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Monte Carlo VaR

Monte Carlo VaR simulates thousands of random market scenarios from an assumed stochastic model, reprices the portfolio in each, and reads VaR off the simulated loss distribution. Unlike historical simulation, it is not limited to scenarios that occurred and can model nonlinear payoffs through full valuation. Its strengths are flexibility and the ability to handle complex, path-dependent instruments. Its costs are heavy computation and model risk: the answer is only as good as the assumed return process and correlations.

Try it yourself

Value at Risk — three ways

How much could this book lose on a bad day? VaR draws a line in the loss tail at confidence 95%. The parametric line assumes a normal curve, VaR = (z·σ·√h − μ·h)·V with z = 1.645 (one-tailed). The historical and Monte-Carlo methods read VaR off a simulated loss distribution instead. Under normality the three roughly agree; fat tails pull the historical figure deeper.

1-day 95% parametric VaRUS$3.29M
-6.0%-4.0%-2.0%0.0%2.0%4.0%6.0%Portfolio return over 1 day (%)VaR cutoff5% loss tail
Parametric (normal) · largestUS$3.29M
Historical (fat tails)US$3.05M
Monte-Carlo (5,000 draws)US$3.27M
z (95%, one-tailed) 1.645Horizon σ√h 2.00%Largest VaR Parametric
Portfolio value VUS$100M
Daily volatility σ2.0%
Daily expected return μ+0.00%
Horizon h1 day
Confidence level c
The three estimates land near US$3.29M, but the historical figure of US$3.05M is about the same, since this sample happens not to bite past the Gaussian tail here. The largest of the three is Parametric.
Discuss: which method would you report to a regulator, and which to your own risk committee?

Why it matters

Monte Carlo is a scenario factory. You specify how markets move, drift, volatility, correlations, then let a random-number generator spin out tens of thousands of possible futures and reprice the book in each. Because you invent the scenarios rather than borrow them from history, you can explore moves that never happened and value the most exotic option exactly. The danger is that you also get to choose the model, and a wrong process or correlation feeds straight through into a confident but mistaken VaR.

Formulas

Simulated price path (geometric Brownian motion)
St+1=St exp⁡ ⁣[(μ−12σ2) Δt+σΔt ε]S_{t+1} = S_t\,\exp\!\left[(\mu - \tfrac{1}{2}\sigma^2)\,\Delta t + \sigma\sqrt{\Delta t}\,\varepsilon\right]
Each draw ε∼N(0,1)\varepsilon \sim N(0,1) generates one scenario. Repricing the portfolio across many such paths builds the simulated P&L distribution.
VaR from the simulated distribution
VaRc=− Percentile(1−c){ ΔV(1),…,ΔV(M) }\mathrm{VaR}_c = -\,\text{Percentile}_{(1-c)}\{\,\Delta V^{(1)}, \dots, \Delta V^{(M)}\,\}
After MM simulated repricings, take the (1−c)(1-c) quantile of the value changes, exactly as in historical simulation but on model-generated scenarios.

Worked examples

Scenario

A risk team runs 10,000 Monte Carlo scenarios on a US$100M book containing options, then sorts the simulated one-day P&L. The 100th-worst outcome is a loss of US$5.8M. State the one-day 99% VaR and one advantage over historical simulation here.

Solution

With 10,000 outcomes the 1st percentile is the 100th-worst, so the one-day 99% VaR is US$5.8M. The advantage in this case is full valuation of the options across model-generated scenarios, including market moves not seen in any historical window, which a plain historical method could never produce. The figure still hinges on the assumed volatility and correlation inputs.

Common mistakes

  • ✗Monte Carlo VaR is assumption-free. It depends entirely on the assumed return process, volatilities, and correlations; a wrong model produces a precise but wrong VaR.
  • ✗More simulations remove model risk. Adding paths shrinks simulation noise but does nothing to fix a misspecified model; accuracy of the inputs is a separate problem.
  • ✗Monte Carlo and historical simulation are interchangeable. Historical simulation replays real past scenarios; Monte Carlo generates scenarios from a model and can explore moves that never occurred.
  • ✗Monte Carlo always gives the most accurate VaR. It is only as accurate as its model; a poorly specified process can be worse than a simple historical estimate.

Revision bullets

  • •Simulate many model-generated scenarios, then reprice
  • •Not limited to scenarios that actually occurred
  • •Handles nonlinear, path-dependent payoffs via full valuation
  • •VaR is the (1−c)(1-c) percentile of simulated P&L
  • •Costs: heavy computation and model risk from the inputs

Quick check

The main advantage of Monte Carlo VaR over historical simulation is that it

The principal drawback of Monte Carlo VaR is

Connected topics

Parametric VaRValuation MethodsHistorical SimMethod Tradeoffs

Sources

  1. Jorion (2007), Ch. 12
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    Presents Monte Carlo VaR, full-valuation repricing, and the role of model risk.
  2. Roncalli (2020), Ch. 2
    Roncalli, T. Handbook of Financial Risk Management. Chapman & Hall/CRC, 2020. ISBN 978-1-138-50187-4.
    Covers Monte Carlo simulation methods for VaR and their dependence on the assumed model.
How to cite this page
Dr. Phil's Quant Lab. (2026). Monte Carlo VaR. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-monte-carlo-var
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