Midterms in Motion: A Deep Dive into Monte Carlo Simulations

Ever wondered how financial institutions predict market fluctuations or price options without divine intervention? Welcome to the world of Monte Carlo simulations, where randomness meets mathematics to paint a probabilistic picture of our financial future.

What's Under the Hood?

At the heart of Monte Carlo simulations lies an algorithm that generates random samples, feeding them into mathematical models to estimate outcomes. The beauty? It can handle complex scenarios like correlated stock prices and option pricing, all while embracing uncertainty.

Triangular Probability Density: A Case in Point

Let's roll up our sleeves and dive into the first question of Mcs Midterm04, focusing on a triangular probability density function (PDF), f(x) = kx for 0 ≤ x ≤ 2, and k(4-x) for 2 ≤ x ≤ 4.

Inverse Transform Method: A Sample at a Time

To generate samples from this PDF using an inverse transform method, we first normalize the PDF to ensure it integrates to 1. Then, we set up the equation F(x) = U, where U is a uniform random variable between 0 and 1. Solving for x gives us our sample.

Acceptance-Rejection Algorithm: A Tale of Two Methods

The acceptance-rejection algorithm introduces a 'proposal' distribution g(x), from which we generate samples more easily than from f(x). For each proposed sample, we accept it with probability min(1, f(x)/g(x)) or reject it and try again. On average, this method requires fewer uniform random variables to generate one sample of X.

Correlated Stock Prices: A Cholesky Decomposition Dance

Now, let's switch gears to correlated stock prices following geometric Brownian motions. To generate (Sat+s, Sbt+s) conditional on knowing (Sat, Sbt), we use the Cholesky decomposition method. We first decompose the correlation matrix into a lower triangular matrix L, then generate standard normal random variables Za and Zb. Our new stock prices become Sat+s = Sat exp((r-0.5sigmaa^2)s + sigmaa L[1,1]Za) and Sbt+s = Sbt exp((r-0.5sigmab^2)s + sigmab L[2,1]Za + sqrt(1-rho)L[2,2]Z_b).