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Accelerating CVA calculations using Quasi Monte Carlo Methods
01 April 2018
One of the most important counterparty credit risk measures is
the credit valuation adjustment (CVA), defined as the present value
of the potential loss due to a counterparty failing to meet their
contractual obligations. Risk neutral pricing states that the
present value is equal to the expected value of the payoff using
risk adjusted probabilities. The payoff in this case is the netted
portfolio value less collateral (floored at zero) at the time of
counterparty default, multiplied by one minus the recovery rate.
The payoff is at counterparty level, potentially path dependent
(collateral, early exercise conditions, lags between fixings and
cash flows), and subject to change. The expectation of high
dimensional, fluid payoffs of this sort must be estimated with
Monte Carlo (MC) simulation (see Gregory 2015 [7]).
Monte-Carlo estimation of an expectation involves randomly
sampling the payoff times according to the risk neutral
probabilities and averaging the results. The estimate approaches
the true expectation with probability 1 with a normally distributed
error with zero mean and standard deviation equal to the standard
deviation of the payoff (a constant) divided by the square root of
the number of replications used [6]. Requiring the error to be on
average 100 times smaller than the standard deviation of the CVA
payoff requires 10,000 replications, a number typically used. This
highlights the main disadvantage of MC: its computational
expense.
This is of particular importance in the context of CVA where
each evaluation of the payoff is also computationally expensive.
Consider a bank with 100,000 trades that uses 200 exposure dates in
the time discretization. One replication of the CVA payoffs across
all counterparties requires roughly 10,000,000 trade prices
(assuming trade maturities are evenly distributed) and thus one MC
CVA estimate using 10,000 paths requires of the order of
100,000,000,000 trade price evaluations. Furthermore, many banks
risk manage these credit adjustments, and to do so requires the
calculation of the derivatives of the CVA with respect to the
market prices of the instruments used to hedge it. Bump and run
techniques require at least one full MC CVA calculation per
derivative. 200 derivatives bring the computational load up to
20,000,000,000,000 trade price evaluations per day.
Not surprisingly, quants have been searching for ways to
accelerate this massive calculation. One successful line of
research uses algorithmic adjoint differentiation (AAD) to compute
the derivatives, reducing the computational burden to a fixed
multiple
(5 to 10 times depending on the problem and memory handling) of the
baseline CVA calculation, no matter how many derivatives are
required (see Capriotti et al. 2011 [4] for more information).
Assuming a conservative fixed multiple of 10, this would reduce
the total number of calculations by a factor of 20, requiring
1,000,000,000,000 trade price evaluations. This dramatic
improvement, however, does not come for free. The implementation of
an AAD enabled system requires large changes to existing code
libraries, requiring a significant upfront investment to implement.
As a consequence, many still compute the derivatives using bump and
run techniques.
In another line of research, Ghamami and Zhang 2014 [5]
highlight that direct and independent simulation of the portfolio
value to each time step diversifies the errors in each time bucket,
leading to a significant reduction of the standard error of the
final sum across time. The benefit of the direct simulation
approach is reduced if simulating to each time step independently
is more computationally expensive than simulating to each step
sequentially using a common simulation path. Highly path dependent
portfolios containing collateral may not benefit as a result, but
the technique looks quite promising for portfolios of
uncollateralized vanillas.
In a similar line of research, Burnett, O'Callaghan and Hulme
2016 [2] note that the computational expense of calculating
valuation adjustment risks (derivatives) vary significantly across
different counterparties, and that the computational expense is
uncorrelated with the size of the adjustment error. This opens up
the possibility to optimally allocate computational resources where
they are needed most, using a different number of paths and/or time
steps for different counterparties and risks. They formalize this
idea by setting up and minimizing the expected unexplained PnL by
varying the number of paths and frequency of time steps allocated
to each counterparty and risk, subject to a computational time
constraint. The acceleration they report computing FVA on a sample
Barclays portfolio is impressive, roughly in line with the
acceleration provided by AAD.
In a forthcoming IHS Markit research paper, we explore yet
another acceleration technique used to price payoffs called quasi
Monte Carlo (QMC). The mechanics are identical to classical Monte
Carlo simulation with the exception that the pseudo random numbers
(PRN) are replaced with carefully selected low discrepancy (number)
sequences (LDS) that are more evenly distributed, with the hope of
improving the convergence rate closer to the optimal . In the
paper, we estimate CVA and CVA sensitivities of several portfolios
of vanilla interest rate swaps, ranging from single currency single
trade portfolios, to nettings sets containing eleven different
currencies, all with a multi-currency, multi-curve extension to the
Hull-White model [8] with deterministic hazard rates.
We find that QMC with Sobol' sequences [9], Broda's 65,536
direction numbers [1], and the Brownian bridge discretization [3],
with on average 1,197 paths produces errors roughly equivalent in
size to classical MC with 10,000 simulations, a factor of 8
acceleration.
The number of paths needed to match classical MC with 10,000
paths varies significantly between portfolios and calculation type,
however, increasing as the portfolio becomes more out of the money
(291 paths for far in the money, 538 paths for at the money, and
2,763 for far out of the money). The gains are most impressive when
the CVA, the CVA sensitivities, and the corresponding standard
errors are the largest (in the money portfolios) and more modest
when the standard errors are the smallest (out of the money
portfolios). For all but the far out of the money portfolio, the
equivalent number of paths increase as more factors are added to
the portfolio (208 paths for single trade single currency, 463
paths for six trade six currency portfolio, and 573 paths for the
eleven trade eleven currency portfolio), and for more complex
calculations (242 for CVA MTM, 282 for CR delta, 431 for IR and FX
delta, and 702 for IR and FX vega). Illustrative results for in the
money and at the money portfolios are presented in tables 1 and 2.
Far out of the money portfolios are the most difficult and erratic,
as indicated by the numbers in table 3.
Table 1: Approximate number of QMC + BB paths
needed to produce CVA and CVA sensitivities with errors roughly
equivalent to classical MC with 10,000
paths for far in the money portfolios of various sizes (one 10 year
fixed rate payer swap in each currency). Fixed rates set to par -
300 basis points.
Table 2: Approximate number of QMC + BB paths
needed to produce CVA and CVA sensitivities with errors roughly
equivalent to classical MC with 10,000 paths for at the money
portfolios of various sizes (one 10 year fixed rate payer swap in
each currency). Fixed rates set to par.
Table 3: Approximate number of QMC + BB paths
needed to produce CVA and CVA sensitivities with errors roughly
equivalent to classical MC with 10,000 paths for far out of the
money portfolios of various sizes (one 10 year fixed rate payer
swap in each currency). Fixed rates set to par + 300 basis
points.
The various acceleration approaches presented above are not
mutually exclusive: they can be used together to compound the
computational savings. One potentially interesting combination we
have just started to explore is to combine the direct simulation
methods proposed by Ghamami 2014 [5] for non-collateralized CVA
calculations with QMC and the Brownian Bridge mechanism, allowing
us to first reduce the effective dimension of each term of the
summed CVA, and second, when combined with a randomization method
such as digital shift ([6]), diversify the errors across the time
axis. We provide some early convergence results in table 4. The
results are very promising indeed.
Table 4: CVA error for an at the money 10 year
USD fixed for float swap with $10,000 notional for various numbers
of simulation paths. Three methodologies are presented. Classical
MC, randomized QMC + BB (RQMCBB) with regular pathwise simulation,
and randomized QMC + BB (RQMCBB) with direct and independent
simulation of the risk factors to each time step.
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