When should I use Monte Carlo simulation instead of parametric VaR, and how does it actually work?

This is one of the most important conceptual questions in FRM Part I Quantitative Analysis. Let me explain both the mechanics and the decision criteria. **When to Use Monte Carlo Over Parametric VaR** The parametric (variance-covariance) method assumes returns are **normally distributed** and that portfolio value changes **linearly** with risk factors. This breaks down when: 1. **Non-linear instruments** — Options, callable bonds, mortgage-backed securities have payoffs that curve with underlying price changes. Delta-normal VaR misses this convexity. 2. **Fat tails** — If the actual return distribution has heavier tails than the normal distribution, parametric VaR underestimates extreme losses. 3. **Complex portfolios** — When you have hundreds of correlated risk factors with non-normal marginal distributions, Monte Carlo handles the complexity naturally. **How Monte Carlo VaR Works — Step by Step:** **Step 1: Define the Risk Factors** Identify all the variables that drive your portfolio's value — equity prices, interest rates, FX rates, volatilities, credit spreads, etc. **Step 2: Specify the Stochastic Process** For each risk factor, define a model for how it evolves. For example, geometric Brownian motion for stock prices: $$S_{t+\Delta t} = S_t \times \exp\left[(\mu - \frac{\sigma^2}{2})\Delta t + \sigma\sqrt{\Delta t} \cdot z\right]$$ where z is drawn from a standard normal distribution. **Step 3: Correlate the Random Draws** Use a **Cholesky decomposition** of the correlation matrix to ensure the simulated risk factor movements reflect real-world dependencies. **Step 4: Simulate Thousands of Scenarios** Generate 10,000+ paths for all risk factors simultaneously. For each scenario, reprice the full portfolio (including options using Black-Scholes or binomial trees). **Step 5: Build the Loss Distribution** Sort the 10,000 portfolio P&L outcomes from worst to best. The VaR at 99% confidence is the 100th worst loss (i.e., the 1st percentile of the distribution). **Quick Example:** Supppose Meridian Capital holds a portfolio of equity options worth $50 million. After running 10,000 Monte Carlo scenarios over a 10-day horizon, the sorted losses look like: - Scenario #1 (worst): −$4.2M - Scenario #100: −$2.8M - Scenario #500: −$1.9M The **10-day 99% VaR** is **$2.8M** (the 100th worst loss out of 10,000). **Limitations:** - Computationally expensive — repricing complex instruments 10,000+ times takes time. - Model risk — results depend on the assumed stochastic processes and correlation structures. - Pseudo-random number quality matters — use variance reduction techniques (antithetic variates, stratified sampling) for stable results. Our FRM Part I course on AcadiFi covers both the theory and practical implementation of Monte Carlo VaR, including exam-style practice problems.