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1. Introduction

Forecasting financial markets is a sophisticated fusion of quantitative precision and global economic nuance. In this quest, the Monte Carlo simulation stands out as a premier statistical instrument, guiding our understanding of future stock prices.

Named after the famous Monte Carlo Casino in Monaco, this method doesn’t bank on luck but is rooted in rigorous probabilistic modelling. Imagine orchestrating thousands of experiments in a controlled environment, with each one unfolding a different story of stock price movement. That’s the power of the Monte Carlo simulation. It allows us to visualize a spectrum of outcomes based on historical data and probabilistic assumptions.

In this article, our approach is twofold: First, we delve into the intricacies of historical and implied volatilities, elucidating their distinctions, implications, and relevance. While historical volatility gauges the fluctuations of past stock prices, implied volatility offers a forward-looking perspective based on market sentiments and option pricing. Next, we employ the Monte Carlo simulation, using both these volatility measures, to extrapolate potential future paths for our chosen stock.

Central to our analysis will be the derivation of probabilities. By observing the simulation’s multitude of potential outcomes, we’ll compute the likelihood of the stock price landing within specific ranges. For instance, what’s the probability of the stock price exceeding a particular threshold or falling below another? Therefore, offering a quantifiable perspective on potential future scenarios.

2. Volatility Analysis

2.1 Historical Volatility

Historical volatility, often termed statistical or realized volatility, quantifies the variability of a stock’s returns over a past period. It gives us an idea of how much a stock’s price deviated from its average value during that timeframe.

Once you have the daily returns, the historical volatility (HV) over N days is calculated as

Where:

• ˉRˉ = Average daily return over the N days

To convert daily volatility to annualized volatility you can use the formula below. This formula is key when you want to estimate volatility over a given horizon.

2.2 Implied Volatility

The implied volatility can be extracted from the Black-Scholes option pricing model:

Where:

• C is the call option price.

• S0​ is the current stock price.

• X is the strike price of the option.

• t is the time to expiration.

• r is the risk-free rate.

• q is the dividend yield.

• N() is the CDF of the standard normal distribution.

It’s worth noting that although the above describes the derivation for a call option, a similar approach is taken for put options. To extract implied volatility (σ), one would reverse engineer the model using the observed market option price.

However, the process can be computationally intensive, especially for those unfamiliar with the nuances of option pricing models. As a result, many traders and investors prefer sourcing the implied volatility value directly from platforms like Yahoo Finance or their trading brokers.

2.3 Historical vs. Implied Volatility

Historical and implied volatilities, though related, serve different purposes:

• Perspective: While historical volatility looks backward, gauging price fluctuations from past data, implied volatility peers forward, encapsulating market expectations of future price variability.

• Application: Historical volatility aids in risk assessment based on past stock behavior. In contrast, implied volatility is pivotal for options pricing and understanding market sentiment about future volatility.

• Determination: Historical volatility is purely statistical, derived from past price data. Implied volatility, however, is extracted from the market prices of options, embodying the collective wisdom of market participants.

3. Monte Carlo Simulations for Price Forecasting

The Monte Carlo simulation leverages probability theory to determine possible outcomes for an uncertain event. In finance, it offers a method to model the evolution of stock prices, based on certain statistical characteristics, primarily the volatility.

3.1 Drift and Volatility

The main components determining stock price movements in our simulation are the drift and the volatility. The drift represents the expected or average rate of return of a stock, adjusted for the time frame of the forecast.

Where:

• μ is the expected stock return.

• σ2 is the variance (volatility squared).

• Δt is the time increment.

Volatility, on the other hand, accounts for the unpredictability and brings randomness into our forecast:

Where:

• ϵ is a random value drawn from a standard normal distribution.​

3.2 Stock Price Path Simulation

To simulate the path of stock prices, we apply the following recursive formula:

Starting from the initial stock price S0​, we compute successive prices S1​,S2​,… using the drift, the volatility term, and the previous stock price.

3.3 Stock Price Path Simulation

The power of the Monte Carlo method emerges when running thousands or even millions of simulations. Each trajectory will offer a unique path for the stock price based on the randomness injected by the volatility term. Collectively, they provide a spectrum of possible outcomes, shaping a probability distribution for future stock prices.

3.4 Stock Price Path Simulation

After obtaining a multitude of stock price paths, we can perform various statistical analyses. For instance, calculating the mean and variance provides insights into the central tendency and dispersion of our forecasts. Moreover, by examining the distribution of the final stock prices from all simulations, we can compute probabilities for the stock landing within certain ranges, as previously discussed.

In conclusion, the Monte Carlo simulation offers a dynamic and probabilistic approach to stock price forecasting. While no model can predict the future with certainty, this method provides a robust framework for understanding potential price trajectories and their associated probabilities.

4. Implementation of Monte Carlo Simulation

In this section, we outline the steps and methodologies used to conduct the previously analyses discussed on Volatility and Monte Carlo.

4.1 Methodology Overview

The simulation uses the Geometric Brownian Motion model, a widely recognized model for stock price dynamics, primarily influenced by drift (deterministic) and volatility (stochastic).

4.2 Data Collection

To begin our simulation, we source the stock’s historical data:

import yfinance as yf

# Download historical data for ASML.AS
symbol = "ASML.AS" 
start_date = "2020-01-01"
end_date = "2023-12-30"
df = yf.download(symbol, start=start_date, end=end_date)

4.3 Computing Daily Returns

With the historical data in place, we compute the stock’s daily returns, serving as the foundation for our simulation:

import numpy as np

# Calculate daily returns
log_returns = np.log(df['Close'] / df['Close'].shift(1))

4.4 Monte Carlo Simulation Setup

Before the simulation starts, we specify key parameters, guiding the forecast’s trajectory:

# Monte Carlo simulation parameters
days_to_forecast = 20
num_simulations = 10000
dt = 1  # 1 trading day

4.5 Volatility Inclusion

Implied volatility is introduced, based on our earlier discussions in the volatility section:

# Volatility for BSM
volatility_bsm = 0.29  # Assuming 29% annualized volatility
print("BMS Implied Volatility: ", volatility_bsm)

4.6 Running the Simulations

Leveraging the calculated logarithmic returns and assumed implied volatility, we create two sets of simulations:

from scipy.stats import norm

# Function to run the Monte Carlo simulation
def run_simulation(volatility, dt, annualized=False):
    ...
    # simulation logic here
    ...

simulated_prices_historical = run_simulation(log_returns.std(), dt)
simulated_prices_bsm = run_simulation(volatility_bsm, dt, annualized=True)
print("Historical Volatility: ", log_returns.std())

4.7 Visualization

The results are visualized through the Monte Carlo confidence cones, showcasing the possible stock price trajectories under both the historical and BSM volatility:

import matplotlib.pyplot as plt

# Plotting logic
fig, axs = plt.subplots(1, 2, figsize=(30, 7))
...
# rest of the plotting logic
plt.show()

This visualization provides a detailed look at potential price paths using two different volatility measures. By overlaying these paths with specified thresholds and confidence intervals, investors can derive meaningful insights into potential future stock movements.

4.8 Understanding Key Parameters

Thresholds:

In the code, we’ve identified two important thresholds for the stock prices. These thresholds might represent critical resistance or support levels from a technical analysis perspective or specific price levels of interest to the investor.

# Define thresholds
threshold1 = 650 #Upper Threshold
threshold2 = 540 #Lower Threshold

4.9 Derived Probabilities

From the simulation results, we can calculate the probabilities of the stock price being above threshold1, below threshold2, and between the two thresholds at the end of our forecast period:

# Probability annotations
above_threshold1_prob = (simulated_prices[-1] > threshold1).sum() / num_simulations
below_threshold2_prob = (simulated_prices[-1] < threshold2).sum() / num_simulations
between_thresholds_prob = 1 - above_threshold1_prob - below_threshold2_prob

These probabilities provide valuable insights:

• The likelihood that the stock price exceeds a favorable price level (above threshold1).

• The risk of the stock price dropping below a concerning level (threshold2).

• The probability that the stock price remains between these two thresholds.

Putting it all together:

import numpy as np
import pandas as pd
import yfinance as yf
import matplotlib.pyplot as plt
from datetime import datetime, timedelta
from scipy.stats import norm

# Download historical data for ASML.AS
symbol = "ASML.AS" 
start_date = "2020-01-01"
end_date = "2023-12-30"
df = yf.download(symbol, start=start_date, end=end_date)

# Calculate daily returns
log_returns = np.log(df['Close'] / df['Close'].shift(1))

# Monte Carlo simulation parameters
days_to_forecast = 20
num_simulations = 10000
dt = 1  # 1 trading day

# Define thresholds
threshold2 = 540 #144 #116
threshold1 = 650 #162 #128

# Volatility for BSM
volatility_bsm = 0.29 # Assuming 29% annualized volatility 

print("BMS Implied Volatility: ", volatility_bsm)

# Run Monte Carlo simulations
def run_simulation(volatility, dt, annualized=False):
    simulated_prices = np.zeros((days_to_forecast, num_simulations))
    simulated_prices[0] = df['Close'][-1]

    # Adjust volatility for annualized estimates
    if annualized:
        volatility = volatility / np.sqrt(252)

    for t in range(1, days_to_forecast):
        random_walk = np.random.normal(loc=log_returns.mean() * dt,
                                       scale=volatility * np.sqrt(dt),
                                       size=num_simulations)
        simulated_prices[t] = simulated_prices[t - 1] * np.exp(random_walk)
    return simulated_prices

simulated_prices_historical = run_simulation(log_returns.std(), dt)
simulated_prices_bsm = run_simulation(volatility_bsm, dt, annualized=True)

print("Historical Volatility: ", log_returns.std())

# Plot the results
fig, axs = plt.subplots(1, 2, figsize=(30, 7))

for simulated_prices, ax, title in zip([simulated_prices_historical, simulated_prices_bsm], axs, ['Historical Volatility', 'BSM Volatility']):
    mean_price_path = np.mean(simulated_prices, axis=1)
    median_price_path = np.median(simulated_prices, axis=1)
    lower_bound_68 = np.percentile(simulated_prices, 16, axis=1)
    upper_bound_68 = np.percentile(simulated_prices, 84, axis=1)
    lower_bound_95 = np.percentile(simulated_prices, 2.5, axis=1)
    upper_bound_95 = np.percentile(simulated_prices, 97.5, axis=1)

    # Define thresholds
    threshold1 = threshold1
    threshold2 = threshold2

    ax.plot(simulated_prices, color='lightgray', alpha=0.1)
    ax.plot(mean_price_path, color='black', label='Mean Price Path')
    ax.plot(median_price_path, color='blue', label='Median Price Path')
    ax.plot(lower_bound_68, color='green', linestyle='--', label='68% confidence interval (Lower)')
    ax.plot(upper_bound_68, color='green', linestyle='--', label='68% confidence interval (Upper)')
    ax.plot(lower_bound_95, color='blue', linestyle='--', label='95% confidence interval (Lower)')
    ax.plot(upper_bound_95, color='red', linestyle='--', label='95% confidence interval (Upper)')

    ax.axhline(y=threshold1, color='purple', linestyle='--', alpha=0.25, label='Threshold 1')
    ax.axhline(y=threshold2, color='orange', linestyle='--', alpha=0.25, label='Threshold 2')

    # Plot settings
    ax.set_xlabel('Days')
    ax.set_ylabel('Stock Price')
    ax.set_title(f'Monte Carlo Confidence Cone for {symbol} using {title}', fontsize = 25)
    ax.legend(fontsize = 15)

    # Add price estimates every 5 days
    for day in range(0, days_to_forecast, 5):
        ax.annotate(f'{lower_bound_68[day]:.2f}', xy=(day, lower_bound_68[day]), xycoords='data',
                    xytext=(-50, 0), textcoords='offset points', color='green')
        ax.annotate(f'{upper_bound_68[day]:.2f}', xy=(day, upper_bound_68[day]), xycoords='data',
                    xytext=(-50, 0), textcoords='offset points', color='green')
        ax.annotate(f'{lower_bound_95[day]:.2f}', xy=(day, lower_bound_95[day]), xycoords='data',
                    xytext=(-50, 0), textcoords='offset points', color='blue')
        ax.annotate(f'{upper_bound_95[day]:.2f}', xy=(day, upper_bound_95[day]), xycoords='data',
                    xytext=(-50, 0), textcoords='offset points', color='red')

    # Add probability annotations
    above_threshold1_prob = (simulated_prices[-1] > threshold1).sum() / num_simulations
    below_threshold2_prob = (simulated_prices[-1] < threshold2).sum() / num_simulations
    between_thresholds_prob = 1 - above_threshold1_prob - below_threshold2_prob

    props = dict(boxstyle='round', facecolor='wheat', alpha=0.5)
    ax.text(0.02, 0.05, f'P(>{threshold1:.2f}): {above_threshold1_prob:.2%}\n'
                       f'P(<{threshold2:.2f}): {below_threshold2_prob:.2%}\n'
                       f'P({threshold2:.2f} - {threshold1:.2f}): {between_thresholds_prob:.2%}',
            transform=ax.transAxes, fontsize=15,
            verticalalignment='bottom', bbox=props)
    
    # Add mean price path labels
    #ax.annotate(f'{mean_price_path[-1]:.2f}', xy=(days_to_forecast - 1, mean_price_path[-1]), xycoords='data',
    #            xytext=(15, 0), textcoords='offset points', color='black')

    # Add median price path labels
    ax.annotate(f'{median_price_path[-1]:.2f}', xy=(days_to_forecast - 1, median_price_path[-1]), xycoords='data',
                xytext=(15, 0), textcoords='offset points', color='blue')
    
    # Add price labels
    ax.annotate(f'{lower_bound_68[-1]:.2f}', xy=(days_to_forecast - 1, lower_bound_68[-1]), xycoords='data',
                xytext=(-15, 0), textcoords='offset points', color='green')
    ax.annotate(f'{upper_bound_68[-1]:.2f}', xy=(days_to_forecast - 1, upper_bound_68[-1]), xycoords='data',
                xytext=(-15, 0), textcoords='offset points', color='green')
    ax.annotate(f'{lower_bound_95[-1]:.2f}', xy=(days_to_forecast - 1, lower_bound_95[-1]), xycoords='data',
                xytext=(-15, 0), textcoords='offset points', color='blue')
    ax.annotate(f'{upper_bound_95[-1]:.2f}', xy=(days_to_forecast - 1, upper_bound_95[-1]), xycoords='data',
                xytext=(-15, 0), textcoords='offset points', color='red')

plt.show()

4.10 Historical vs. Implied Volatility: Results Comparison

Our simulation provided two sets of potential future stock price paths, one based on historical volatility and the other on implied volatility. Here’s a brief comparison:

1. Historical Volatility: Derived from the stock’s past price movements, our simulation gives us a projection rooted in its historical behavior. In environments where market conditions remain consistent, this could offer a reliable forecast.

2. BSM Implied Volatility: This represents market expectations. In scenarios where there are anticipated future events or changes in market conditions, the implied volatility may provide a more relevant projection.

By comparing these two simulations, we gain a more comprehensive perspective:

• If both simulations align closely, it indicates a consensus between past behavior and market expectations.

• Divergence between the two suggests that there are external factors or market expectations not captured by the historical data.

The side-by-side visual representation in the plots further illustrates these dynamics, providing a robust tool for investors to weigh the potential risks and rewards.

5. Concluding Remarks

Financial market dynamics are complex, oscillating between historical trends and ever-evolving market sentiments. Our Monte Carlo simulation, leveraging both historical and implied volatility, offers a rich tapestry of potential outcomes for the stock prices . But like all models, its strength lies not just in the predictions it renders but also in the insights it offers.

1. Appreciating Uncertainty: One of the most salient takeaways from our analysis is the range of potential outcomes. This range encapsulates the inherent uncertainty in stock price movements, emphasizing the need for risk management.

2. Informed Decision Making: The simulation offers a a spectrum of possibilities. Investors can use these insights to inform their decisions, whether it’s setting stop-loss orders, determining entry and exit points, or simply adjusting their portfolio’s risk profile.

3. The Importance of Both Historical and Implied Volatilities: Our analysis underlined the significance of considering both past stock movements and market sentiments. While the historical data offers a look into how the stock behaved under past market conditions, implied volatility provides a window into market expectations. By juxtaposing these two, investors can capture a more holistic view of potential stock movements.

In conclusion, tools like the Monte Carlo simulation empower investors to peer into a range of potential tomorrows. It’s a step away from mere speculation and a leap towards informed, data-driven decision-making