Uncollateralized netting sets or netting sets with weak credit support annexes can incur large valuation adjustment (xVA) charges due their impact on a dealer's counterparty credit risk exposure (CVA), need for funding (FVA and MVA), and capital constraints (KVA). Whereas the unadjusted value of derivatives is often determined and risk-managed with custom-tailored local model and calibration approaches, valuation adjustments are netting set and higher-level metrics that require the joint modeling of a multitude of risk factors. The high dimensionality of the risk factor space and the fluidity of the portfolios involved necessitates that the xVAs are almost exclusively estimated using Monte Carlo simulation.

The computational burden of this calculation is well known, and can be easily understood by considering the number of trade valuations, which is proportional to the size of the portfolio (>100k), the number of future exposure dates (>200), the number of simulations paths needed to obtain a certain level of accuracy (1k-10k), and the number of sensitivities needed to hedge the risk (>200). A sizeable portion of xVA quants' and/or software engineers' time is spent thinking about how to reduce the overall time it takes to perform these calculations. A lot of progress has been made, based on the use of large-scale distributed computing systems, GPU computing, adjoint algorithmic differentiation, trade compression, highly optimized mathematical libraries (e.g. MKL) and pricers, varying the number of simulations per sensitivity and counterparty, etc. But faster is always better, and the search for speed continues to this day. Indeed, in the latest edition of Wilmott magazine [1], IHS Markit's Pouya Bastani, Stefano Renzitti, and Steven Sivorot explore whether quasi-random methods can be used to accelerate this calculation by reducing the number of simulation paths needed to achieve the given accuracy requirement.

They present how many quasi-Monte Carlo paths are needed to estimate CVA and CVA sensitivities at the same level of accuracy as those obtained when using 10,000 pseudo-random paths. They focus on portfolios of receiver interest rate swaps with varying characteristics, such as the number of currencies, moneyness, and collateral, and apply local and global simulation models. Local models only capture the risk factors underlying a particular netting set while global models simultaneously capture the risk factors for all netting sets combined. The latter are typically used for cross-netting set consistency, operational simplicity, and/or entity-level calculations. The global model covers 36 currencies in the tests presented. In all cases, a cross-currency Hull-White model is used to simulate the risk factors, with one interest rate factor per currency, and one factor per exchange rate. Hazard rates are assumed to be independent from the exposures, and thus do not impact the simulation performance.

The results reported when using quasi random numbers (Sobol' sequences from Broda's 65536 generator [2]) with a Brownian bridge path construction technique are quite impressive, a selection of which is displayed in Plot 1. The results are presented as acceleration factors of quasi-Monte Carlo over pseudo-Monte Carlo (CVA) calculations, i.e. an acceleration factor of one indicates that the same number of paths are needed to obtain the same level of accuracy, an acceleration factor of two indicates half the number of paths are needed to obtain the same level of accuracy, etc. More specifically, the findings suggest that acceleration factors are such that

1. between 25 and 70 times fewer paths are needed for at-the-money (ATM) uncollateralized portfolios and local models;
2. between 15 and 25 times fewer paths are needed for ATM collateralized portfolios and local models; and
3. between 4 and 5 times fewer paths are needed for ATM collateralized and uncollateralized portfolios when using a global model.

Plot 1: CVA acceleration factor when using quasi-random numbers with the Brownian bridge path construction over pseudo-random numbers. Portfolios contain one ATM interest rate swap per currency. Local models are portfolio specific and represent the number of risk factors implied by the number of currencies, i.e. #currencies*2-1 factors per time step, whereas the global model always simulates 36 currencies, i.e. 71 factors per time step.

As impressive as the results are, they do vary significantly across model and portfolio characteristics:

• portfolios with fewer currencies (and factors) perform far better than portfolios with many currencies when using a local model,
• uncollateralized portfolios perform far better than collateralized portfolios when using a local model,
• local models significantly outperform global models,
• global models show less variability to the number of portfolio factors and the presence of collateral.
• in-the-money (ITM) portfolios perform far better than ATM portfolios, which perform far better than out-of-the money (OTM) portfolios (not shown here, see paper for details)

Nevertheless, in almost all the cases tested, when using the same number of paths, the quasi-random numbers with the Brownian bridge path construction produced more accurate results than the pseudo-random numbers with and without antithetic sampling. In this situation, it does not seem to be so much a question of if they are better, but rather by how much. Holding the number of paths, and thus computational time, fixed, it seems clear that with the class of portfolios and models tested, quasi-Monte Carlo with the Brownian bridge path construction should be used over pseudo-Monte Carlo and pseudo-Monte Carlo with antithetic sampling.

When it comes to reducing the number of simulation paths, and hence the computational time, the choice is more difficult and ultimately depends on the user's objective. If their objective is to minimize the sum of all errors across all netting sets and xVAs, subject to a computational time constraint (and in this case using the same number of paths per netting set and risk), then with the portfolios and models tested, a significant reduction in the number of paths is indeed possible, but the accuracy improvement is no longer uniform across netting sets and risk measures. Larger benefits do occur for non-collateralized ITM portfolios, but OTM and/or collateralized netting sets may come out slightly worse.

A more conservative option is to set the number of paths such that the error of the least, or one of the least performant quasi-Monte Carlo scenarios (collateralized or OTM non-collateralized portfolios), is equivalent to the error obtained using pseudo-Monte Carlo. The number-of-paths reduction will be smaller in this case, but most portfolios other than the most difficult ones will also be more accurate. Whatever the choice, quasi-Monte Carlo methods are another promising technique that banks can test on their specific portfolio and models to determine if they can be used to either 1) improve their xVA accuracy for a fixed computational budget or 2) accelerate their xVA calculations.

Author: Stefano Renzitti, Data Analytics, Executive Director, Financial Risk Analytics, IHS Markit

To read the full research paper, which first appeared in Wimott Magazine here, please download the following PDF.

[1] Renzitti, S., Bastani, P., and Sivorot, S.. 2020. Accelerating CVA and CVA Sensitivities Using Quasi-Monte Carlo Methods. Wilmott Magazine 108, 78-93.

[2] http://www.broda.co.uk/

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