3rd articles completed and reviewed

Author: steven-ja

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

@@ -121,7 +121,7 @@ def monte_carlo_option_pricing(S0, K, T, r, sigma, num_simulations, num_steps):
 
 # Example usage
 S0 = 100  # Initial stock price
-K = 100   # Strike price
+K = 98.5   # Strike price
 T = 1     # Time to maturity (in years)
 r = 0.05  # Risk-free rate
 sigma = 0.2  # Volatility
@@ -154,9 +154,11 @@ plt.ylabel("Frequency")
 plt.show()
 ```
 
-{{< img src="\posts\finance\monte_carlo\Black-Scholes\images\simulation_path.png" align="center" title="Results">}}
+{{< img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_path.png" align="center">}}
 
-{{< img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png" align="center" title="Results">}}
+---
+
+{{< img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png" align="center" title="Histogram">}}
 
 These visualizations show the range of possible stock price paths and the distribution of final stock prices, providing insight into the option's potential outcomes.
 
@@ -173,8 +175,8 @@ def black_scholes_call(S0, K, T, r, sigma):
     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:.2f}")
-print(f"Monte Carlo price: {price:.2f}")
+print(f"Black-Scholes price: {bs_price:.3f}")
+print(f"Monte Carlo price: {price:.3f}")
 print(f"Difference: {abs(bs_price - price):.4f}")
 ```
 

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

@@ -7,6 +7,9 @@
 
 import numpy as np
 import matplotlib.pyplot as plt
+import seaborn as sns
+sns.set_style('whitegrid')
+plt.style.use("fivethirtyeight")
 
 def monte_carlo_option_pricing(S0, K, T, r, sigma, num_simulations, num_steps):
     dt = T / num_steps
@@ -35,15 +38,15 @@ def 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.figure(figsize=(12, 8))
+plt.plot(paths[:100, :].T, linewidth=1)
 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.figure(figsize=(12, 8))
 plt.hist(paths[:, -1], bins=100)
 plt.title("Distribution of Final Stock Prices")
 plt.xlabel("Stock Price")

content/posts/finance/monte_carlo/Black-Scholes/index.md

@@ -494,7 +494,7 @@ <h2 id="4-implementing-monte-carlo-simulation-in-python">4. Implementing Monte C
 </span></span><span style="display:flex;"><span>
 </span></span><span style="display:flex;"><span><span style="color:#75715e"># Example usage</span>
 </span></span><span style="display:flex;"><span>S0 <span style="color:#f92672">=</span> <span style="color:#ae81ff">100</span>  <span style="color:#75715e"># Initial stock price</span>
-</span></span><span style="display:flex;"><span>K <span style="color:#f92672">=</span> <span style="color:#ae81ff">100</span>   <span style="color:#75715e"># Strike price</span>
+</span></span><span style="display:flex;"><span>K <span style="color:#f92672">=</span> <span style="color:#ae81ff">98.5</span>   <span style="color:#75715e"># Strike price</span>
 </span></span><span style="display:flex;"><span>T <span style="color:#f92672">=</span> <span style="color:#ae81ff">1</span>     <span style="color:#75715e"># Time to maturity (in years)</span>
 </span></span><span style="display:flex;"><span>r <span style="color:#f92672">=</span> <span style="color:#ae81ff">0.05</span>  <span style="color:#75715e"># Risk-free rate</span>
 </span></span><span style="display:flex;"><span>sigma <span style="color:#f92672">=</span> <span style="color:#ae81ff">0.2</span>  <span style="color:#75715e"># Volatility</span>
@@ -519,9 +519,7 @@ <h2 id="5-visualization-and-analysis">5. Visualization and Analysis</h2>
 </span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>xlabel(<span style="color:#e6db74">&#34;Stock Price&#34;</span>)
 </span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>ylabel(<span style="color:#e6db74">&#34;Frequency&#34;</span>)
 </span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>show()
-</span></span></code></pre></div><img src="%5cposts%5cfinance%5cmonte_carlo%5cBlack-Scholes%5cimages%5csimulation_path.png"
-    
-        alt="Results"
+</span></span></code></pre></div><img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_path.png"
 
 
 
@@ -531,9 +529,10 @@ <h2 id="5-visualization-and-analysis">5. Visualization and Analysis</h2>
 
 >
 
+<hr>
 <img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png"
 
-        alt="Results"
+        alt="Histogram"
 
 
 
@@ -554,8 +553,8 @@ <h2 id="6-comparison-with-analytical-solutions">6. Comparison with Analytical So
 </span></span><span style="display:flex;"><span>    <span style="color:#66d9ef">return</span> S0 <span style="color:#f92672">*</span> norm<span style="color:#f92672">.</span>cdf(d1) <span style="color:#f92672">-</span> K <span style="color:#f92672">*</span> np<span style="color:#f92672">.</span>exp(<span style="color:#f92672">-</span>r <span style="color:#f92672">*</span> T) <span style="color:#f92672">*</span> norm<span style="color:#f92672">.</span>cdf(d2)
 </span></span><span style="display:flex;"><span>
 </span></span><span style="display:flex;"><span>bs_price <span style="color:#f92672">=</span> black_scholes_call(S0, K, T, r, sigma)
-</span></span><span style="display:flex;"><span>print(<span style="color:#e6db74">f</span><span style="color:#e6db74">&#34;Black-Scholes price: </span><span style="color:#e6db74">{</span>bs_price<span style="color:#e6db74">:</span><span style="color:#e6db74">.2f</span><span style="color:#e6db74">}</span><span style="color:#e6db74">&#34;</span>)
-</span></span><span style="display:flex;"><span>print(<span style="color:#e6db74">f</span><span style="color:#e6db74">&#34;Monte Carlo price: </span><span style="color:#e6db74">{</span>price<span style="color:#e6db74">:</span><span style="color:#e6db74">.2f</span><span style="color:#e6db74">}</span><span style="color:#e6db74">&#34;</span>)
+</span></span><span style="display:flex;"><span>print(<span style="color:#e6db74">f</span><span style="color:#e6db74">&#34;Black-Scholes price: </span><span style="color:#e6db74">{</span>bs_price<span style="color:#e6db74">:</span><span style="color:#e6db74">.3f</span><span style="color:#e6db74">}</span><span style="color:#e6db74">&#34;</span>)
+</span></span><span style="display:flex;"><span>print(<span style="color:#e6db74">f</span><span style="color:#e6db74">&#34;Monte Carlo price: </span><span style="color:#e6db74">{</span>price<span style="color:#e6db74">:</span><span style="color:#e6db74">.3f</span><span style="color:#e6db74">}</span><span style="color:#e6db74">&#34;</span>)
 </span></span><span style="display:flex;"><span>print(<span style="color:#e6db74">f</span><span style="color:#e6db74">&#34;Difference: </span><span style="color:#e6db74">{</span>abs(bs_price <span style="color:#f92672">-</span> price)<span style="color:#e6db74">:</span><span style="color:#e6db74">.4f</span><span style="color:#e6db74">}</span><span style="color:#e6db74">&#34;</span>)
 </span></span></code></pre></div><p>The difference between the two methods gives us an idea of the Monte Carlo simulation&rsquo;s accuracy.</p>
 <blockquote>

content/posts/finance/monte_carlo/Black-Scholes/options.py

@@ -7,6 +7,9 @@
 
 import numpy as np
 import matplotlib.pyplot as plt
+import seaborn as sns
+sns.set_style('whitegrid')
+plt.style.use("fivethirtyeight")
 
 def monte_carlo_option_pricing(S0, K, T, r, sigma, num_simulations, num_steps):
     dt = T / num_steps
@@ -35,15 +38,15 @@ def 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.figure(figsize=(12, 8))
+plt.plot(paths[:100, :].T, linewidth=1)
 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.figure(figsize=(12, 8))
 plt.hist(paths[:, -1], bins=100)
 plt.title("Distribution of Final Stock Prices")
 plt.xlabel("Stock Price")