Monte Carlo simulation estimates uncertain outcomes by repeatedly drawing random samples. In financial analysis, it turns historical returns into many possible future prices or portfolio results, helping quantify risk when a single forecast is inadequate. section 7.2.1
A practical financial workflow typically:
- Loads historical prices and calculates percentage returns. section 7.2.1
- Estimates average return and volatility from those observations. section 7.2.1
- Samples many possible returns, then converts them into simulated prices or portfolio outcomes. section 7.2.1 section 7.6.1
- Summarizes the distribution with a percentile, such as a 95% Value at Risk estimate. section 7.2.1
Apple one-day VaR
Using one year of Apple prices, estimate return mean and volatility, generate 10,000 normally sampled returns, and compare simulated prices with the latest price. The 5th percentile of simulated losses provides a 95% VaR estimate; the example indicates losses above $5.50 occurred in roughly 5% of simulations. Historical assumptions do not capture unpredictable Black Swan events. section 7.2.1 section 7.7
Listing 7.1 Monte Carlo simulation for VaR on Apple
import numpy as np
import yfinance as yf
import matplotlib.pyplot as plt
ticker = "AAPL" #1
confidence_level = 0.95 #1
num_simulations = 10000 #1
time_horizon = 1 #1
stock = yf.Ticker(ticker) #2
hist = stock.history(period="1y") #2
returns = hist['Close'].pct_change().dropna() #2
mu = returns.mean() #3
sigma = returns.std() #3
last_price = hist['Close'].iloc[-1] #4
sim_returns = np.random.normal(mu,
sigma,
num_simulations) #4
sim_prices = last_price * (1 + sim_returns) #4
threshold = np.percentile(sim_prices - last_price,
(1 - confidence_level) * 100) #5
plt.hist(simulated_prices - last_price, bins=50,
alpha=0.75, color="blue", edgecolor="black") #6
plt.axvline(threshold, color="red",
linestyle="dashed", linewidth=2) #6
plt.title(f"Monte Carlo Simulated "
f"P&L Distribution for {ticker}") #6
plt.xlabel("Profit/Loss") #6
plt.ylabel("Frequency") #6
plt.show() #6
print(f"{confidence_level * 100}% Monte Carlo "
f"VaR for {ticker}: ${-var_threshold:.2f}") #6
Inspect the executable Python pattern for downloading Apple data, sampling returns, calculating the percentile threshold, and plotting simulated losses. section 7.2.1
