option python file added

Author: steven-ja

content/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png

@@ -0,0 +1,68 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Sun Jun 23 11:30:47 2024
+
+@author: stefa
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+def monte_carlo_option_pricing(S0, K, T, r, sigma, num_simulations, num_steps):
+    dt = T / num_steps
+    paths = np.zeros((num_simulations, num_steps + 1))
+    paths[:, 0] = S0
+    
+    for t in range(1, num_steps + 1):
+        z = np.random.standard_normal(num_simulations)
+        paths[:, t] = paths[:, t-1] * np.exp((r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * z)
+    
+    option_payoffs = np.maximum(paths[:, -1] - K, 0)
+    option_price = np.exp(-r * T) * np.mean(option_payoffs)
+    
+    return option_price, paths
+
+# Example usage
+S0 = 100  # Initial stock price
+K = 98.5   # Strike price
+T = 1     # Time to maturity (in years)
+r = 0.05  # Risk-free rate
+sigma = 0.2  # Volatility
+num_simulations = 10000
+num_steps = 252  # Number of trading days in a year
+
+price, paths = monte_carlo_option_pricing(S0, K, T, r, sigma, num_simulations, num_steps)
+print(f"Estimated option price: {price:.3f}")
+
+## Visualization
+plt.figure(figsize=(10, 6))
+plt.plot(paths[:100, :].T)
+plt.title("Sample Stock Price Paths")
+plt.xlabel("Time Steps")
+plt.ylabel("Stock Price")
+plt.show()
+plt.savefig("images/simulation_path.png", dpi=300)
+
+plt.figure(figsize=(10, 6))
+plt.hist(paths[:, -1], bins=100)
+plt.title("Distribution of Final Stock Prices")
+plt.xlabel("Stock Price")
+plt.ylabel("Frequency")
+plt.vlines(S0, 0, 350, colors='r', linestyles='--', label=r'$S_0$')
+plt.legend()
+plt.show()
+plt.savefig("images/simulation_histogram.png", dpi=300)
+
+
+from scipy.stats import norm
+
+def black_scholes_call(S0, K, T, r, sigma):
+    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
+    d2 = d1 - sigma * np.sqrt(T)
+    return S0 * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)
+
+
+bs_price = black_scholes_call(S0, K, T, r, sigma)
+print(f"Black-Scholes price: {bs_price:.3f}")
+print(f"Monte Carlo price: {price:.3f}")
+print(f"Difference: {abs(bs_price - price):.4f}")