Monte Carlo exposure simulation

Monte Carlo exposure simulation is the standard computational framework for quantifying credit exposure and xVA. Despite being expensive, Monte Carlo is preferred because it is completely general: it can handle high dimensionality (multiple currencies, yield curves, FX rates), path dependency (margin modelling, Bermudan exercise), and the portfolio-level aggregation required for netting. The simulation generates joint realisations of all relevant risk factors at discrete future time points, enabling the revaluation of all trades in a netting set and the computation of metrics such as expected positive exposure, potential future exposure, and the inputs to credit value adjustment (CVA), funding value adjustment (FVA), KVA, and MVA.

There are two main implementation approaches: path-wise and direct. In a path-wise approach, each simulation generates an entire trajectory of risk factors on a fixed time grid (typically 50—200 points), enabling the capture of path-dependent features like margin and the natural computation of PFE profiles. In a direct approach, default times are sampled directly (not bucketed on a grid) and portfolio revaluation is done at each sampled time; this can offer better convergence for pure xVA calculations (which require only a single integrated value) and greater P&L stability. Both approaches should converge to the same result, but path-wise simulation is more common in practice due to its flexibility.

A typical setup involves around 10,000 simulations with quasi-random sequences for variance reduction, 100 time steps, potentially hundreds of counterparties and tens of trades each — yielding on the order of 10 billion trade revaluations. This creates enormous computational demands. Practical optimisations include: highly-optimised analytic pricing functions, cash flow bucketing (combining cash flows within a time bucket into a single equivalent payment), grid-based look-ups for complex products, American Monte Carlo (Longstaff-Schwartz) for path-dependent exercise decisions, and machine learning surrogates for fast revaluation and Greeks. Scaling adjustments can correct systematic differences between the simulation’s simplified pricing and front-office valuations. Risk-factor models are typically parsimonious (e.g., Hull-White one-factor for interest rates, lognormal for FX) to keep calibration, simulation, and correlation estimation tractable across asset classes.

Key Details

• Typical scale: ~10,000 simulations $\times$ ~100 time steps $\times$ ~250 counterparties $\times$ ~40 trades = ~10 billion revaluations
• Path-wise simulation uses a fixed time grid; direct simulation samples default times freely
• Exposure data has three dimensions: transaction ($k$), simulation ($s$), time step ($t$), giving future value $V(k,s,t)$
• Aggregated value: $V_{\text{Agg}}(s,t)={\sum }_{k=1}^{K}V(k,s,t)$
• Roll-off risk: discrete events (maturities, cash flows, exercise dates) can cause exposure jumps missed by coarse grids
• Margin modelling requires additional “look-back” grid points at $t-\text{MPoR}$ intervals
• Risk-neutral calibration is standard for CVA/FVA; physical measure may be used for PFE/IMM; KVA/MVA create mixed-measure challenges

Textbook References

The xVA Challenge (Gregory, 2020)

• Section 15.3.1 (pp. 419—421): Path-wise vs direct simulation; typical number of simulations; roll-off risk and time-grid design (Figures 15.6—15.8)
• Section 15.3.2 (pp. 421—423): Revaluation bottleneck; cash flow bucketing; American Monte Carlo; machine learning for pricing; scaling adjustments (Figure 15.9)
• Section 15.3.3 (pp. 423—429): Risk-neutral vs physical measure — drift, volatility, correlation calibration; BCBS requirements; P-measure vs Q-measure consistency (Figure 15.14)
• Section 15.3.4 (pp. 429—430): Aggregation hierarchy for xVA; three-dimensional data representation $V(k,s,t)$
• Section 15.4.1—15.4.5 (pp. 430—439): Model choices — HW1F for rates, lognormal FX, correlation estimation; balance between parsimony and realism

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