Abdalla Harem | September 25th, 2025 | 9 Minutes Read

Predicting stock prices is inherently uncertain, but a Monte Carlo simulation can help by generating thousands of possible future price paths based on historical data. In this post, we’ll use Python to forecast Apple Inc. (AAPL) stock prices over the next year, using historical data from August 15 to September 25, 2025, and visualize the results in an interactive chart.

AAPL Stock Price Prediction: How It Works?

The Monte Carlo simulation models AAPL stock prices as a geometric Brownian motion, which assumes prices follow a random walk with a drift (average return) and volatility. Here’s the process:

1. Load Data: We use adjusted closing prices from Yahoo Finance to account for dividends and splits.
2. Calculate Parameters:
   • Log Returns: Computed as ln(price_t / price_{t-1}).
   • Mean Return (mu): Annualized by multiplying the average daily log return by 252 (trading days per year).
   • Volatility (sigma): Annualized by multiplying the standard deviation of log returns by √252.
3. Run Simulation: Starting from the latest price ($256.05), we simulate 1,000 price paths over 252 days using the formula:
   price_t = price_{t-1} × exp((mu – 0.5 × sigma²) × dt + sigma × √dt × Z),
   where Z is a random number from a standard normal distribution, and dt = 1/252.
4. Visualize Results: We display the paths, mean price, and confidence interval in an interactive chart.

The Python Code

Below is the Python code to perform the simulation. A suggested tool to run this code can be Google Colab.

import pandas as pd
import numpy as np
from io import StringIO

# Historical AAPL data (adjusted closing prices from Aug 15 to Sep 25, 2025)
csv_data = '''Date,Open,High,Low,Close,Adj Close,Volume
"Sep 25, 2025",253.21,256.36,251.71,256.05,256.05,33897693
"Sep 24, 2025",255.22,255.74,251.04,252.31,252.31,42265200
"Sep 23, 2025",255.88,257.34,253.58,254.43,254.43,60275200
"Sep 22, 2025",248.30,256.64,248.12,256.08,256.08,105517400
"Sep 19, 2025",241.23,246.30,240.21,245.50,245.50,163741300
"Sep 18, 2025",239.97,241.20,236.65,237.88,237.88,44249600
"Sep 17, 2025",238.97,240.10,237.73,238.99,238.99,46508000
"Sep 16, 2025",237.18,241.22,236.32,238.15,238.15,63421100
"Sep 15, 2025",237.00,238.19,235.03,236.70,236.70,42699500
"Sep 12, 2025",229.22,234.51,229.02,234.07,234.07,55824200
"Sep 11, 2025",226.88,230.45,226.65,230.03,230.03,50208600
"Sep 10, 2025",232.19,232.42,225.95,226.79,226.79,83440800
"Sep 9, 2025",237.00,238.78,233.36,234.35,234.35,66313900
"Sep 8, 2025",239.30,240.15,236.34,237.88,237.88,48999500
"Sep 5, 2025",240.00,241.32,238.49,239.69,239.69,54870400
"Sep 4, 2025",238.45,239.90,236.74,239.78,239.78,47549400
"Sep 3, 2025",237.21,238.85,234.36,238.47,238.47,66427800
"Sep 2, 2025",229.25,230.85,226.97,229.72,229.72,44075600
"Aug 29, 2025",232.51,233.38,231.37,232.14,232.14,39418400
"Aug 28, 2025",230.82,233.41,229.34,232.56,232.56,38074700
"Aug 27, 2025",228.61,230.90,228.26,230.49,230.49,31259500
"Aug 26, 2025",226.87,229.49,224.69,229.31,229.31,54575100
"Aug 25, 2025",226.48,229.30,226.23,227.16,227.16,30983100
"Aug 22, 2025",226.17,229.09,225.41,227.76,227.76,42477800
"Aug 21, 2025",226.27,226.52,223.78,224.90,224.90,30621200
"Aug 20, 2025",229.98,230.47,225.77,226.01,226.01,42263900
"Aug 19, 2025",231.28,232.87,229.35,230.56,230.56,39402600
"Aug 18, 2025",231.70,233.12,230.11,230.89,230.89,37476200
"Aug 15, 2025",234.00,234.28,229.34,231.59,231.59,56038700
'''

df = pd.read_csv(StringIO(csv_data))
df['Date'] = pd.to_datetime(df['Date'], format='%b %d, %Y')
df = df.sort_values('Date')

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

# Annualized parameters
mu = log_returns.mean() * 252
sigma = log_returns.std() * np.sqrt(252)

# Starting price
S0 = df['Adj Close'].iloc[-1]

# Simulation parameters
num_days = 252  # 1 trading year
num_sims = 1000  # Number of simulation paths
dt = 1.0 / 252

# Run simulation
np.random.seed(0)  # For reproducibility
prices = np.full((num_days + 1, num_sims), S0)
for i in range(1, num_days + 1):
    rand = np.random.standard_normal(num_sims)
    prices[i] = prices[i-1] * np.exp((mu - 0.5 * sigma ** 2) * dt + sigma * np.sqrt(dt) * rand)

# Results from final day
final_prices = prices[-1]
mean_price = np.mean(final_prices)
median_price = np.median(final_prices)
percentile_5 = np.percentile(final_prices, 5)
percentile_95 = np.percentile(final_prices, 95)

# Print results
print(f"Annualized mean return (mu): {mu:.4f}")
print(f"Annualized volatility (sigma): {sigma:.4f}")
print(f"Starting price (S0): {S0:.2f}")
print(f"Mean forecasted price in 1 year: {mean_price:.2f}")
print(f"Median forecasted price in 1 year: {median_price:.2f}")
print(f"5th percentile forecasted price: {percentile_5:.2f}")
print(f"95th percentile forecasted price: {percentile_95:.2f}")

Results and Interactive Visualization

The simulation yields the following key statistics:

• Annualized Mean Return (mu): 0.9036 (indicating a strong upward trend in the recent data)
• Annualized Volatility (sigma): 0.2580
• Starting Price (S0): $256.05 (as of September 25, 2025)
• Mean Forecasted Price in 1 Year: $634.84
• Median Forecasted Price in 1 Year: $619.05
• 5th Percentile Forecasted Price: $408.23
• 95th Percentile Forecasted Price: $914.97

To visualize these results, the chart below shows 50 of the 1,000 simulated price paths (blue lines), the mean price path (red line), and the 5th to 95th percentile range (shaded gray area). Hover over the chart to see price values for specific days.

The Visualized Results:

The code below was used to display the above visualized results:

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>AAPL Monte Carlo Simulation Chart</title>
    <script src="https://cdn.jsdelivr.net/npm/chart.js@3.9.1/dist/chart.min.js"></script>
    <style>
        body {
            font-family: -apple-system, BlinkMacSystemFont, 'Roboto', 'Segoe UI', Arial, sans-serif;
            margin: 20px;
            background-color: #f4f4f4;
            color: #333;
            display: flex;
            flex-direction: column;
            align-items: center;
            line-height: 1.6;
        }
        h1 {
            font-size: 1.8em;
            font-weight: 600;
            text-align: center;
            margin-bottom: 20px;
        }
        .chart-container {
            max-width: 800px;
            width: 100%;
            overflow-x: auto;
            padding: 10px;
            background-color: #fff;
            box-shadow: 0 2px 8px rgba(0, 0, 0, 0.1);
            border-radius: 8px;
        }
        canvas {
            width: 100%;
            min-width: 300px; /* Minimum width for small screens */
        }
        .chart-note {
            font-size: 0.9em;
            color: #555;
            text-align: center;
            margin-top: 10px;
        }
        /* Mobile and tablet adjustments */
        @media (max-width: 768px) {
            h1 {
                font-size: 1.5em;
            }
            .chart-container {
                max-width: 100%;
                padding: 5px;
            }
            canvas {
                transform: scale(0.9); /* Slightly reduce scale on mobile */
                transform-origin: top left;
            }
        }
        @media (max-width: 480px) {
            h1 {
                font-size: 1.2em;
            }
            .chart-container {
                padding: 5px;
            }
            canvas {
                transform: scale(0.85);
            }
            .chart-note {
                font-size: 0.8em;
            }
        }
    </style>
</head>
<body>
    <h1>AAPL Stock Price Monte Carlo Simulation</h1>
    <div class="chart-container">
        <canvas id="priceChart"></canvas>
    </div>
    <p class="chart-note">Tap or hover on the chart for detailed price information</p>
    <script>
        // Simulation parameters (matching Python code)
        const numDays = 252;
        const numSims = 1000;
        const startPrice = 256.05;
        const mu = 0.9036;
        const sigma = 0.2580;
        const dt = 1.0 / 252;

        // Simple seeded random number generator for reproducibility
        function mulberry32(a) {
            return function() {
                let t = a += 0x6D2B79F5;
                t = Math.imul(t ^ t >>> 15, t | 1);
                t ^= t + Math.imul(t ^ t >>> 7, t | 61);
                return ((t ^ t >>> 14) >>> 0) / 4294967296;
            };
        }

        // Box-Muller transform for standard normal random numbers
        function randomNormal(seed) {
            const rand = mulberry32(seed);
            const u = rand(), v = rand();
            return Math.sqrt(-2.0 * Math.log(u)) * Math.cos(2.0 * Math.PI * v);
        }

        // Generate price paths
        const prices = Array(numDays + 1).fill().map(() => Array(numSims).fill(startPrice));
        for (let i = 1; i <= numDays; i++) {
            for (let j = 0; j < numSims; j++) {
                const rand = randomNormal(j * i);
                prices[i][j] = prices[i-1][j] * Math.exp((mu - 0.5 * sigma * sigma) * dt + sigma * Math.sqrt(dt) * rand);
            }
        }

        // Generate x-axis labels in months (252 days ≈ 12 months)
        const months = Array.from({length: 253}, (_, i) => (i / 21).toFixed(1));

        // Chart.js configuration
        const ctx = document.getElementById('priceChart').getContext('2d');
        new Chart(ctx, {
            type: 'line',
            data: {
                labels: months,
                datasets: [
                    // 50 simulated price paths
                    ...Array.from({length: 50}, (_, i) => ({
                        label: i === 0 ? 'Simulated Paths' : `Path ${i + 1}`,
                        data: prices.map(row => row[i].toFixed(2)),
                        borderColor: 'rgba(0, 128, 255, 0.1)',
                        borderWidth: 1,
                        pointRadius: 0,
                        fill: false,
                        // Only show "Simulated Paths" for the first path in legend
                        showInLegend: i === 0
                    })),
                    // Mean price path
                    {
                        label: 'Mean Price',
                        data: prices.map(row => (row.reduce((a, b) => a + b, 0) / 1000).toFixed(2)),
                        borderColor: '#FF0000',
                        borderWidth: 2,
                        pointRadius: 0,
                        fill: false,
                        showInLegend: true
                    },
                    // 95th percentile
                    {
                        label: '5th-95th Percentile Range',
                        data: prices.map(row => {
                            const sorted = [...row].sort((a, b) => a - b);
                            return sorted[Math.floor(1000 * 0.95)].toFixed(2);
                        }),
                        borderColor: 'rgba(0, 0, 0, 0)',
                        backgroundColor: 'rgba(128, 128, 128, 0.2)',
                        fill: '-1',
                        showInLegend: true
                    },
                    // 5th percentile
                    {
                        label: '5th Percentile',
                        data: prices.map(row => {
                            const sorted = [...row].sort((a, b) => a - b);
                            return sorted[Math.floor(1000 * 0.05)].toFixed(2);
                        }),
                        borderColor: 'rgba(0, 0, 0, 0)',
                        backgroundColor: 'rgba(128, 128, 128, 0.2)',
                        fill: false,
                        showInLegend: false
                    }
                ]
            },
            options: {
                responsive: true,
                plugins: {
                    title: {
                        display: true,
                        text: 'AAPL Stock Price Paths (1 Year)',
                        font: { size: 18 }
                    },
                    legend: {
                        display: true,
                        position: 'bottom',
                        labels: {
                            // Filter to ensure only one "Simulated Paths" label appears
                            filter: (item, chartData) => {
                                return item.text === 'Simulated Paths' || item.text === 'Mean Price' || item.text === '5th-95th Percentile Range';
                            }
                        }
                    }
                },
                scales: {
                    x: {
                        title: { display: true, text: 'Time (Months)' }
                    },
                    y: {
                        title: { display: true, text: 'Price (USD)' },
                        min: 0,
                        max: 1500,
                        ticks: {
                            callback: function(value) {
                                return '$' + value;
                            }
                        }
                    }
                }
            }
        });
    </script>
</body>
</html>

This chart consolidates the simulation results, showing the spread of possible price paths, the expected mean path, and the 90% confidence interval.

Notes and Improvements

• Short Data Window: The simulation uses one month of data, leading to a high mean return (0.9036) due to a recent upward trend. For more reliable results, use 5+ years of historical data.
• Fetching Real-Time Data: Replace the CSV data with a library like yfinance:

import yfinance as yf
df = yf.download('AAPL', start='2020-01-01', end='2025-09-25')

• Model Limitations: The geometric Brownian motion assumes constant volatility and log-normal returns, which may not capture extreme market events. Consider models like GARCH for better accuracy.
• Enhancements: Add dividends, transaction costs, or alternative stochastic processes to improve realism.

Conclusion

Using a Monte Carlo simulation, we forecasted AAPL stock prices with a median projection of $619.05 and a wide range ($408.23 to $914.97) over the next year. The interactive chart above makes it easy to see the range of outcomes. Try running the code with different stocks or timeframes, and explore tools like xAI’s Grok on grok.com or the X platform for further analysis.

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