Overall Statistics Total Trades 569 Average Win 0.92% Average Loss -0.65% Compounding Annual Return 10.760% Drawdown 13.900% Expectancy 0.693 Net Profit 240.209% Sharpe Ratio 0.965 Loss Rate 30% Win Rate 70% Profit-Loss Ratio 1.41 Alpha 0.186 Beta -5.984 Annual Standard Deviation 0.091 Annual Variance 0.008 Information Ratio 0.785 Tracking Error 0.091 Treynor Ratio -0.015 Total Fees \$616.19
```"""

Aim: Get a better position sizing than  [target_vol / realized_vol_{t-1}]
(where the realized_vol is calculated over a fixed lookback period, e.g. past 20 days)
using a more 'adaptive' volatility that varies its lookback period according to market conditions.

The simplest method is to use the R-squared of the regression of prices vs time:
1. high R-squared indicates a trending market
-> use short lookback periods to capture sudden changes in volatilities;
2. low R-squared instead iimplies a rangebound/mean-reverting market
-> lengthen lookbacks since vol will revert to historical means.

To translate the R_squared value into the alpha for an exponential moving average,
the following exponential function is used (motivation: returns supposed lognormal):

raw_alpha =  exp[-10. * (1 - R_squared(price vs. time, period=20)]
alpha = min(raw_alpha, 0.5)

The cap (0.5) effectively limits the lookback to 3 days, since alpha := 2 / (1 + lookback).

Such a capped aplha is used in an EMA of the squared returns for the past 20 days.

Finally the (theorical) daily exposure is:

target_vol / sqrt( EMA_{t-1}(squared rturns, alpha) * 252)

and target_vol is an annualised target vol, say 20%.

To limit excessive trading, I only rebalace if theoretical exposure changes above a certain threshold (say 5%).

Application hereby:
long SPY (or similar) with a daily position sizing

A more interesting use of this position sizing scheme is when using algorithms with
long periodical rebalacings, say monthly or quarterly.
"""
import numpy as np
import pandas as pd
from datetime import datetime, timedelta
from scipy.stats import linregress

def __init__(self):
self.symbols = ['SPY',
'TLT',
'GLD'
]

self.back_period = 21 * 3 + 1     # 3 months

self.vol_period = 21    # days for calc vol
self.target_vol = 0.2
self.lev = 1.5          # max lev from ratio targ_vol / real_vol

self.delta = 0.05       # min rebalancing

self.w = 1. / len(self.symbols)
self.x = np.asarray(range(self.vol_period))

def Initialize(self):

self.SetCash(100000)
self.SetStartDate(2006,1,1)
self.SetEndDate(datetime.now().date() - timedelta(1))
self.SetBrokerageModel(BrokerageName.InteractiveBrokersBrokerage,
AccountType.Margin)

# register and replace 'tkr symbol' with 'tkr object'
for i, tkr in enumerate(self.symbols):

self.Schedule.On(self.DateRules.EveryDay(self.symbols[0]),
self.TimeRules.AfterMarketOpen(self.symbols[0], 1),
Action(self.rebalance))

def rebalance(self):

# get all weights
weight = self.pos_sizing()

tot_port = self.Portfolio.TotalPortfolioValue

for tkr in self.symbols:

# gauge if needs to trade (new weight vs. current one > self.delta)
curr_weight = self.Portfolio[tkr.Value].Quantity * self.Securities[tkr.Value].Price  / tot_port
new_weight = weight[tkr.Value]
shall_trade = abs(float(new_weight) - float(curr_weight)) > self.delta

self.SetHoldings(tkr, new_weight)
self.Log("tkr: %s and weight: %s"  %(str(tkr), str(new_weight) ) )

def pos_sizing(self):

# get daily returns for period = self.back_period
prices = self.History(self.symbols, self.back_period, Resolution.Daily)["close"].unstack(level=0)     # .dropna(axis=1)
daily_rtrn = prices.pct_change().dropna() # or: np.log(self.price / self.price.shift(1)).dropna()

pos = {}

# calculate alpha for EWM
for tkr in self.symbols:

_rsq = self.rsquared(self.x, np.asarray(prices[tkr.Value])[-self.vol_period:])

alpha_raw = np.exp(-10. * (1. - _rsq))
alpha_ = min(alpha_raw, 0.5)

vol = daily_rtrn[tkr.Value].ewm(alpha=alpha_).std() # alpha = 2/(span+1) = 1-exp(log(0.5)/halflife)
ann_vol = vol.tail(1) * np.sqrt(252)

self.Log("rsqr: %s, alpha_raw: %s, ann_vol = %s" %(str(_rsq), str(alpha_raw), str(ann_vol)) )

pos[tkr.Value] = (self.target_vol / ann_vol).clip(0.0, self.lev)  * self.w  # NB: self.w = 1/no_assets

return pos

def rsquared(self, x, y):
# slope, intercept, r_value, p_value, std_err
_, _, r_value, _, _ = linregress(x, y)
return r_value**2                        ```