"""
Adaptive Volatility (position sizing)

credit attribution:
    David Varadi
    https://cssanalytics.wordpress.com/2017/11/15/adaptive-volatility/

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

class AdaptiveVolatility(QCAlgorithm):

    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.symbols[i] = self.AddEquity(tkr, Resolution.Daily).Symbol

        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
            
            if shall_trade: 
                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