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.