Kelly return ratio plot and Monte Carlo simulation

Kelly.py

# -*- coding: utf-8 -*-
"""
Created on Mon Sep 21 14:56:39 2020

Kelly Criterion functions

@author: johnx
"""

import numpy as np
import matplotlib.pyplot as plt

def plotReturnRatio(p,w):
    """
    

    Parameters
    ----------
    p : TYPE Real, 0 <= p <= 1
        Probability of winning
    w : TYPE Real, 0 < w
        Winnings per bet

    Returns
    -------
    Plot of return ratio for bet fraction b, 0 <= b <= 1

    """
    
    # Return ratio function
    R = lambda b,p,w: (1-b+b*w)**p * (1-b)**(1-p)
    
    # Bet fractions
    b = np.linspace(0,1,num = 100)
    
    # Return ratio for each bet fraction
    r = [R(beta,p,w) for beta in b]
    
    # Optimal b and r
    b_max = (p*w - 1)/(w-1)
    r_max = R(b_max,p,w)
    
    # Plot curve
    plt.plot(b,r)
    plt.plot(b_max,r_max,color = 'red', marker = 'o')
    plt.xlabel('Bet fraction, b')
    plt.ylabel('Return ratio, R')
    plt.title('Kelly Criterion')
               
def KellyMC(p,w):
    """
    

    Parameters
    ----------
    p : TYPE Real, 0 <= p <= 1
        Probability of winning.
    w : TYPE Real, 0 < w
        Winnings per bet.

    Returns
    -------
    S : Money on hand after each bet for 3 different betting strategies

    """
    
    # Number of MC experiments
    nMC = 100
    
    # Betting fraction error
    b_err = 0.05
    
    # Series of win/loss experiments
    nWL = int(np.round(p*nMC))
    WL = np.random.permutation(np.concatenate((np.ones(nWL),np.zeros(nMC-nWL)),axis=0))
    
    # Net value on hand after each experiment
    S = np.zeros((3,nMC))

    # Optimal betting fraction
    b = (p*w-1)/(w-1)

    # Bet fractions
    B = np.array([b - b_err, b, b + b_err])
    
    # Play each hand (1st hand assumes initial stash = 1)
    S[:,0] = 1 - B + w*B*WL[0]
    for k in range(1,nMC):
        S[:,k] = S[:,k-1] * (1 - B + w*B*WL[k])
        
    # Plot results
    games = np.array(range(1,nMC+1)).transpose()
    plt.plot(games,S[0,:], label = "Bet-")
    plt.plot(games,S[1,:], label = "Optimal")
    plt.plot(games,S[2,:], label = "Bet+")
    plt.xlabel('Game number')
    plt.ylabel('Net value')
    plt.legend('loc','upper left')
    plt.title('Monte Carlo Kelly Experiment')
    plt.grid()
    plt.legend(["Bet-","Optimal","Bet+"])
    
    return S