Algorithms

[miron]

Algorithms used in Algorithmic Trading

Renaissance Technologies (Jim Simons, James Ax, Elwyn Berlekamp)

• Mean Reversion Model (historica on its own) + Trend Following

• Kernel Methods:

  • Linear Regression
  • High-Dimensional Regression

• Kelly Criterion Investment Formula

"""
f fraction of assets to apply to security.
p probability that investment increases in value.
q probability that investment decreases in value (q=1-p).
a fraction that is lost in a negative outcome. If security price falls 10%, a=0.1
b fraction that is gained in a positive outcome. If security price rises 10%, b=0.1.
"""
def kelly(p, b):
    f =  (p/a) - (q/b)
    return f

• A* Search

Moving Averages

• Simple Moving Average (SMA)
  Momentum Trading or Trend following:
  When moving average 50 falls below moving average of 200, sell (and short futures), otherwise, buy and go long.

import pandas as pd  
import numpy as np
pd.DataFrame.rolling(window = 7, min_periods = 7).mean()
pd.DataFrame.rolling(window = 25).mean()
pd.DataFrame.rolling(window = 99).mean()

# Common Movin Averages:
price.rolling(window = 50)
price.rolling(window = 200)

price.rolling(window = 5)
price.rolling(window = 10)
price.rolling(window = 20)
price.rolling(window = 100)

# Fibonacci:
price.rolling(window =  (5, 8, 13, 21 ...) )

month_ret = pd.DataFrame.resample('M', kind = 'period').last().pct_change().dropna()
for years in [1, 3, 5, 10, 20]:
      # Anualized Mean Returns For Last ... Years.
      pd.DataFrame['month_ret'].rolling(years * 12).mean() * 12 
      # Anualized Risk For Last ... Years.
      pd.DataFrame['month_ret'].rolling(years * 12).std() * np.sqrt(12) 

• Exponential Moving Average (EMA)
  reacts faster to recent price changes
  Typical short term are 12 and 26 days, long term 50 and 200 days

price.ewm(span = 10, min_periods= 10).mean()

• Double Exponential Moving Average (DEMA)
  DEMA = 2*EMA - EMA(EMA)
• Expanding Windows
  Fixed lower bound
  Mean of price up to this point in time

price.expanding().mean()

Rolling Correlation

returns['BTC'].rolling(36).corr(returns['ETH'])

Returns

• Rolling Returns
  Over 12 Month period

assets_close.resample('M').last().pct_change().rolling(12).mean().dropna()

• Logarithmic Returns
  I.e. price is 130, 110, 120, 90, 80:
  returns are -15%, 9%, -25%, -11%
  mean: -12.13%

  130 * np.exp(4*-0.1213) = 80

mean_log_ret =  np.log(df / df.shift(1)).dropna().mean()

# Actual Price
investment * np.exp(periods * mean_log_ret) 

• Geometric Average

   # R=Return
   # n=Count of the numbers in the series
   ((1+R1) * (1+R2) * (1+R3) * (1+Rn))**(1/n)-1

I.e. returns are respectively
30%, 10%, 20%, -10%, and -80%
The geometric average annual return == 4.3%

(1.3 *1.1 * 1.2 * 0.9 * 0.8)**(1/5)-1

Equal Weighted

• Equal Weighted Portfolio

assets_qty = len(assets.columns)
weights = [1/assets_qty for i in range(assets_qty)]
returns.dot(weights)

Other

• Confidence Interval
  https://en.wikipedia.org/wiki/Confidence_interval