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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
between 25 and 70 times fewer paths are needed for at-the-money
(ATM) uncollateralized portfolios and local models;
between 15 and 25 times fewer paths are needed for ATM
collateralized portfolios and local models; and
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.
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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