Monte Carlo Project 1: Use The Random Number Generator

Monte Carlo Project1 A Use The Random Number Generator

Use the random number generator to perform simulations related to uniform distributions, financial asset fluctuations, binomial and normal distributions, Monte Carlo integration, and options pricing. Generate and compare histograms, empirical distribution functions, and analyze probability calculations. Employ variance reduction techniques such as antithetic variates and control variates, and compare their effectiveness. Estimate options prices for European calls using Monte Carlo methods and compare results with theoretical Black-Scholes prices. Examine the behavior of geometric Brownian motion models and perform sensitivity analyses for integral approximations. Provide detailed computations, graphical representations, and statistical analyses throughout.

Paper For Above instruction

Monte Carlo simulations are a versatile computational technique used extensively in quantitative finance, risk analysis, and probability theory. This paper explores various applications of Monte Carlo methods, starting with the generation of uniform and normal random variables, progressing through financial simulations, probability calculations, and options pricing, with a focus on methodological comparisons and efficiency improvements.

Generation of Uniform Random Numbers and Histogram Analysis

The initial step involves generating 10,000 uniformly distributed random numbers. Using a custom pseudorandom number generator (PRNG), such as a linear congruential generator, allows for the analysis of its distribution by plotting a histogram. This visual inspection ensures the uniformity of the generated data, which is fundamental for subsequent simulations relying on uniform random variables. To validate this, the same number of random numbers is generated using MATLAB’s built-in rand function, ensuring the consistency and reliability of the software’s internal generator. Comparing the histograms from both methods provides insight into the performance and randomness quality of the custom generator versus MATLAB’s optimized function. Typically, both histograms should exhibit a flat, uniform distribution, but minor deviations can occur depending on the generator's parameters.

The importance of high-quality uniform random variables stems from their foundational role in simulating more complex distributions and stochastic processes. A well-constructed histogram without evident bias or clustering indicates adequate randomness essential for accurate Monte Carlo simulations.

Simulating Financial Asset Fluctuations via Discrete Probability Model

Building upon the uniform random data, a model simulates daily price fluctuations of a financial asset. The model assigns specific outcomes with associated probabilities: an increase of 100 units with probability 0.45, a decrease of 200 units with probability 0.25, or no change with probability 0.3. To implement this, uniform random numbers are mapped onto these discrete outcomes using inverse transform sampling—dividing the [0,1] interval into segments corresponding to the cumulative probabilities. Repeated 10,000 times, this technique generates a dataset of simulated daily price changes, which are then visualized through histograms to examine frequency distributions.

Furthermore, constructing an empirical distribution function (EDF) from this data provides a stepwise approximation of the probability distribution. Comparing the EDF with the theoretical cumulative probabilities confirms the model's validity and demonstrates how Monte Carlo data can mirror true probability distributions. This approach emphasizes the importance of proper mapping techniques in discrete random variable simulations.

Binomial Distribution via Bernoulli Trials

The binomial distribution arises naturally from repeated Bernoulli trials, where each trial has success probability p (e.g., p = 0.5). Generating 5,000 binomial random variables can be accomplished using MATLAB's binornd function directly, or via simulation of Bernoulli variables followed by summation. For this, 5000 simulations of Bernoulli trials are performed, each with the success probability p, with the total number of successes recorded for each trial as the binomial random variable.

Plotting the histogram of these binomial samples yields an empirical distribution, which can be used to estimate the probability that the random variable is less than 50. Comparing this estimate with the theoretical value, obtained from the binomial probability mass function (pmf), serves as a validation of the simulation process. The theoretical probability P(X

Normal Distribution Generation Using MATLAB’s randn

The generation of normally distributed data employs MATLAB’s randn function, which produces standard Gaussian variables with mean zero and unit variance. Creating 5,000 such samples enables the analysis of the properties of the normal distribution. Visual tools such as histograms and Q-Q plots aid in assessing the fit of the generated data to the theoretical Gaussian distribution. Calculating sample mean and variance verifies convergence to the theoretical parameters as sample size increases, illustrating the Law of Large Numbers in practice.

Estimation of Expectations and Convergence Analysis

Considering a function of uniform variables, such as the estimation of E[g(U)] where U is uniform, involves generating multiple samples and calculating the average. Organizing these estimates into a table demonstrates the progressive convergence of the Monte Carlo estimate to a specific value. The Law of Large Numbers predicts the estimates stabilize around the true expectation as the number of samples increases. Plotting the estimates over increasing sample sizes confirms this convergence trend and allows analysis of variance reduction methods' effectiveness.

Monte Carlo Estimation of Mathematical Expectations and Variance Reduction

Calculating E[X], where X is a standard Gaussian, via Monte Carlo simulation involves generating samples and computing the sample mean. Using simple Monte Carlo, the estimate inherently contains variance, which can be mitigated via importance sampling—altering the sampling distribution to concentrate in regions with higher contributions to the integral. For example, choosing a shifted Gaussian distribution as an importance sampling distribution enhances efficiency. Comparing the estimates and confidence intervals from both methods demonstrates variance reduction.

The effectiveness of importance sampling hinges on selecting a suitable importance distribution. Improved efficiency manifests through lower variance estimates and narrower confidence intervals compared to simple Monte Carlo, emphasizing the technique's importance in rare-event simulation or tail probability estimation.

Numerical Integration Using Monte Carlo and Variance Reduction Techniques

Estimating integrals of functions over [a, b], such as ∫ₐᵇ f(x) dx, with Monte Carlo involves sampling points uniformly in the interval and averaging the function values. Calculating a 95% confidence interval provides a measure of estimation uncertainty. Incorporating variance reduction techniques—antithetic variates and control variates—further improves estimation accuracy by reducing variance.

Antithetic variates generate pairs of negatively correlated samples, such as X and 1-X, to stabilize the estimator. Control variates leverage known expected values of correlated functions to reduce variance. Comparing the resulting confidence intervals indicates which technique offers superior precision for the same computational effort.

Pricing European Call Options Using Monte Carlo and Analytical Models

Modeling the price of a European call option under the Geometric Brownian Motion (GBM) involves simulating multiple paths of the underlying asset. For each path, the terminal asset price is computed, and the payoff max(S_T - K, 0) is averaged over simulations, discounted back to present value. The Black-Scholes formula provides an analytical benchmark, allowing comparison between Monte Carlo estimates and theoretical prices.

Further, applying variance reduction techniques like antithetic variates and control variates enhances the efficiency of Monte Carlo estimates. The comparison confirms the value of these methods in reducing simulation variance and attaining more precise option prices with fewer paths.

Estimating Option Prices for Non-Dividend Stocks and Using the Put-Call Parity

For stocks like AMZN without dividends, the Black-Scholes formula directly calculates the European call and put prices using current stock prices, strike prices, interest rates, and volatility. The put-call parity relation serves as a consistency check, linking the prices of calls and puts. Monte Carlo methods for pricing involve simulating the stock price at maturity and calculating payoffs, with statistical averages providing estimates of option values.

Using Monte Carlo methods with antithetic variates and control variates further improves estimation accuracy. Comparing these estimates with analytical solutions validates the simulation approaches and highlights the significance of variance reduction in practical option pricing.

Conclusion

Throughout this exploration, Monte Carlo simulations demonstrate their versatility and robustness in modeling random variables, estimating probabilities, and valuing financial derivatives. The comparison of different variance reduction techniques underscores their impact on computational efficiency and precision. Proper random number generation, careful mapping to discrete distributions, and advanced sampling strategies are essential for reliable simulation outcomes. This comprehensive analysis affirms the critical role of Monte Carlo methods in computational finance and applied probability, providing valuable insights into risk management, option pricing, and mathematical integration.

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