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

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plt.show()
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```
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{{< img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_path.png" align="center" title="Paths">}}
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{{< img src="/posts/finance/monte_carlo/black-scholes/images/simulation_path.png" align="center" title="Paths">}}
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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.
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{{< img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png" align="center" title="Histogram">}}
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{{< img src="/posts/finance/monte_carlo/black-scholes/images/simulation_histogram.png" align="center" title="Histogram">}}
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## 6. Comparison with Analytical Solutions
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public/posts/finance/monte_carlo/black-scholes/index.html

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</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>xlabel(<span style="color:#e6db74">&#34;Stock Price&#34;</span>)
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</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>ylabel(<span style="color:#e6db74">&#34;Frequency&#34;</span>)
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</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>show()
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</span></span></code></pre></div><img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_path.png"
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</span></span></code></pre></div><img src="/posts/finance/monte_carlo/black-scholes/images/simulation_path.png"
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alt="Paths"
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<img src="/posts/finance/monte_carlo/Black-Scholes/images/simulation_histogram.png"
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<p>These visualizations show the range of possible stock price paths and the distribution of final stock prices, providing insight into the option&rsquo;s potential outcomes.</p>
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<img src="/posts/finance/monte_carlo/black-scholes/images/simulation_histogram.png"
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alt="Histogram"
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<p>These visualizations show the range of possible stock price paths and the distribution of final stock prices, providing insight into the option&rsquo;s potential outcomes.</p>
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<h2 id="6-comparison-with-analytical-solutions">6. Comparison with Analytical Solutions</h2>
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<p>To validate our Monte Carlo results, we can compare them with the Black-Scholes analytical solution:</p>
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<div class="highlight"><pre tabindex="0" style="color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#f92672">from</span> scipy.stats <span style="color:#f92672">import</span> norm
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