4.2.4 Monte Carlo VaR (Simulate the Factors, Full Revaluation)

Basic Prerequisites: Numerical Methods and Statistics & Inference (Monte Carlo, CLT).

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1. Intuition & Practical Objective

Monte Carlo VaR is the most powerful and most expensive of the three methods. The practical objective in one line: simulate thousands (or hundreds of thousands) of plausible future moves of the risk factors, revalue the portfolio under each - or under an approximation - and read the quantile of the simulated losses.

Why bother when parametric is fast and historical is simple? Because MC controls the distribution you simulate from and handles arbitrary nonlinearity by full revaluation. Two things neither of the other two can do:

• Parametric forces normality and linearity; MC can simulate fat-tailed, skewed, or regime-switching factor processes (Student-$t$, jumps, GARCH) - you choose the model.
• Historical replays the past verbatim; MC can stress the current portfolio under what could happen (the factor model’s distribution), including scenarios no window has seen.

Glasserman Ch 9 frames it sharply (and the flat file gets it right): use the real-world distribution of $\Delta S$ for the loss probability (you want actual-risk probabilities), but the risk-neutral distribution to revalue an option (its price is a risk-neutral expectation - the RN measure is where the price lives). Mixing these two measures is the classic error.

The cost is raw compute: $m$ scenarios × (revaluation cost per portfolio), so a 100,000-option portfolio under 1M scenarios is heavy. That is exactly why the delta–gamma approximation exists (06) and why variance reduction (antithetic, importance sampling) is a core MC tool.

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2. Mathematical Ground Truth & Derivations

The MC VaR estimator. Let $L_i=-\Delta V_i$ be the portfolio loss under scenario $i$, where scenario $i$ draws a factor move $\Delta S_i$ from the chosen model and revalues the portfolio. Sort the losses and take the $\alpha$-quantile of the empirical distribution:

$${\widehat{\text{VaR}}}_{\alpha }^{(MC)}=L_{(\lceil m(1-\alpha )\rceil )},$$

exactly the HS quantile formula of 03 - the difference is where the scenarios come from (simulated vs replayed-from-the-past).

The factor model. For $d$ Gaussian factors, $\Delta S\sim N(0,\Sigma )$, sample via Cholesky: $\Delta S=AZ$, $A{A}^T=\Sigma$, $Z\sim N(0,I)$ (Glasserman Ch 2; Hull Ch 21 §21.6). For non-normal factors, replace the normal with a fat-tailed model (Glasserman Ch 9 §9.3: Student-$t$, jumps) - this is MC’s superpower over parametric.

Revaluation: full vs partial - the variance/error tradeoff.

• Full revaluation: value $V(S+\Delta S)$ under each scenario, $\Delta V={\Sigma }_{\text{pos}}[V_k(S+\Delta S)-V_k(S)]$. Unbiased for the model, but slow.
• Partial (delta/delta–gamma): approximate $\Delta V\approx {\delta }^T\Delta S+\frac{1}{2}\Delta S^T\Gamma \Delta S$, faster but biased for options. Glasserman Ch 9 uses the delta–gamma both as a control variate and as the sampling engine for importance sampling (twist the quadratic, revalue the true portfolio).

The quantile’s sampling error (Glasserman Ch 9 §9.1). Same honest formula as HS:

$$\sqrt{m}({\hat{x}}_p-x_p)\Rightarrow N(0,\frac{p(1-p)}{f(x_p{)}^2}),p=1-\alpha .$$

To halve the absolute uncertainty in the VaR you need $4\times$ the scenarios - the $\sigma /\sqrt{m}$ law. And since $f(x_p)$ is small in the tail, $f(x_p{)}^2$ in the denominator makes tail quantiles diverge slowly - exactly where variance reduction pays. Antithetic variates (the Glasserman Ch 4 Ch 4 §4.2) and importance sampling for rare tails (Ch 9 §9.2) attack exactly this.

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3. Computational Implementation - MC VaR with honest error bars, stdlib only

We simulate the shared 2-asset portfolio under 200k scenarios and report both the VaR and its standard error (so no one mistakes MC for a precision instrument). The 3,405.97 reproduces the hub MC column; the se is the honest $\sqrt{p(1-p)}/(\sqrt{m}f(x_p))$ quantile error from §2.

(The ±13 is honest: even 200k scenarios only pin the 99% VaR to ~±$13 - the same raw quantile noise HS carries in 03. Halving that band costs 4× the scenarios; importance sampling for the far tail is the lever (Glasserman Ch 9 §9.2).)

Where MC earns its keep - full revaluation beats delta-normal on an option. Same idea, one FX put. Because the put is only mildly convex over one day, delta-normal is close but not exact; full revaluation is the benchmark and delta-gamma is the cheap fix (both taken to page 06):

The delta-normal shortcut and full revaluation differ by only ~7% here - because this put’s gamma is modest. The whole point of delta–gamma and full revaluation is the straddle-style case (delta≈0, big gamma) where delta-normal is catastrophically wrong; that lives in 06 · Delta–Gamma & Backtesting.

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4. Failure Modes & First-Principles Breakdowns

1. Garbage-in, garbage-out on the factor model. MC with normal factors and the wrong correlation is just an expensive confirmation of a bad assumption. The distribution you simulate from is the model - choose it consciously (normality here is a hypothesis, not a law; see EVT).
2. Real-world vs risk-neutral confusion. For loss probability use the real-world measure; for revaluing an option use the risk-neutral price. Using RN revalue prices for loss probs (or vice versa) misstates both (Glasserman Ch 9; the flat file’s warning).
3. The $\sigma /\sqrt{m}$ tail is cruel. Halving VaR error = $4\times$ scenarios; tail quantiles need the most and converge slowest. Without variance reduction, “just raise $m$” is the classic brute-force error.
4. Numerical error in revaluation. Full revaluation of a path-dependent or American option needs an accurate pricer; a crude/dirty pricer injects its own bias bigger than the MC error (numerical methods: Numerical Methods).
5. MC ≠ historical ≠ parametric - they answer the same question with different models. Running only MC and assuming it’s “the truth” repeats the window/model mistake in a heavier coat. All three are biased estimates of the same quantile; only backtesting tells you which model fits the realized data (06 · Backtesting).

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Hull, Options, Futures, and Other Derivatives
• RiskMetrics / J.P. Morgan, RiskMetrics Technical Document (1996)

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6. Connected Graph Bridges

• Base: Numerical Methods (MC, variance reduction, quasi-MC) · Probability & Measure (CLT, martingales, risk-neutral measure).
• Back: 03 · Historical Simulation.
• Forward: 05 · Failure Modes · 06 · Delta–Gamma & Backtesting.
• Measure: Black–Scholes–Merton (risk-neutral revaluation).