copula, tau)

# simulate the empirical distribution function for returns of two paired stocks
self.ecdf_x, self.ecdf_y  = ECDF(x), ECDF(y)

# run linear regression over the two history return series
self.coef = stats.linregress(x,y).slope

def _parameter(self, family, tau):

'''
estimate the parameters for three kinds of Archimedean copulas
According to association between Archimedean copulas and the Kendall rank correlation measure
'''

if  family == 'clayton':
return 2*tau/float(1-tau)

elif family == 'frank':

'''
debye = quad(integrand, sys.float_info.epsilon, theta)[0]/theta  is first order Debye function
frank_fun is the squared difference
Minimize the frank_fun would give the parameter theta for the frank copula
'''

integrand = lambda t: t/(np.exp(t)-1)  # generate the integrand
frank_fun = lambda theta: ((tau - 1)/4.0  - (quad(integrand, sys.float_info.epsilon, theta)[0]/theta - 1)/theta)**2

return minimize(frank_fun, 4, method='BFGS', tol=1e-5).x

elif family == 'gumbel':
return 1/float(1-tau)

def _lpdf_copula(self, family, theta, u, v):

# Estimate the log probability density function of three kinds of Archimedean copulas

if  family == 'clayton':
pdf = (theta+1) * ((u**(-theta)+v**(-theta)-1)**(-2-1/theta)) * (u**(-theta-1)*v**(-theta-1))

elif family == 'frank':
num = -theta * (np.exp(-theta)-1) * (np.exp(-theta*(u+v)))
denom = ((np.exp(-theta*u)-1) * (np.exp(-theta*v)-1) + (np.exp(-theta)-1))**2
pdf = num/denom

elif family == 'gumbel':
if u == 1: u = 0.9999
if v == 1: v = 0.9999
A = (-np.log(u))**theta + (-np.log(v))**theta
c = np.exp(-A**(1/theta))
pdf = c * (u*v)**(-1) * (A**(-2+2/theta)) * ((np.log(u)*np.log(v))**(theta-1)) * (1+(theta-1)*A**(-1/theta))

return np.log(pdf)

def _misprice_index(self, family, theta, u, v):

'''
Calculate mispricing index for every day in the trading period by using estimated copula
Mispricing indices are the conditional probability P(U < u | V = v) and P(V < v | U = u)
'''

if  family == 'clayton':
self.MI_u_v = v**(-theta-1) * (u**(-theta)+v**(-theta)-1)**(-1/theta-1) # P(U<u|V=v)
self.MI_v_u = u**(-theta-1) * (u**(-theta)+v**(-theta)-1)**(-1/theta-1) # P(V<v|U=u)

elif family == 'frank':
A = (np.exp(-theta*u)-1) * (np.exp(-theta*v)-1) + (np.exp(-theta*v)-1)
B = (np.exp(-theta*u)-1) * (np.exp(-theta*v)-1) + (np.exp(-theta*u)-1)
C = (np.exp(-theta*u)-1) * (np.exp(-theta*v)-1) + (np.exp(-theta)-1)
self.MI_u_v = B/float(C)
self.MI_v_u = A/float(C)

elif family == 'gumbel':
A = (-np.log(u))**theta + (-np.log(v))**theta
C_uv = np.exp(-A**(1/theta))   # C_uv is gumbel copula function C(u,v)
self.MI_u_v = C_uv * (A**((1-theta)/theta)) * (-np.log(v))**(theta-1) * (1.0/v)
self.MI_v_u = C_uv * (A**((1-theta)/theta)) * (-np.log(u))**(theta-1) * (1.0/u)

def _pair_selection(self):

tick_syl =  [["QQQ","XME","TNA","FAS","XLF","EWC","QLD"],
["XLK","EWG","TLT","FAZ","XLU","EWA","QID"]]

logreturn={}

for i in range(2):
syl = [self.AddSecurity(SecurityType.Equity, x, Resolution.Daily).Symbol.Value for x in tick_syl[i]]
history = self.History(syl, self.lookbackdays,Resolution.Daily)
# generate the log return series of paired stocks
close = history['close'].unstack(level=0)
df_logreturn = (np.log(close) - np.log(close.shift(1))).dropna()
for j in tick_syl[i]:
logreturn[j] = df_logreturn[j]
# estimate coefficients of different correlation measures
tau_coef,pr_coef,sr_coef= [],[],[]
for i in range(len(tick_syl[i])):
tik_x, tik_y= logreturn[tick_syl[0][i]], logreturn[tick_syl[1][i]]
tau_coef.append(kendalltau(tik_x, tik_y)[0])
pr_coef.append(pearsonr(tik_x, tik_y)[0])
sr_coef.append(spearmanr(tik_x, tik_y)[0])
index_max = tau_coef.index(max(tau_coef))
self.ticker = [tick_syl[0][index_max],tick_syl[1][index_max]]

self.Log("tau" + str(tau_coef))
self.Log("pr" + str(pr_coef))
self.Log("sr" + str(sr_coef))
self.Log(str(self.ticker))                        ```