Monte Carlo VaR

Historical simulation can only replay the days that happened: 750 of them here, and only 8 in the 1% tail. Monte Carlo draws as many days as you can afford from a model of the risk factors. On RiskAtoms Bank's desk, 10,000 simulated days take well under a second. The catch is twofold: the answer is only as good as the model, and it moves every time you change the seed.

Monte Carlo VaRVaR read off the P&Ls of many simulated days, each drawn from a statistical model of the risk factors and revalued in full. keeps the full revaluation of L3.2.10: Historical simulation in practice but replaces the history with a model. Click through the recipe:

Correlated draws: the Cholesky factor

Independent normals are easy to generate; correlated ones are not. The trick is a matrix square root. A Cholesky factorA lower-triangular matrix L with L L′ = Σ. Multiplying independent standard normals by L gives draws with covariance Σ. $L$ of the covariance matrix satisfies $LL^\top = \Sigma$, and then $x = Lz$ has covariance $\Sigma$. For the UK index, the US index and UK vol:

Read $L_3$ row by row. The UK index move is just 1% × $z_1$. The US index gets 0.0066 × $z_1$ (its link to the UK) plus its own 0.0088 × $z_2$. UK vol loads −0.60 on $z_1$, so it rises when the UK index falls. The full 8 × 8 factor used for the desk:

Lognormal prices

The equity, FX and oil moves are log returns, so a simulated price is $S = S_0\,e^{x}$: a lognormalA price whose log return is normal: S = S₀eˣ. It cannot go negative and is skewed to the right. price (M-C07: The lognormal distribution). It cannot go below zero, and it is skewed: at the 1st and 99th percentiles the UK index ends the day at 7,816 and 8,188, a fall of 184 points against a rise of 188, and its mean is 8,000.40, a touch above 8,000. Rates, the spread and implied vol are moved by normal amounts in bp or vol points.

The desk's Monte Carlo VaR

Two comparisons tell the story. Valued with deltas, the simulated paths give £1.944m, almost exactly delta-normal's £1.944m: on linear positions Monte Carlo just reproduces the formula, with noise. Revalued in full they give £1.848m: the puts' gamma cushions the big falls. And Monte Carlo sits well below historical simulation's £2.274m because the model has calm volatilities and normal tails. Feed it the stressed spell's covariance and it gives £4.160m. More paths will never fix the model.

How accurate is it?

Another seedThe starting value of a pseudo-random number generator. The same seed gives the same draws, so a run can be repeated exactly. gives another answer. The standard errorThe standard deviation of an estimate across repeated runs: how much the answer would change with a different seed. of a simulated 99% quantile from $N$ paths is about $\sqrt{\alpha(1-\alpha)/N}\,/\,f(q)$, with $f$ the density of the P&L at the quantile. With the run's own standard deviation (£842,436) and a normal density, the 10,000-path VaR has a standard error of £31.5k: a 95% interval from £1.787m to £1.910m. A seed study confirms the formula:

Four times the paths halve the error. The module's run (£1.848m) sits about one standard error below the seed study's average, which is exactly the kind of difference a seed makes. To get the standard error down to 1% of the VaR would take about 28,953 paths.

Run time

The module's run is 10,000 paths × 8 positions = 80,000 revaluations; the Try it below times it in your browser. A real trading book has thousands of positions, some priced by models that are slow themselves, so banks run fewer paths, approximate the slow pricers (delta-gamma or pre-computed grids), share paths across desks and use variance reductionWays to get the same accuracy with fewer paths: antithetic draws, control variates, importance sampling, quasi-random numbers.. M6.2: Market risk modelling covers those techniques.

Covariance of the draws. With $z \sim N(0, I)$, so $\mathbb{E}[zz^\top] = I$, and $x = Lz$:

Finding L. Write out $LL^\top = \Sigma$ for a 3 × 3 matrix and solve entry by entry, top-left first:

For the example: $l_{11} = \sqrt{0.0001} = 0.01$, $l_{21} = 0.000066 / 0.01 = 0.0066$, $l_{22} = \sqrt{0.000121 - 0.0066^2} = 0.0088$, $l_{31} = -0.006 / 0.01 = -0.60$, $l_{32} = (-0.00484 - (-0.60)(0.0066))/0.0088 = -0.10$ and $l_{33} = \sqrt{0.64 - 0.36 - 0.01} = 0.5196$. Each square root must have a positive argument, which is exactly the condition that Σ is positive definite.

Standard error of the quantile. Let $q$ be the true 1% quantile of the P&L. The number of paths below it is binomial (M-C04: The binomial distribution) with mean $N(1-\alpha)$ and standard deviation $\sqrt{N\alpha(1-\alpha)}$, so the share below it varies by $\sqrt{\alpha(1-\alpha)/N}$. Near $q$, a change $\Delta q$ in the cut-off moves that share by about $f(q)\,\Delta q$. Equating the two:

The error shrinks only with $\sqrt N$, and it grows as α moves into the tail, where $f(q)$ is small.

Run 1,000 paths a few times and watch the dots spread. Switch to 20,000 and run again: the strip tightens by about √20 ≈ 4.5. Then switch to the stressed covariance. The paths run in chunks between frames, so the page stays responsive.

Basel's internal models rules have never prescribed one method: historical simulation, Monte Carlo and parametric models are all allowed if the model meets the qualitative and quantitative standards and passes validation and backtesting. FRTB keeps that freedom for expected shortfall (M4.3: Market risk capital (FRTB) and CVA). A bank using simulation should document its number of paths and its seeds, and be able to show that the simulation error is small. Variance reduction and fat-tailed simulation models are in M6.2: Market risk modelling.

• Monte Carlo VaR draws many days from a model of the factors, revalues the book in full on each, and reads off the quantile.
• A Cholesky factor (LL′ = Σ) turns independent normals into correlated moves: x = Lz.
• Log-return factors give lognormal prices S₀eˣ: never negative, skewed up.
• On the desk: £1.85m from 10,000 paths, below delta-normal (the puts' gamma) and below historical simulation (calm normal model).
• The standard error of the quantile is about √(α(1−α)/N) / f(q): four times the paths halve it.
• More paths reduce noise, never model error.

Monte Carlo VaR

VaR read off the P&Ls of many simulated days, each drawn from a statistical model of the risk factors and revalued in full.

Path

One simulated day: a draw of all the factor moves, and the book's P&L for it.

Cholesky factor

A lower-triangular matrix L with L L′ = Σ. Multiplying independent standard normals by L gives draws with covariance Σ.

Seed

The starting value of a pseudo-random number generator. The same seed gives the same draws, so a run can be repeated exactly.

Standard error

The standard deviation of an estimate across repeated runs: how much the answer would change with a different seed.

Lognormal price

A price whose log return is normal: S = S₀eˣ. It cannot go negative and is skewed to the right.

Variance reduction

Ways to get the same accuracy with fewer paths: antithetic draws, control variates, importance sampling, quasi-random numbers.