The variance-covariance method bought its speed with a costly compromise: it approximates every position's P&L as linear in the underlying risk factors, and assumes the portfolio total ends up normally distributed. That's a fine approximation for a book of stocks and bonds — and a bad one for a book of options, whose payoffs curve. Monte Carlo VaR refuses to approximate at all: simulate as many scenarios as you like from whatever model you trust, then fully reprice the actual portfolio — curvature, optionality, and all — under each one.

The method, step by step

1. Choose (or fit) a model for how the risk factors move — correlated normal shocks, fatter-tailed Student-t shocks, or a full stochastic process for each factor.
2. Simulate a large number K of scenarios from that model — commonly 10,000 or more, since the far tail needs plenty of scenarios to be well represented.
3. For each scenario, fully reprice every position in the portfolio under the simulated risk-factor values — running an option through Black–Scholes or a binomial tree exactly as if that scenario had actually happened — to get the scenario's true P&L, curvature included.
4. Sort the K simulated P&Ls and read VaR off the appropriate percentile — the identical last step as historical simulation, except the scenarios now come from a fitted model instead of the market's own past.

That combination — a model you control, plus exact repricing rather than a linear shortcut — is what makes Monte Carlo the most flexible of the three methods. It is also, by a wide margin, the most computationally expensive: revaluing a large exotic book tens of thousands of times is real work, which is exactly why desks don't retire the fast variance-covariance number — they run both.

Worked example: where the linear approximation quietly fails

A desk is short a large book of options with a combined delta worth -\$300k per 1% move in the underlying, and negative gamma — the position loses extra money on large moves in either direction, a cost a delta-only approximation cannot see. Compare the variance-covariance method's linear-only estimate of P&L against a Monte Carlo engine's full repricing, scenario by scenario:

Near zero the two columns agree closely — the linear approximation is a good local fit. But out in the tail, at an +8% move, the delta-only estimate understates the true loss by \$2{,}656k - \$2{,}400k = \$256k, over 10% of the true figure. That gap is entirely the negative gamma the linear approximation cannot see, and it lands exactly where VaR is trying to measure risk — in the tail. Feed the delta-only column into a variance-covariance VaR and it will report a materially smaller, and wrong, number.

Watch the gap widen

Both curves below are cumulative distributions of a short-gamma book's simulated daily P&L, built from the same underlying scenarios — one column reads off the linear (delta-only) approximation, the other off a full repricing that includes a gamma cost. Drag the gamma slider to zero and the two curves sit on top of each other — a purely linear book, and the two methods agree perfectly. Increase gamma, and watch the full-repricing curve peel away into a fatter left tail exactly where VaR is read.