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<h5class="card-title">Monte Carlo Simulation for Option Pricing</h5>
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<pclass="card-text post-summary">1. Introduction In the dynamic world of finance, options play a crucial role in risk management, speculation, and portfolio optimization. An option is a contract that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) within a specific time frame. The challenge lies in accurately pricing these financial instruments, given the uncertainties in market movements.</p>
<description>1. Introduction In the dynamic world of finance, options play a crucial role in risk management, speculation, and portfolio optimization. An option is a contract that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) within a specific time frame. The challenge lies in accurately pricing these financial instruments, given the uncertainties in market movements.</description>
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<title>MSFT Stock Prediction using LSTM or GRU</title>
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@@ -412,15 +412,16 @@ <h1>Monte Carlo Simulation for Option Pricing</h1>
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<h2id="1-introduction">1. Introduction</h2>
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<p>In the dynamic world of finance, options play a crucial role in risk management, speculation, and portfolio optimization. An option is a contract that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) within a specific time frame. The challenge lies in accurately pricing these financial instruments, given the uncertainties in market movements.</p>
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<p>Traditional analytical methods, while powerful, often struggle with complex option structures or realistic market conditions. This is where Monte Carlo simulation steps in, offering a flexible and robust approach to option pricing. By leveraging the power of computational methods, Monte Carlo simulations can handle a wide array of option types and market scenarios, making it an indispensable tool in a quantitative analyst’s toolkit.</p>
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<p>For further explanation about <em>options pricing</em>, check <ahref="https://www.investopedia.com/articles/optioninvestor/07/options_beat_market.asp" target="_blank" rel="noopener">Investopedia</a>.</p>
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<h2id="2-the-black-scholes-model">2. The Black-Scholes Model</h2>
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<p>Before diving into Monte Carlo methods, it’s crucial to understand the Black-Scholes model, a cornerstone in option pricing theory. Developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, this model revolutionized the field of quantitative finance.</p>
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<h3id="limitations-of-the-black-scholes-model">Limitations of the Black-Scholes Model</h3>
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<p>While groundbreaking, the Black-Scholes model has several limitations:</p>
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<p><strong>Constant Volatility</strong>: The model assumes volatility is constant, which doesn’t hold in real markets where volatility can change dramatically.</p>
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<p><strong>Log-normal Distribution</strong>: It assumes stock returns are normally distributed, which doesn’t account for the fat-tailed distributions observed in reality.</p>
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<p><strong>Continuous Trading</strong>: The model assumes continuous trading is possible, which isn’t realistic in practice.</p>
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<p><strong>No Dividends</strong>: It doesn’t account for dividends, which can significantly affect option prices.</p>
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<p><strong>European Options Only</strong>: The original model only prices European-style options, not American or exotic options.</p>
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<p><strong>Risk-free Rate</strong>: It assumes a constant, known risk-free rate, which can vary in reality.</p>
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<li><strong>Constant Volatility</strong>: The model assumes volatility is constant, which doesn’t hold in real markets where volatility can change dramatically.</li>
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<li><strong>Log-normal Distribution</strong>: It assumes stock returns are normally distributed, which doesn’t account for the fat-tailed distributions observed in reality.</li>
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<li><strong>Continuous Trading</strong>: The model assumes continuous trading is possible, which isn’t realistic in practice.</li>
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<li><strong>No Dividends</strong>: It doesn’t account for dividends, which can significantly affect option prices.</li>
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<li><strong>European Options Only</strong>: The original model only prices European-style options, not American or exotic options.</li>
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<li><strong>Risk-free Rate</strong>: It assumes a constant, known risk-free rate, which can vary in reality.</li>
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<p>These limitations highlight why more flexible approaches like Monte Carlo simulation are valuable in option pricing.</p>
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<h2id="3-monte-carlo-simulation-theoretical-background">3. Monte Carlo Simulation: Theoretical Background</h2>
<p>Monte Carlo methods use repeated random sampling to obtain numerical results. In the context of option pricing, we simulate many possible price paths for the underlying asset and then calculate the option’s payoff for each path.</p>
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<h3id="application-to-option-pricing">Application to Option Pricing</h3>
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<p>For option pricing, we model the stock price movement using a stochastic differential equation:</p>
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<pretabindex="0"><code>dS = μSdt + σSdW
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</code></pre><p>Where:</p>
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<p>$$
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dS = \mu Sdt + \sigma SdW
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$$</p>
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<p>Where:</p>
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<ul>
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<li>S: Stock price</li>
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<li>μ: Expected return</li>
@@ -491,14 +482,14 @@ <h2 id="4-implementing-monte-carlo-simulation-in-python">4. Implementing Monte C
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</span></span><spanstyle="display:flex;"><span> dt <spanstyle="color:#f92672">=</span> T <spanstyle="color:#f92672">/</span> num_steps
</span></span><spanstyle="display:flex;"><span><spanstyle="color:#66d9ef">for</span> t <spanstyle="color:#f92672">in</span> range(<spanstyle="color:#ae81ff">1</span>, num_steps <spanstyle="color:#f92672">+</span><spanstyle="color:#ae81ff">1</span>):
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</span></span><spanstyle="display:flex;"><span> z <spanstyle="color:#f92672">=</span> np<spanstyle="color:#f92672">.</span>random<spanstyle="color:#f92672">.</span>standard_normal(num_simulations)
</span></span></code></pre></div><p>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.</p>
<p>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.</p>
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<h2id="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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</span></span><spanstyle="display:flex;"><span>print(<spanstyle="color:#e6db74">f</span><spanstyle="color:#e6db74">"Monte Carlo price: </span><spanstyle="color:#e6db74">{</span>price<spanstyle="color:#e6db74">:</span><spanstyle="color:#e6db74">.2f</span><spanstyle="color:#e6db74">}</span><spanstyle="color:#e6db74">"</span>)
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