Credit Valuation Adjustment Compression by Genetic Optimization

Marc Chataigner; Stéphane Crépey

doi:10.3390/risks7040100

Credit Valuation Adjustment Compression by Genetic Optimization

Chataigner, Marc;Crépey, Stéphane 2019-09-29 00:00:00 risks Article Credit Valuation Adjustment Compression by Genetic Optimization Marc Chataigner and Stéphane Crépey * LaMME, Univ Evry, CNRS, Université Paris-Saclay, CEDEX, 91037 Évry, France; marc.chataigner@univ-evry.fr * Correspondence: stephane.crepey@univ-evry.fr Received: 4 August 2019; Accepted: 23 September 2019; Published: 29 September 2019 Abstract: Since the 2008–2009 ﬁnancial crisis, banks have introduced a family of X-valuation adjustments (XVAs) to quantify the cost of counterparty risk and of its capital and funding implications. XVAs represent a switch of paradigm in derivative management, from hedging to balance sheet optimization. They reﬂect market inefﬁciencies that should be compressed as much as possible. In this work, we present a genetic algorithm applied to the compression of credit valuation adjustment (CVA), the expected cost of client defaults to a bank. The design of the algorithm is ﬁne-tuned to the hybrid structure, both discrete and continuous parameter, of the corresponding high-dimensional and nonconvex optimization problem. To make intensive trade incremental XVA computations practical in real-time as required for XVA compression purposes, we propose an approach that circumvents portfolio revaluation at the cost of disk memory, storing the portfolio exposure of the night so that the exposure of the portfolio augmented by a new deal can be obtained at the cost of computing the exposure of the new deal only. This is illustrated by a CVA compression case study on real swap portfolios. Keywords: counterparty risk; credit valuation adjustment (CVA); XVA (X-valuation adjustments) compression; genetic algorithm MSC: 91B30; 91G20; 91G40; 91G60 JEL Classiﬁcation: D52; D53; G01; G13; G24 1. Introduction XVAs, where VA stands for valuation adjustment and X is a catch-all letter to be replaced by C for credit, F for funding, M for margin, or K for capital, have been implemented by banks in reaction to the regulatory changes aroused by 2008 ﬁnancial turmoils. They monetize counterparty risk and its funding and capital consequences by add-ons to derivative entry prices sourced from clients. According to the cost-of-capital XVA approach of Albanese and Crépey (2019), accounting for the impossibility for a bank to replicate the jump-to-default related cash ﬂows, the ﬁnal, all-inclusive XVA formula reads CVA + FVA + MVA + KVA, (1) to be sourced by the bank from clients on an incremental run-off basis at every new deal. As stated by the Basel Committee on Banking Supervision (2015), major counterparty credit losses on OTC (Over-the-counter) derivative portfolios in 2008 arose from XVA accounting losses rather than from actual client defaults. In particular, a bank incurs a CVA loss when the market perceives a deterioration of the credit risk of a client. This has motivated the creation of XVA desks for dealing with these risks. Risks 2019, 7, 100; doi:10.3390/risks7040100 www.mdpi.com/journal/risks Risks 2019, 7, 100 2 of 21 In this paper, we deal with CVA compression, i.e., the minimization of the CVA of a client portfolio by the introduction of an incremental trade, subject to the constraint of not altering too much the market risk of the portfolio. In the ﬁnancial derivative industry, the term compression term is generally applied in the context of “trade compression”, i.e., the reduction of the gross notional of positions in the market. Trade compression aims notably at reducing certain capital requirements, the number of transactions, and their amount (see Section 5.3 of (Gregory 2015)). As reﬂected by the proliferation of related industry presentations , this kind of balance sheet optimization is very active in top tier banks at the moment. XVAs reﬂect market inefﬁciencies that should be compressed as much as possible. Here, we focus on CVA compression for concreteness and simplicity, but the developed XVA compression methodology is generic. It could and should be applied to further XVA metrics, as soon as these are available with sufﬁcient speed, for computation at the portfolio level, and accuracy, for numerical signiﬁcance of the results at the incremental trade level (see Section 5 in (Albanese et al. 2018), which emphasizes the XVA compression perspective on the pricing and risk management of ﬁnancial derivatives in the post-2008–2009 global ﬁnancial crisis era, and cf. (Kondratyev and Giorgidze 2017), which uses a genetic algorithm for determining an optimal trade-off between MVA compression and transaction costs). The complexity of XVA compression problems stems, in particular, from the hybrid nature of the state space of the corresponding optimization problems. Indeed, a new trade (ﬁnancial derivative) is described by a combination of continuous and discrete parameters. This rules out the use of standard convex optimization schemes for such problems. Instead, we are lead to the use of metaheuristic algorithms. In this paper, we show how a genetic algorithm with penalization can efﬁciently ﬁnd a CVA offsetting trade, while limiting the impact of the trade on the market exposure proﬁle. The latter is necessary for staying in line with the separation of mandates between the XVA desks, in charge of managing counterparty risk, and the other, dubbed “clean”, trading desks of the bank, in charge of hedging the market risk of the bank positions. The other XVA compression challenge is execution time, with intensive valuation of the involved XVA metrics as a bottleneck. The XVA metrics are primarily deﬁned at the portfolio level: Time-0 XVAs can be formulated as expectations of nonlinear functionals of the bank derivative portfolio exposure, i.e., “clean” valuation (or “mark-to-market” (MtM) ignoring counterparty risk) of the bank portfolio, assessed at randomly sampled times and scenarios. Each new deal gives rise to XVA add-ons computed as the corresponding trade incremental XVA amounts, i.e., the differences between the XVAs of the portfolios including and excluding the new deal. To make intensive trade incremental XVA computations practical in real-time as required for XVA compression purposes, our proposed MtM store-and-reuse approach circumvents clean revaluation at the cost of disk memory, storing the portfolio exposure of the night so that the exposure of the portfolio augmented by a new deal can be obtained at the cost of computing the exposure of the new deal only. Outline and Contributions The paper is outlined as follows. Section 2 formulates the penalized CVA compression problem and introduces the related genetic optimization algorithm. Section 3 is about two key acceleration techniques in this regard. Section 4 presents a numerical case study on real swap portfolios. Section 5 concludes. The main contributions of the paper are the design of a parallelized genetic algorithm for the CVA compression task, the MtM store-and-reuse acceleration technique for trade incremental XVA computations, and the numerical CVA compression case study on real swap portfolios. See also, e.g., David Bachelier ’s panel discussion Capital and Margin Optimisation, Quantminds International 2018 conference, Lisbon, 16 May 2018. Risks 2019, 7, 100 3 of 21 More broadly, this paper enriches the literature on the use of genetic (also called evolutionary) optimization algorithms in ﬁnance. Cont and Hamida (2005) applied evolutionary algorithms to investigate a set of co-calibrated model parameterizations in order to assess the associated model risk. Kroha and Friedrich (2014) compared different genetic algorithms for automatic trading. Jin et al. (2019) applied evolutionary algorithms to optimal investment and consumption stochastic control problems. For wider reviews of genetic algorithms in ﬁnance, we refer the readers to (Chen 2012; Drake and Marks 2002). We refer the reader to the end of the paper for a list of the main abbreviations. 2. CVA Compression Modeling 2.1. Credit Valuation Adjustment We consider a complete stochastic basis (W,F,P), for a reference market ﬁltration (ignoring the default of the bank itself) F = (F ) , satisfying the usual conditions, and a risk-neutral pricing t t2R measure P, calibrated to market quotes of fully collateralized transactions. All the processes of interest are F adapted and all the random times of interest are F stopping times. This holds at least after so-called reduction of all the data to F, starting from a larger ﬁltration G including the default of the bank itself as a stopping time, assuming immersion from F into G for simplicity (see (Albanese and Crépey 2019) for details). The P expectation and (F ,P) conditional expectation are denoted by E and E , respectively. In developed markets, the overnight indexed swap (OIS) rate is together the reference remuneration rate for posted collateral and the best market proxy for a risk-free rate. We denote r ds by r = (r ) an F progressive OIS rate process and we write b = e for the corresponding t t2R risk-neutral discount factor. By clean valuation or mark-to-market of a contract (or portfolio), we mean the (trade additive) risk-neutral conditional expectation of its OIS discounted future promised cash ﬂows, ignoring counterparty risk and its capital and funding implications. We consider a bank engaged in bilateral trading with a single corporate counterparty (client) with default time and recovery rate t and R , respectively. This setup, which is chosen for simplicity, c c is consistent with a common situation where credit risk budget is assigned at each counterparty level within the bank. We denote by MtM the corresponding mark-to-market process of the client portfolio to the bank. The (time 0) CVA of the bank is its expected discounted loss in case of client default, i.e., 1 + CVA = E 1 b b (1 R )MtM . (2) t c ft Tg t t c c Assuming deterministic interest rates, this can be rewritten as CVA = (1 R ) b EPE(t)P(t 2 dt), (3) c t c where the expected positive exposure (EPE) is deﬁned as EPE(t) = E(MtM js = t ) . (4) s jt =t Equation (3) is popular with practitioners because it allows obtaining the CVA as the integral of the EPE against the client CDS (Credit default swap) curve. However, it is only really practical in simplistic models where the market and credit sides of the problem are independent, so that EPE(t) = E(MtM ). However, a key CVA modeling issue is wrong-way risk, i.e., the risk of adverse dependence between market and credit (see (Brigo and Vrins 2018; Crépey and Song 2016 2017; Glasserman and Yang 2018; Hull and White 2012; Iben Taarit 2018; Li and Mercurio 2015; Pykhtin 2012)). Risks 2019, 7, 100 4 of 21 Assuming the client default time endowed with an intensity g , a more ﬂexible formula is g du c + 0 (5) CVA = (1 R )E b e g MtM ds. c s s s Under a credit support agreement (CSA), MtM should be replaced by (MtMC) in all equations above, where C is the collateral posted by the counterparty. Obviously, collateral can mitigate the EPE and the CVA considerably. In the data of our case study there is no CSA, i.e., C = 0. Non-linearity of MtM with respect to the portfolio payoff components imposes CVA calculations at the counterparty portfolio (netting set) level. Similar approaches apply to FVA computations, with analogous comments, whereas the MVA can be computed based on quantile regression for the embedded dynamic initial margin calculations (see (Crépey et al. 2019)). In any case, the numerical bottleneck of XVA computations lies in intensive MtM calculations. 2.2. Fitness Criterion By the augmented and initial portfolios, respectively, we mean the portfolio of the bank inclusive and exclusive of a newly considered deal with the client. The aim of an XVA compression problem is to ﬁnd a new trade that minimizes the corresponding XVA metric of the augmented portfolio. This is equivalent to minimize the incremental CVA, which we denote by Z Z R R T T s s c c augm g du c + g du c init + u u 0 0 DCVA = (1 R )E b e g (MtM ) ds (1 R )E b e g (MtM ) ds c s s c s s s s 0 0 (6) augm g du c + init + = (1 R )E b e g (MtM ) (MtM ) ds, c s s s s where the indices init and augm refer to the initial portfolio and augmented portfolio, respectively. We emphasize that trade incremental CVA computations require two portfolio-wide calculations: one without the new trade and another one including it. Minimizing an XVA metric is most easily obtained through a signiﬁcant deformation of the portfolio exposure process (especially in the context of this work of a portfolio with a single counterparty). However, an XVA compression procedure should not affect too much the market risk of the portfolio, because market risk is the mandate of the clean desks of the bank, who, in particular, are subject to trading limits. This motivates the addition of a penalization to the incremental XVA criterion. In our case study, the incremental deal will consist of an interest rate swap. As such product is mostly sensitive to interest rate moves, a natural penalization is then in terms of its DV01 (dollar value of an 01), i.e., the variation of its mark-to-market (at time 0) under a parallel shift of the yield curve by one basis point (=10 ). More precisely, an interest rate swap exchanges one leg indexed on a ﬂoating interest rate against one leg paying a ﬁxed interest rate, called swap rate. The swap is said to be payer (respectively, receiver) for the party that pays (respectively, receives) the ﬂoating payments. A monocurrency swap exchanges both legs in the same currency. It is mainly sensitive to the ﬂuctuations of the corresponding ﬂoating interest rate term structure. DV01 measures the associated risk as the difference between the prices of the swap under the baseline (actual market data observed in the real market) and for a bumped yield curve deﬁned as the concatenation of the money market rates, forward rates, and swap rates, on the relevant (successive) time segments. Bumping the yield curve typically means adding 10 to each tenor of this curve and updating the other reference curves (zero coupon rates, forward rates, etc.) accordingly. Focusing on the CVA metric in this paper, we obtain the following ﬁtness minimization problem: minimize f (x) = DCVA(x) + ajDV01(x)j, (7) x2A Risks 2019, 7, 100 5 of 21 where x parameterizes a new deal (swap) to be found in a suitable search space A (see Section 4.1), DCVA(x) is its incremental CVA (cf. Equation (6)), DV01(x) is its DV01, and a is a penalization parameter controlling the trade-off between CVA reduction and market risk proﬁle preservation. By solving Equation (7), we aim at identifying a new deal which, if added to the current client portfolio of the bank, would diminish its counterparty risk without impacting too much its market risk. Note that, for scaling reasons (with, in particular, market penalization), we address the XVA compression problem in terms of trade incremental (as opposed to augmented portfolio) XVA numbers. A new deal is determined by a combination of quantitative (e.g., notional, maturity, etc.) and qualitative (e.g., currency, long or short direction, etc.) parameters, so that no gradient or Hessian is available for the ﬁtness function f in Equation (7). Moreover, one is interested in exploring a variety of local minima of f , to see different trading opportunities emerge from the optimization procedure. Furthermore, we can guess that some (crucial) parameters need be learned ﬁrst, such as currency or maturity; other parameters, such as notional, can be reﬁned in a second stage. All these features lead us to addressing Equation (7) by means of a genetic optimization algorithm. 2.3. Genetic Optimization Algorithm Genetic optimization algorithms belong to the class of derivative-free optimizers, which is surveyed and benchmarked numerically in (Rios and Sahinidis 2013) (including the CMA-ES and DAKOTA/EA genetic algorithms). The idea of genetic (or evolutionary) optimization algorithms is to evolve a population of individuals through cycles of modiﬁcation (mutation) and selection in order to improve the performance of its individuals, as measured by a given ﬁtness function. In addition, so-called crossover is used to enhance the search in parameter space. To the best of our knowledge, evolutionary algorithms were ﬁrst explicitly introduced in (Turing 1950, chp. 7 Learning Machines, p. 456). See the classical monographs by the authors of (Back 1996; Goldberg 1989; Holland 1975). They then experienced the general artiﬁcial intelligence disgrace and comeback before and after the 2000s. However, they always stayed an active ﬁeld of research, seen from different perspectives, such as particle ﬁltering, MCMC, or sequential monte carlo methods (see (Del Moral 2004)). Beyond its ﬁnancial applications reviewed at the end of Section 1, genetic optimization has been used in many different ﬁelds, such as mechanics (Verma and Lakshminiarayanan 2006), calibration of neural networks hyperparameters (Young et al. 2015), or operational research (Larranaga et al. 1999). Genetic optimization offers no theoretically guaranteed rate of convergence, but it is often found the most efﬁcient approach in practice for dealing with hybrid (partly continuous, partly discrete/combinatorial, hence without well deﬁned gradient and Hessian), nonconvex (in the sense of one local minimum, at least, for each set of values of the discrete parameters), and high-dimensional optimization problems such as Equation (7). At each iteration, the ﬁtness f (x) is computed for each individual (also named chromosome) x of an initial population (a set of chromosomes). The values returned by the objective function are used for selecting chromosomes from the population. Among numerous selection methods (see (Blickle and Thiele 1995)), we can quote ﬁtness proportionate selection, ranking proportionate selection (in order to avoid the overrepresentation of the chromosomes with the highest ﬁtness values), and tournament selection (selection of the best among randomly drawn chromosomes). The common intention of these selection methods is to sample in priority individuals with the best ﬁtness values. A genetic algorithm is dubbed elitist if the selection operator always keeps the chromosome with the best ﬁtness value. Otherwise (as in our case), there is no guarantee that the best visited chromosome is contained in the population corresponding to the ﬁnal iteration. The mutation stage is intended to maintain some diversity inside the population, in order to avoid the algorithm being trapped by local minima. A mutation randomly changes one gene, i.e., one component (e.g., in our case, the notional of a new swap) of a chromosome. Risks 2019, 7, 100 6 of 21 Selection and mutation play opposite roles: a focus on ﬁtness leads to a quicker convergence toward a local minimum; conversely, a too heavily mutated population results in a slow random research. In addition, a crossover operator plays the role of a reproduction inside the algorithm. The principle of crossover is to build two children chromosomes from parent chromosomes. A distribution (often the same as the one used for selection) is chosen for picking chromosomes from a population of the previous iteration and for recombining pairs of selected chromosomes. Children share gene values of their parents but a gene value from one parent cannot be inherited by both children. A crossover mask decides for each gene in which parent a child can copy the gene version. One of the most popular crossover masks is single point crossover (see Appendix A). The role of the crossover operator is paradoxical, as crossover can be seen as a combination of mutations, which increase the genetic diversity, while crossover also promotes chromosomes with higher ﬁtnesses. Crossover aims at beneﬁting of a presupposed proximity of best solutions. The above operators are applied iteratively until a suitable stopping condition is satisﬁed. The most basic one is a ﬁxed number of iteration, but customized criteria may also be used to limit further the number of iterations. For instance, the algorithm can be interrupted when the minimum (or sometimes even the maximum) ﬁtness value within the population at the beginning of an iteration is below a predeﬁned threshold. See Algorithm 1 and Figure 1 for the algorithm in pseudo-code and skeleton forms, denoting by r the mutation rate, i.e., the percentage of individuals in a population affected by a mutation, and by r the crossover rate, i.e., the percentage of individuals affected by crossover recombination. The behavior of a genetic optimization algorithm is essentially determined by the choice of the selection operator, the number of solutions affected by a mutation, and the number of chromosomes affected by a crossover. See the work of Tabassum and Kuruvilla (2014) for a user guide to the main genetic algorithm ingredients and Carvalho et al. (2011) for applications of genetic optimization algorithms to benchmark functions. Algorithm 1: Pseudo-code of an optimization genetic algorithm Data: An initial population P of size P and the associated ﬁtness values for each init chromosome, a crossover rate r , and a mutation rate r . c m Initialization; while a stopping condition is not satisﬁed do Save b(1 r ) Pc chromosomes, chosen by an appropriate selection method from P , c init in P ; Save br Pc chromosomes, chosen by an appropriate selection method from selected c r P P , in P ; Recombine, uniformly without replacement, b c pairs from init crossover P ; Merge P and P in P ; Mutate randomly br Pc in P ; crossover crossover selected mutated m mutated for Each chromosome c in P do mutated Compute the ﬁtness value of c; end end Result: A new population and the associated ﬁtness values. Risks 2019, 7, 100 7 of 21 Start Sampling Population of size P Fitness Evaluation Population with associated ﬁtness values. Stop Termination test Crossover Selection r P Selected population Population of b c of size b(1 r ) Pc recombined pairs c Mutation Population with br Pc mutated chromosomes Figure 1. Skeleton of an optimization genetic algorithm. 3. Acceleration Techniques Without suitable acceleration techniques, the above CVA compression approach is not workable in real time on realistic banking portfolios: in the examples of Section 4, a naive (desktop) implementation requires about 20 h of computations. This becomes even more problematic for hyperparameters tuning (such as a, crossover rate r , etc.). Hyperparameters are generally chosen with grid search, random search (see (Bergstra et al. 2011)), Bayesian optimization (see (Snoek et al. 2012)) or even evolutionary algorithms again (see (Young et al. 2015)). In any case, their calibration is greedy in terms of overall genetic algorithm execution. In this section, we deal with the two following acceleration techniques, which may be used simultaneously: a MtM store-and-reuse approach for trade incremental XVA computations, speeding up the unitary evaluation of the ﬁtness function; and a parallelization of the genetic algorithm accelerating the ﬁtness evaluation at the level of the population. Risks 2019, 7, 100 8 of 21 3.1. MtM Store-and-Reuse Approach for Trade Incremental XVA Computations Most of the time in portfolio-wide XVA calculations is spent in clean valuation (i.e., mark-to-market MtM) computations: by comparison, simulation of the risk factors or of the collateral is typically negligible. Our case study is based on the CVA metric. As observed after Equation (6), by lack of trade-additivity of the (portfolio-wide) CVA, trade incremental XVA computations require two portfolio-wide calculations: one without the new trade and another one including it. However, it is possible to store the (including MtM) paths simulated for the initial portfolio and reuse them each time we want to compute a new trade incremental XVA. Then, each trade incremental XVA computation only requires the forward simulation of the mark-to-market process of the new deal. The corresponding MtM store-and-reuse approach to trade incremental XVA computations circumvents repeated valuations at the cost of disk memory. It exploits the trade additivity of clean valuation by recording the MtM paths of the initial portfolio on a disk. For every new deal, the augmented portfolio exposure is obtained by adding, along the paths of the risk factors, the mark-to-market of the initial portfolio and of the new deal. This augmented portfolio exposure is then plugged into the XVA engine. An optimally implemented MtM store-and-reuse approach brings down trade incremental XVA computations to the time of generating the clean price process of the trade itself, instead of the one of the augmented portfolio as a whole. Another advantage of this approach is its compliance with desk segregation: As far as clean valuation is concerned, the XVA desks just use the pricers of the clean desks. Hence, the MtM process plugged into the XVA computations is consistent with the one used for producing the market risk hedging sensitivities. However, such an approach comes at the costs of memory disk (obviously), but also data slippage as, for consistency, it requires anchoring all the trade incremental XVA computations at the market data and parameters corresponding to the generation of the initial portfolio exposure. In practice, an MtM process at the overall portfolio level can only be generated during night runs, between two market sessions. Moreover, we have to distinguish between ﬁrst-order (or ﬁrst-generation) XVAs, which are options on the MtM process, and higher-order (or second-generation) XVAs (see (Crépey et al. 2019)), which can be viewed as compound options of order two or more on the MtM process. Second-generation XVAs may also involve conditional risk measures, e.g., conditional value-at-risk for the dynamic initial margin calculations that are required for MVA computations, as opposed to conditional expectations only in the case of ﬁrst generation XVAs. A Monte Carlo simulation diffuses risk factors X (such as interest rates, credit spreads, etc.) along drivers Z (such as Brownian motions, Poisson processes, etc.), according to a model formulated as a Markovian system of stochastic differential equations, starting from some given initial condition X for all risk factors, suitably discretized in time and space. Modulo calibration, X can be identiﬁed with the time 0 market data. We denote by Y a suitable estimate of a process Y at all (outer) nodes of a Monte Carlo XVA engine. In particular, MtM is the fully discrete counterpart of the MtM process of the initial portfolio, namely the clean value of the portfolio at future exposure dates in a time grid and for different scenario paths. At ﬁrst sight, an MtM store-and-reuse approach is unsuitable for second-order XVAs, such as the MVA and the KVA (but also the CVA in the case of a CSA where the bank receives so-called initial margin), Indeed, in their case, the principle of swapping computations against storage would require to store not one portfolio exposure MtM, but a whole family of resimulated, future conditional portfolio exposures, (at least, over a certain time horizon), which seems hardly feasible in practice. However, even in the case of second-order XVA metrics, an MtM store and reuse approach can be For details regarding the initial margin and the MVA, see (Crépey et al. 2019, sets. 5.2 and 6.4). Risks 2019, 7, 100 9 of 21 implemented with the help of appropriate regression techniques (at the cost of an additional regression error; see (Crépey et al. 2019)). Formalizing the above discussion, the conditions for a straightforward and satisfactory application of the MtM store-and-reuse approach to a given XVA metric are as follows, referring by indices init, incr, and augm to the initial portfolio, the new deal, and the augmented portfolio: 1. (No nested resimulation of the portfolio exposure required) The formula for the corresponding (portfolio-wide, time-0) XVA metric should be estimatable without nested resimulation, only based on the portfolio exposure rooted at (0, X ). A priori, additional simulation level makes impractical the MtM store-and-reuse idea of swapping execution time against storage. incr init [ [ 2. (Common random numbers) MtM should be based on the same paths of the drivers as MtM . Otherwise, numerical noise (or variance) would arise during MtM aggregation. incr 3. (Lagged market data) MtM should be based on the same time, say 0, and initial condition init X (including, modulo calibration, market data), as MtM . This condition ensures a consistent init incr augm [ [ [ aggregation of MtM and MtM into MtM . These conditions have the following implications: 1. The ﬁrst seems to ban second-order generation XVAs, such as CVA in presence of initial margin, but these can in fact be included with the help of appropriate regression techniques. 2. The second implies storing the driver paths that were simulated for the purpose of obtaining init incr MtM ; it also puts a bound on the accuracy of the estimation of MtM , since the number of Monte Carlo paths is imposed by the initial run. Furthermore, the XVA desks may want to account for some wrong way risk dependency between the portfolio exposure and counterparty credit risk (see Section 2.1); approaches based on correlating the default intensity and the market exposure in Equation (5) are readily doable in the present framework, provided the trajectories of the drivers and/or risk factors are shared between the clean and XVA desks. 3. The third induces a lag between the market data (of the preceding night) that are used in the incr incr computation of MtM and the exact MtM process; when the lag on market data becomes unacceptably high (because of time ﬂow and/or volatility on the market), a full reevaluation of the portfolio exposure is required. Figure 2 depicts the embedding of an MtM store-and-reuse approach into the trade incremental XVA engine of a bank. Market Data Database Data from the night init Initial run: MtM , Clean desks binit Clean Pricers Incremental run: incr bincr MtM , Z init Incremental run : Z XVA desks augm augm [ b MtM , Z augm Aggregate XVA Engine DataBase [ XVA Incremental run: init init [ b MtM , Z Figure 2. MtM store-and-reuse implementation of a trade incremental XVA engine with drivers Z. Risks 2019, 7, 100 10 of 21 3.2. Parallelization of the Genetic Algorithm Most of the XVA compression computational time is spent in the evaluation of the incremental XVA metric involved in the ﬁtness criterion visible in Equation (7). The MtM store-and-reuse approach allows reducing the complexity of such trade incremental XVA computations to trade (as opposed to portfolio) size. However, to achieve XVA compression in real time, this is not enough; another key step is the parallelization of the genetic algorithm that is used for solving Equation (7). The genetic algorithm is a population based method, which implies to maintain a population of individuals (tentative new deals) through each iteration of the algorithm. The calculation of the objective function, for a given individual, does not depend on the ﬁtness value of the other individuals. Therefore we can vectorize the computation of the ﬁtness values within the population. Provided a suitable parallel architecture is available, a perfectly distributed genetic algorithm makes the execution time independent of the population size P (see Algorithm 1 and Figure 1). This makes an important difference with other metaheuristic optimization algorithm, such as simulated annealing or stochastic hill climbing, which only evaluate one or very few solutions per iteration, but need much more iterations to converge toward a good minimum (see (Adler 1993; Janaki Ram et al. 1996)). As discussed in (Pardalos et al. 1995), the above parallelization of the ﬁtness function evaluation, for a given population, should not be confused with a parallel genetic algorithm in the sense of an independent evolution of several smaller populations. In our context, where individuals only represent incremental trades, a parallelization of population ﬁtness evaluation is compatible with an MtM store-and-reuse approach for the trade incremental XVA computations. Combining the two techniques results in an XVA compression time independent of the sizes of the initial portfolio of the bank and of the population of the genetic algorithm used for the optimization, which represents an XVA compression computation time gain factor of the order of Number of trades in the initial portfolio population size. 4. Case Study In the remainder of the paper, we present CVA compression results on real swap portfolios , using an additional swap for the CVA compression. We aim at addressing questions such as: Which type of swap is suitable for achieving the compression of the CVA, in the context of a given initial portfolio? How does the compression distort the portfolio exposure, with or without penalization? To ease the implementation of the MtM store-and-reuse approach, we assume no CSA (cf. Section 3.1). 4.1. New Deal Parameterization A swap is parameterized by its notional, its maturity, its direction, and its currency. The quantitative parameters are encoded through grids of values: 5 7 5 Notional: From 10 to 10 by step of 10 dollars. Maturity: From 1 to 20 years by step of 1 year, 30 years and 50 years. The qualitative parameters are encoded as enumerations of values: Currency: Euro, US dollar, GBP or Yen. Direction: A binary variable for payer or receiver. The underlying interest rate and FX models are proprietary and cannot be disclosed in the paper. We use a deterministic credit spread model for the counterparty, calibrated to the CDS term structure of the latter. Risks 2019, 7, 100 11 of 21 Moreover, we impose the additional swap to be at par so that it can be entered at no cost, which is equally desirable from the bank and the client perspectives. The above parameterization deﬁnes a discrete search space A with 100 22 4 2 = 1.76 10 elements. 4.2. Design of the Genetic Algorithm We address the optimization problem in Equation (7) by a genetic algorithm as per Section 2.3. The new deal space A in Equation (7) is viewed as a space of chromosomes x, the genes (deal parameters) of which evolve randomly along the iterations of the algorithm as detailed in Section 2.3. In the theoretical literature on genetic algorithms, an individual is represented as a bit string. In practice, however, bit string representation of parameters does not give enough control on the mutation distribution. Namely, in bit string representation, mutations affect all bits uniformly, whereas we might want to mutate some parameters more frequently (the quantitative parameters, in particular, as the algorithm tends to quickly identify the relevant values of the qualitative parameters). Hence, we rather model our individuals x by a variable string, a choice also made in (Kondratyev and Giorgidze 2017). We choose rank proportionate selection to avoid ﬁtness scaling issues. More precisely, if we have a population P = f1, . . . , Pg of P individuals and the associated ﬁtnesses ( f ) , then the probability i i2P to select chromosome i is 2rank( f ) p = , P(P + 1) where rank is a function that ranks chromosomes according to their ﬁtness value (returning one for the highest value, in the context of a minimization problem). Regarding the crossover operator, we use a uniform crossover mask, i.e., the choice of gene inherited from one parent or another is drawn with a uniform probability. Regarding mutations, the probability to mutate a gene is proportional to the number of alleles (values) that it can take. The mutation operator then selects uniformly a new gene allele (value). In our experiments, the notional of the new swap can take 100 different values, its currency 4 values, its position 2 values, and its maturity 22 values. Hence, when a chromosome is selected for 100 22 mutation, the probability to mutate each of its genes is equal to for the notional, for the maturity, 128 128 4 2 for the currency, and for the position. Indeed, a more frequent mutation of notional and 128 128 maturity parameters are desirable. Diversity for currency and position is ensured at the initialization of the algorithm (with a large population) and maintained across the iterations thanks to the crossover operator. A prerequisite for a successful implementation is a reasonable speciﬁcation of the search space A. Hyperparameters strongly impact the behavior of the algorithm. In the case of XVA compression, which is time-consuming, searching good values for the hyperparameters by a grid search method would be too demanding computationally. In our numerics, the mutation rate r is set to 20% and the crossover rate r to 50%. In the genetic algorithms literature, the crossover rate is often close to one, but for problems with few genes (i.e., components of x, or parameters, only four in our case), it is recommended to select a smaller value. With parallelization in mind (see Section 3.2), we prefer to decrease the number of iterations even if it implies exploring more solutions. We set the genetic algorithm population size to P = 100 individuals and we limit the number of iterations to 5. Hence, we value the ﬁtness function on 600 tentative new swaps x. 4.3. Results in the Case of Payer Portfolio Without Penalization First, we consider a portfolio only composed of payer swaps. The expected exposure (EE) and the expected positive and negative exposures (EPE and ENE), i.e., EMtM and EMtM , are shown as a function of time t in Figure 3, which illustrates the asymmetric market risk proﬁle of the portfolio. Risks 2019, 7, 100 12 of 21 Figure 3. Market risk proﬁle of the portfolio (payer portfolio without penalization). Our ﬁrst point is to verify that the algorithm without penalization, i.e., for a = 0 in Equation (7), will select a receiver swap with a maturity comparable to those of the swap of the initial portfolio. Table 1 reports after each iteration the three best solutions (from top to bottom) ever found since the beginning of the algorithm (in terms of the ﬁtness criterion in Equation (7) with a = 0, i.e., DCVA). A negative incremental CVA means that the new swap decreases the counterparty risk of the bank. The initial portfolio CVA amounts to 34,929e. We also report the jDV01js of the augmented portfolios in order to be able to assess the impact of the penalization in our next experiment. Table 1. Evolution of optimal solutions after each iteration (payer portfolio without penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 10 4,800,000 1.6471 GBP Receive 8019 23.0 4484 0 10 4,700,000 1.6471 GBP Receive 7948 22.8 4390 10 4,600,000 1.6471 GBP Receive 7872 22.5 4297 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 1 12 5,400,000 1.7036 GBP Receive 9163 26.2 5957 16 3,900,000 0.6377 JPY Receive 8760 25.1 6137 14 6,600,000 1.3416 EUR Receive 21,680 62.1 8626 2 17 5,100,000 1.4623 EUR Receive 19,729 56.5 7875 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 14 6,600,000 1.3416 EUR Receive 21,680 62.1 8626 3 17 5,100,000 1.4623 EUR Receive 19,729 56.5 7875 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 17 3,300,000 1.4623 EUR Receive 27,300 78.2 5096 4 12 6,100,000 1.2203 EUR Receive 25,382 72.7 6959 11 5,600,000 1.147 EUR Receive 23,009 65.9 5908 17 3,300,000 1.4623 EUR Receive 27,300 78.2 5096 5 12 6,100,000 1.2203 EUR Receive 25,382 72.7 6959 12 5,100,000 1.2203 EUR Receive 25,264 72.3 5818 As shown in Figure 4, a stabilization of the algorithm is observed after four iterations, on a new swap leading to a CVA gain of about 27,300e, i.e., about 78% of the initial portfolio CVA. The maturity and the notional are found the two most sensitive genes in the optimization. The maturity of the swap Risks 2019, 7, 100 13 of 21 is chosen by the algorithm so as to reduce the exposure peak: The decrease of the exposure in the ﬁrst eight years of the portfolio is visible in terms of EPE proﬁle in Figure 5 and of CVA proﬁle in Figure 6. Figure 4. Fitness value as a function of iteration number (payer portfolio without penalization). Figure 5. Market risk proﬁle of the portfolio before and after optimization (payer portfolio without penalization). Term structure obtained by integrating the EPE proﬁle against the CDS curve of the counterparty from time 0 to an increasing upper bound t T (cf. Equation (3)). Risks 2019, 7, 100 14 of 21 Figure 6. CVA proﬁle before and after optimization (payer portfolio without penalization). 4.4. Results in the Case of Payer Portfolio With Penalization We keep the same initial portfolio but we now penalize our objective function by the jDV01j of the new swap, setting the regularization parameter a to one in Equation (7). As shown below, this choice achieves a good balance between the two terms DCVA and aDV01 in Equation (7). In the present context of a payer portfolio, jDV01j control and CVA gain are two antagonistic targets. This may explain why the algorithm seems to struggle in ﬁnding a stable solution: indeed, the last iteration still decreases the ﬁtness signiﬁcantly (see Figure 7). Figure 7. Fitness as a function of iteration number (payer portfolio with penalization). During the execution (see Table 2), the algorithm ﬁrst optimizes the CVA and then (in iteration 5) reduces the jDV01j. This is due to the difference of order of magnitude between DCVA and jDV01j (recalling a = 1): DCVA is more important, hence the algorithm only takes care of the penalization once DCVA has been compressed. Risks 2019, 7, 100 15 of 21 Table 2. Evolution of optimal solutions after each iteration (payer portfolio with penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 10 4,500,000 1.6471 GBP Receive 7790 22.3 4218 0 10 4,600,000 1.6471 GBP Receive 7871 22.5 4311 10 4,700,000 1.6471 GBP Receive 7947 22.8 4405 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 1 10 4,500,000 1.6471 GBP Receive 7790 22.3 4217 10 4,600,000 1.6471 GBP Receive 7871 22.5 4311 14 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 2 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 17 6,100,000 1.4531 EUR Receive 15,038 43.1 9466 14 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 3 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 9 4,500,000 0.9584 EUR Receive 10,454 29.9 3945 10 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 4 11 6,600,000 1.3999 EUR Receive 18,825 53.9 9207 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 11 2,900,000 1.3811 EUR Receive 25,059 71.7 4039 5 18 1,500,000 1.48 EUR Receive 18,258 52.3 2442 17 1,500,000 1.4531 EUR Receive 16,553 47.4 2327 In the end, the gains in CVA are of the same order of magnitude as in the case without penalization (92% of the CVA gain without penalization), but for about 20% of jDV01j less than before. The second and third best solutions also achieve a great CVA gain, while diminishing the jDV01j by a factor three with respect to the nonpenalized case. By comparison with the unpenalized case (cf. Tables 1 and 2), the trades identiﬁed by the algorithm have a lower maturity or a smaller notional, hence a smaller jDV01j. Figures 8 and 9 show the corresponding market risk and CVA proﬁles before and after the optimization. Figure 8. Market risk profile of portfolio before and after optimization (payer portfolio with penalization). Risks 2019, 7, 100 16 of 21 Figure 9. CVA proﬁle before and after optimization (payer portfolio with penalization). 4.5. Results in the Case of a Hybrid Portfolio With Penalization Next, we challenge our algorithm with a more balanced initial portfolio, as shown in Figure 10 (to be compared with Figure 3). The initial CVA is now 6410e. We set the regularization parameter a in Equation (7) to 0.3, as opposed to 1 in the previous case, in view of the lower CVA of the initial portfolio. Figure 10. Market risk proﬁle of the portfolio (hybrid portfolio with penalization). As visible in Table 3 and Figure 11, the stabilization of the algorithm occurs after three iterations, showing that, for the hybrid portfolio, jDV01j penalization and DCVA play less antagonistic roles. This is obtained by a relatively small notional and a maturity limited to 9 years, versus 11 years in the previous case of a payer portfolio with penalization. The corresponding market risk and CVA proﬁles, before and after the optimization, are displayed in Figures 12 and 13. Figure 12 explains the choices operated by the algorithm: as we restrict our incremental strategy to one swap, the algorithm limits the EPE until the ﬁrst positive peak before 2026. A better strategy, but one outside our search space A, would be to add a second swap with entry date in 2028 and end date in 2037. Risks 2019, 7, 100 17 of 21 Table 3. Evolution of optimal solutions after each iteration (hybrid portfolio with penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 1 6,000,000 0.025 JPY Receive 14 -0.2 609 0 1 6,100,000 0.025 JPY Receive 14 0.2 619 1 6,300,000 0.025 JPY Receive 14 0.2 640 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 1 6 2,300,000 0.586 EUR Receive 1166 18.2 1370 9 700,000 1.608 GBP Receive 820 12.8 595 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 2 6 2,300,000 0.586 EUR Receive 1166 18.2 1370 9 700,000 1.608 GBP Receive 82 12.8 595 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 3 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 7 2,700,000 0.7225 EUR Receive 1628 25.4 1865 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 4 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 7 2,700,000 0.7225 EUR Receive 1628 25.4 1865 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 5 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 9 2,500,000 0.9584 EUR Receive 1942 30.3 2192 Figure 11. Fitness value as a function of iteration number (hybrid portfolio with penalization). Risks 2019, 7, 100 18 of 21 Figure 12. Market risk proﬁle of portfolio before and after optimization (hybrid portfolio with penalization). Figure 13. CVA proﬁle before and after optimization (hybrid portfolio with penalization). 5. Conclusions There exists a trade-off between CVA compression and DV01 penalization, which have antagonistic inﬂuences on the incremental exposure. Provided the search space for incremental trades is adequately chosen and parameterized, genetic optimization can result in signiﬁcant CVA gains and, under DV01 penalization, this can be achieved without too much impact on the market risk of the bank position. On the portfolios considered in our case studies, with 10–20 trades, a basic XVA compression run on a standard PC without the acceleration techniques of Section 3 takes about 20 h. The time gain resulting from an MtM store-and-reuse implementation of the trade incremental XVA computations as per Section 3.1 primarily depends on the size of the initial portfolio, but also on the maturity, and complexity more generally (vanilla vs. callable or path-dependent, etc.), of the constituting trades. Likewise, the time gain resulting from a parallel implementation of the genetic algorithm as per Section 3.2 primarily depends on the population size P, but it can be deteriorated by grid latency, hardware limitation, or data ﬂow management features. In our simulations, an MtM store-and-reuse implementation of the trade incremental XVA computations reduces the XVA compression time to about 7 h. A further parallel implementation of the genetic optimization algorithm lowers the execution time to about 1 h. The case study of this paper is only a ﬁrst step toward more complex optimizations. One could thus enlarge the search space with, e.g., crosscurrency swaps. In this case, the market risk penalization Risks 2019, 7, 100 19 of 21 should be revisited to penalize other risk factors, beyond interest rate risk that is already accounted for by jDV01j. The penalization could also be reﬁned with a focus on forward mark-to-market, i.e., market risk in the future (our current jDV01j penalization only controls spot market risk). CVA compression strategies involving several additional trades could be implemented. A ﬁrst step toward such a multi-variate, multi-trade, compression would be an iterated application of single-trade XVA compressions, whereby, after each compression, the optimally augmented portfolio becomes the initial portfolio for the next compression. The beneﬁt of such an iterative approach would be the ability to work with a search space A (or a sequence of them) of constant size, as opposed to a global search space A that would need to grow exponentially with the number of new trades in the case of a single multi-trade compression cycle. Additional XVA metrics, and ultimately the all-inclusive XVA add-on in Equation (1), should be included in the compression (which, in particular, would allow one to identify possible XVA cuts across different netting sets). Author Contributions: Conceptualization, M.C. and S.C.; methodology, M.C.; software, M.C.; validation, M.C.; formal analysis, M.C. investigation, M.C. and S.C.; writing—original draft preparation, S.C.; writing—review and editing, S.C.; supervision, S.C.; project administration, S.C.; and funding acquisition, S.C. Funding: This research was conducted with the support of the Research Initiative “Modélisation des marchés actions, obligations et dérivés” ﬁnanced by HSBC France under the aegis of the Europlace Institute of Finance. The views and opinions expressed in this presentation are those of the author alone and do not necessarily reﬂect the views or policies of HSBC Investment Bank, its subsidiaries or afﬁliates. The PhD thesis of Marc Chataigner is co-funded by a public grant as part of investissement d’avenir project, reference ANR-11-LABX-0056-LLH LabEx LMH. Acknowledgments: The authors would like to thank Hugo Lebrun, Rémi Ligoureau, and Guillaume Macey (HSBC France) for their assistance in the conduct of this study. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Abbreviations The following abbreviations are used in this manuscript: CDS Credit default swap CVA Credit valuation adjustment DV01 Dollar value of an 01 EE Expected exposure EPE Expected positive exposure ENE Expected negative exposure FVA Funding valuation adjustment KVA Capital valuation adjustment MtM Mark-to-market MVA Margin valuation adjustment OIS Overnight indexed swap OTC Over-the-counter XVA Generic “X” valuation adjustment Appendix A. Single Point Crossover Let ( p , p ) be a pair of chromosomes chosen as parents and let (c , c ) denote the children. 1 2 1 2 We assume that each chromosome has four genes A, B, C, D, that p has gene versions f A , B , C , D g 1 1 1 1 1 and p has gene versionsf A , B , C , D g. For a single point crossover, we draw uniformly an integer i 2 2 2 2 2 such the ﬁrst i genes for c are inherited from p and the remaining genes are transferred from p to c , 1 1 2 1 and symmetrically so for c . 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Risks Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/credit-valuation-adjustment-compression-by-genetic-optimization-PG1ocLNKK1

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risks Article Credit Valuation Adjustment Compression by Genetic Optimization Marc Chataigner and Stéphane Crépey * LaMME, Univ Evry, CNRS, Université Paris-Saclay, CEDEX, 91037 Évry, France; marc.chataigner@univ-evry.fr * Correspondence: stephane.crepey@univ-evry.fr Received: 4 August 2019; Accepted: 23 September 2019; Published: 29 September 2019 Abstract: Since the 2008–2009 ﬁnancial crisis, banks have introduced a family of X-valuation adjustments (XVAs) to quantify the cost of counterparty risk and of its capital and funding implications. XVAs represent a switch of paradigm in derivative management, from hedging to balance sheet optimization. They reﬂect market inefﬁciencies that should be compressed as much as possible. In this work, we present a genetic algorithm applied to the compression of credit valuation adjustment (CVA), the expected cost of client defaults to a bank. The design of the algorithm is ﬁne-tuned to the hybrid structure, both discrete and continuous parameter, of the corresponding high-dimensional and nonconvex optimization problem. To make intensive trade incremental XVA computations practical in real-time as required for XVA compression purposes, we propose an approach that circumvents portfolio revaluation at the cost of disk memory, storing the portfolio exposure of the night so that the exposure of the portfolio augmented by a new deal can be obtained at the cost of computing the exposure of the new deal only. This is illustrated by a CVA compression case study on real swap portfolios. Keywords: counterparty risk; credit valuation adjustment (CVA); XVA (X-valuation adjustments) compression; genetic algorithm MSC: 91B30; 91G20; 91G40; 91G60 JEL Classiﬁcation: D52; D53; G01; G13; G24 1. Introduction XVAs, where VA stands for valuation adjustment and X is a catch-all letter to be replaced by C for credit, F for funding, M for margin, or K for capital, have been implemented by banks in reaction to the regulatory changes aroused by 2008 ﬁnancial turmoils. They monetize counterparty risk and its funding and capital consequences by add-ons to derivative entry prices sourced from clients. According to the cost-of-capital XVA approach of Albanese and Crépey (2019), accounting for the impossibility for a bank to replicate the jump-to-default related cash ﬂows, the ﬁnal, all-inclusive XVA formula reads CVA + FVA + MVA + KVA, (1) to be sourced by the bank from clients on an incremental run-off basis at every new deal. As stated by the Basel Committee on Banking Supervision (2015), major counterparty credit losses on OTC (Over-the-counter) derivative portfolios in 2008 arose from XVA accounting losses rather than from actual client defaults. In particular, a bank incurs a CVA loss when the market perceives a deterioration of the credit risk of a client. This has motivated the creation of XVA desks for dealing with these risks. Risks 2019, 7, 100; doi:10.3390/risks7040100 www.mdpi.com/journal/risks Risks 2019, 7, 100 2 of 21 In this paper, we deal with CVA compression, i.e., the minimization of the CVA of a client portfolio by the introduction of an incremental trade, subject to the constraint of not altering too much the market risk of the portfolio. In the ﬁnancial derivative industry, the term compression term is generally applied in the context of “trade compression”, i.e., the reduction of the gross notional of positions in the market. Trade compression aims notably at reducing certain capital requirements, the number of transactions, and their amount (see Section 5.3 of (Gregory 2015)). As reﬂected by the proliferation of related industry presentations , this kind of balance sheet optimization is very active in top tier banks at the moment. XVAs reﬂect market inefﬁciencies that should be compressed as much as possible. Here, we focus on CVA compression for concreteness and simplicity, but the developed XVA compression methodology is generic. It could and should be applied to further XVA metrics, as soon as these are available with sufﬁcient speed, for computation at the portfolio level, and accuracy, for numerical signiﬁcance of the results at the incremental trade level (see Section 5 in (Albanese et al. 2018), which emphasizes the XVA compression perspective on the pricing and risk management of ﬁnancial derivatives in the post-2008–2009 global ﬁnancial crisis era, and cf. (Kondratyev and Giorgidze 2017), which uses a genetic algorithm for determining an optimal trade-off between MVA compression and transaction costs). The complexity of XVA compression problems stems, in particular, from the hybrid nature of the state space of the corresponding optimization problems. Indeed, a new trade (ﬁnancial derivative) is described by a combination of continuous and discrete parameters. This rules out the use of standard convex optimization schemes for such problems. Instead, we are lead to the use of metaheuristic algorithms. In this paper, we show how a genetic algorithm with penalization can efﬁciently ﬁnd a CVA offsetting trade, while limiting the impact of the trade on the market exposure proﬁle. The latter is necessary for staying in line with the separation of mandates between the XVA desks, in charge of managing counterparty risk, and the other, dubbed “clean”, trading desks of the bank, in charge of hedging the market risk of the bank positions. The other XVA compression challenge is execution time, with intensive valuation of the involved XVA metrics as a bottleneck. The XVA metrics are primarily deﬁned at the portfolio level: Time-0 XVAs can be formulated as expectations of nonlinear functionals of the bank derivative portfolio exposure, i.e., “clean” valuation (or “mark-to-market” (MtM) ignoring counterparty risk) of the bank portfolio, assessed at randomly sampled times and scenarios. Each new deal gives rise to XVA add-ons computed as the corresponding trade incremental XVA amounts, i.e., the differences between the XVAs of the portfolios including and excluding the new deal. To make intensive trade incremental XVA computations practical in real-time as required for XVA compression purposes, our proposed MtM store-and-reuse approach circumvents clean revaluation at the cost of disk memory, storing the portfolio exposure of the night so that the exposure of the portfolio augmented by a new deal can be obtained at the cost of computing the exposure of the new deal only. Outline and Contributions The paper is outlined as follows. Section 2 formulates the penalized CVA compression problem and introduces the related genetic optimization algorithm. Section 3 is about two key acceleration techniques in this regard. Section 4 presents a numerical case study on real swap portfolios. Section 5 concludes. The main contributions of the paper are the design of a parallelized genetic algorithm for the CVA compression task, the MtM store-and-reuse acceleration technique for trade incremental XVA computations, and the numerical CVA compression case study on real swap portfolios. See also, e.g., David Bachelier ’s panel discussion Capital and Margin Optimisation, Quantminds International 2018 conference, Lisbon, 16 May 2018. Risks 2019, 7, 100 3 of 21 More broadly, this paper enriches the literature on the use of genetic (also called evolutionary) optimization algorithms in ﬁnance. Cont and Hamida (2005) applied evolutionary algorithms to investigate a set of co-calibrated model parameterizations in order to assess the associated model risk. Kroha and Friedrich (2014) compared different genetic algorithms for automatic trading. Jin et al. (2019) applied evolutionary algorithms to optimal investment and consumption stochastic control problems. For wider reviews of genetic algorithms in ﬁnance, we refer the readers to (Chen 2012; Drake and Marks 2002). We refer the reader to the end of the paper for a list of the main abbreviations. 2. CVA Compression Modeling 2.1. Credit Valuation Adjustment We consider a complete stochastic basis (W,F,P), for a reference market ﬁltration (ignoring the default of the bank itself) F = (F ) , satisfying the usual conditions, and a risk-neutral pricing t t2R measure P, calibrated to market quotes of fully collateralized transactions. All the processes of interest are F adapted and all the random times of interest are F stopping times. This holds at least after so-called reduction of all the data to F, starting from a larger ﬁltration G including the default of the bank itself as a stopping time, assuming immersion from F into G for simplicity (see (Albanese and Crépey 2019) for details). The P expectation and (F ,P) conditional expectation are denoted by E and E , respectively. In developed markets, the overnight indexed swap (OIS) rate is together the reference remuneration rate for posted collateral and the best market proxy for a risk-free rate. We denote r ds by r = (r ) an F progressive OIS rate process and we write b = e for the corresponding t t2R risk-neutral discount factor. By clean valuation or mark-to-market of a contract (or portfolio), we mean the (trade additive) risk-neutral conditional expectation of its OIS discounted future promised cash ﬂows, ignoring counterparty risk and its capital and funding implications. We consider a bank engaged in bilateral trading with a single corporate counterparty (client) with default time and recovery rate t and R , respectively. This setup, which is chosen for simplicity, c c is consistent with a common situation where credit risk budget is assigned at each counterparty level within the bank. We denote by MtM the corresponding mark-to-market process of the client portfolio to the bank. The (time 0) CVA of the bank is its expected discounted loss in case of client default, i.e., 1 + CVA = E 1 b b (1 R )MtM . (2) t c ft Tg t t c c Assuming deterministic interest rates, this can be rewritten as CVA = (1 R ) b EPE(t)P(t 2 dt), (3) c t c where the expected positive exposure (EPE) is deﬁned as EPE(t) = E(MtM js = t ) . (4) s jt =t Equation (3) is popular with practitioners because it allows obtaining the CVA as the integral of the EPE against the client CDS (Credit default swap) curve. However, it is only really practical in simplistic models where the market and credit sides of the problem are independent, so that EPE(t) = E(MtM ). However, a key CVA modeling issue is wrong-way risk, i.e., the risk of adverse dependence between market and credit (see (Brigo and Vrins 2018; Crépey and Song 2016 2017; Glasserman and Yang 2018; Hull and White 2012; Iben Taarit 2018; Li and Mercurio 2015; Pykhtin 2012)). Risks 2019, 7, 100 4 of 21 Assuming the client default time endowed with an intensity g , a more ﬂexible formula is g du c + 0 (5) CVA = (1 R )E b e g MtM ds. c s s s Under a credit support agreement (CSA), MtM should be replaced by (MtMC) in all equations above, where C is the collateral posted by the counterparty. Obviously, collateral can mitigate the EPE and the CVA considerably. In the data of our case study there is no CSA, i.e., C = 0. Non-linearity of MtM with respect to the portfolio payoff components imposes CVA calculations at the counterparty portfolio (netting set) level. Similar approaches apply to FVA computations, with analogous comments, whereas the MVA can be computed based on quantile regression for the embedded dynamic initial margin calculations (see (Crépey et al. 2019)). In any case, the numerical bottleneck of XVA computations lies in intensive MtM calculations. 2.2. Fitness Criterion By the augmented and initial portfolios, respectively, we mean the portfolio of the bank inclusive and exclusive of a newly considered deal with the client. The aim of an XVA compression problem is to ﬁnd a new trade that minimizes the corresponding XVA metric of the augmented portfolio. This is equivalent to minimize the incremental CVA, which we denote by Z Z R R T T s s c c augm g du c + g du c init + u u 0 0 DCVA = (1 R )E b e g (MtM ) ds (1 R )E b e g (MtM ) ds c s s c s s s s 0 0 (6) augm g du c + init + = (1 R )E b e g (MtM ) (MtM ) ds, c s s s s where the indices init and augm refer to the initial portfolio and augmented portfolio, respectively. We emphasize that trade incremental CVA computations require two portfolio-wide calculations: one without the new trade and another one including it. Minimizing an XVA metric is most easily obtained through a signiﬁcant deformation of the portfolio exposure process (especially in the context of this work of a portfolio with a single counterparty). However, an XVA compression procedure should not affect too much the market risk of the portfolio, because market risk is the mandate of the clean desks of the bank, who, in particular, are subject to trading limits. This motivates the addition of a penalization to the incremental XVA criterion. In our case study, the incremental deal will consist of an interest rate swap. As such product is mostly sensitive to interest rate moves, a natural penalization is then in terms of its DV01 (dollar value of an 01), i.e., the variation of its mark-to-market (at time 0) under a parallel shift of the yield curve by one basis point (=10 ). More precisely, an interest rate swap exchanges one leg indexed on a ﬂoating interest rate against one leg paying a ﬁxed interest rate, called swap rate. The swap is said to be payer (respectively, receiver) for the party that pays (respectively, receives) the ﬂoating payments. A monocurrency swap exchanges both legs in the same currency. It is mainly sensitive to the ﬂuctuations of the corresponding ﬂoating interest rate term structure. DV01 measures the associated risk as the difference between the prices of the swap under the baseline (actual market data observed in the real market) and for a bumped yield curve deﬁned as the concatenation of the money market rates, forward rates, and swap rates, on the relevant (successive) time segments. Bumping the yield curve typically means adding 10 to each tenor of this curve and updating the other reference curves (zero coupon rates, forward rates, etc.) accordingly. Focusing on the CVA metric in this paper, we obtain the following ﬁtness minimization problem: minimize f (x) = DCVA(x) + ajDV01(x)j, (7) x2A Risks 2019, 7, 100 5 of 21 where x parameterizes a new deal (swap) to be found in a suitable search space A (see Section 4.1), DCVA(x) is its incremental CVA (cf. Equation (6)), DV01(x) is its DV01, and a is a penalization parameter controlling the trade-off between CVA reduction and market risk proﬁle preservation. By solving Equation (7), we aim at identifying a new deal which, if added to the current client portfolio of the bank, would diminish its counterparty risk without impacting too much its market risk. Note that, for scaling reasons (with, in particular, market penalization), we address the XVA compression problem in terms of trade incremental (as opposed to augmented portfolio) XVA numbers. A new deal is determined by a combination of quantitative (e.g., notional, maturity, etc.) and qualitative (e.g., currency, long or short direction, etc.) parameters, so that no gradient or Hessian is available for the ﬁtness function f in Equation (7). Moreover, one is interested in exploring a variety of local minima of f , to see different trading opportunities emerge from the optimization procedure. Furthermore, we can guess that some (crucial) parameters need be learned ﬁrst, such as currency or maturity; other parameters, such as notional, can be reﬁned in a second stage. All these features lead us to addressing Equation (7) by means of a genetic optimization algorithm. 2.3. Genetic Optimization Algorithm Genetic optimization algorithms belong to the class of derivative-free optimizers, which is surveyed and benchmarked numerically in (Rios and Sahinidis 2013) (including the CMA-ES and DAKOTA/EA genetic algorithms). The idea of genetic (or evolutionary) optimization algorithms is to evolve a population of individuals through cycles of modiﬁcation (mutation) and selection in order to improve the performance of its individuals, as measured by a given ﬁtness function. In addition, so-called crossover is used to enhance the search in parameter space. To the best of our knowledge, evolutionary algorithms were ﬁrst explicitly introduced in (Turing 1950, chp. 7 Learning Machines, p. 456). See the classical monographs by the authors of (Back 1996; Goldberg 1989; Holland 1975). They then experienced the general artiﬁcial intelligence disgrace and comeback before and after the 2000s. However, they always stayed an active ﬁeld of research, seen from different perspectives, such as particle ﬁltering, MCMC, or sequential monte carlo methods (see (Del Moral 2004)). Beyond its ﬁnancial applications reviewed at the end of Section 1, genetic optimization has been used in many different ﬁelds, such as mechanics (Verma and Lakshminiarayanan 2006), calibration of neural networks hyperparameters (Young et al. 2015), or operational research (Larranaga et al. 1999). Genetic optimization offers no theoretically guaranteed rate of convergence, but it is often found the most efﬁcient approach in practice for dealing with hybrid (partly continuous, partly discrete/combinatorial, hence without well deﬁned gradient and Hessian), nonconvex (in the sense of one local minimum, at least, for each set of values of the discrete parameters), and high-dimensional optimization problems such as Equation (7). At each iteration, the ﬁtness f (x) is computed for each individual (also named chromosome) x of an initial population (a set of chromosomes). The values returned by the objective function are used for selecting chromosomes from the population. Among numerous selection methods (see (Blickle and Thiele 1995)), we can quote ﬁtness proportionate selection, ranking proportionate selection (in order to avoid the overrepresentation of the chromosomes with the highest ﬁtness values), and tournament selection (selection of the best among randomly drawn chromosomes). The common intention of these selection methods is to sample in priority individuals with the best ﬁtness values. A genetic algorithm is dubbed elitist if the selection operator always keeps the chromosome with the best ﬁtness value. Otherwise (as in our case), there is no guarantee that the best visited chromosome is contained in the population corresponding to the ﬁnal iteration. The mutation stage is intended to maintain some diversity inside the population, in order to avoid the algorithm being trapped by local minima. A mutation randomly changes one gene, i.e., one component (e.g., in our case, the notional of a new swap) of a chromosome. Risks 2019, 7, 100 6 of 21 Selection and mutation play opposite roles: a focus on ﬁtness leads to a quicker convergence toward a local minimum; conversely, a too heavily mutated population results in a slow random research. In addition, a crossover operator plays the role of a reproduction inside the algorithm. The principle of crossover is to build two children chromosomes from parent chromosomes. A distribution (often the same as the one used for selection) is chosen for picking chromosomes from a population of the previous iteration and for recombining pairs of selected chromosomes. Children share gene values of their parents but a gene value from one parent cannot be inherited by both children. A crossover mask decides for each gene in which parent a child can copy the gene version. One of the most popular crossover masks is single point crossover (see Appendix A). The role of the crossover operator is paradoxical, as crossover can be seen as a combination of mutations, which increase the genetic diversity, while crossover also promotes chromosomes with higher ﬁtnesses. Crossover aims at beneﬁting of a presupposed proximity of best solutions. The above operators are applied iteratively until a suitable stopping condition is satisﬁed. The most basic one is a ﬁxed number of iteration, but customized criteria may also be used to limit further the number of iterations. For instance, the algorithm can be interrupted when the minimum (or sometimes even the maximum) ﬁtness value within the population at the beginning of an iteration is below a predeﬁned threshold. See Algorithm 1 and Figure 1 for the algorithm in pseudo-code and skeleton forms, denoting by r the mutation rate, i.e., the percentage of individuals in a population affected by a mutation, and by r the crossover rate, i.e., the percentage of individuals affected by crossover recombination. The behavior of a genetic optimization algorithm is essentially determined by the choice of the selection operator, the number of solutions affected by a mutation, and the number of chromosomes affected by a crossover. See the work of Tabassum and Kuruvilla (2014) for a user guide to the main genetic algorithm ingredients and Carvalho et al. (2011) for applications of genetic optimization algorithms to benchmark functions. Algorithm 1: Pseudo-code of an optimization genetic algorithm Data: An initial population P of size P and the associated ﬁtness values for each init chromosome, a crossover rate r , and a mutation rate r . c m Initialization; while a stopping condition is not satisﬁed do Save b(1 r ) Pc chromosomes, chosen by an appropriate selection method from P , c init in P ; Save br Pc chromosomes, chosen by an appropriate selection method from selected c r P P , in P ; Recombine, uniformly without replacement, b c pairs from init crossover P ; Merge P and P in P ; Mutate randomly br Pc in P ; crossover crossover selected mutated m mutated for Each chromosome c in P do mutated Compute the ﬁtness value of c; end end Result: A new population and the associated ﬁtness values. Risks 2019, 7, 100 7 of 21 Start Sampling Population of size P Fitness Evaluation Population with associated ﬁtness values. Stop Termination test Crossover Selection r P Selected population Population of b c of size b(1 r ) Pc recombined pairs c Mutation Population with br Pc mutated chromosomes Figure 1. Skeleton of an optimization genetic algorithm. 3. Acceleration Techniques Without suitable acceleration techniques, the above CVA compression approach is not workable in real time on realistic banking portfolios: in the examples of Section 4, a naive (desktop) implementation requires about 20 h of computations. This becomes even more problematic for hyperparameters tuning (such as a, crossover rate r , etc.). Hyperparameters are generally chosen with grid search, random search (see (Bergstra et al. 2011)), Bayesian optimization (see (Snoek et al. 2012)) or even evolutionary algorithms again (see (Young et al. 2015)). In any case, their calibration is greedy in terms of overall genetic algorithm execution. In this section, we deal with the two following acceleration techniques, which may be used simultaneously: a MtM store-and-reuse approach for trade incremental XVA computations, speeding up the unitary evaluation of the ﬁtness function; and a parallelization of the genetic algorithm accelerating the ﬁtness evaluation at the level of the population. Risks 2019, 7, 100 8 of 21 3.1. MtM Store-and-Reuse Approach for Trade Incremental XVA Computations Most of the time in portfolio-wide XVA calculations is spent in clean valuation (i.e., mark-to-market MtM) computations: by comparison, simulation of the risk factors or of the collateral is typically negligible. Our case study is based on the CVA metric. As observed after Equation (6), by lack of trade-additivity of the (portfolio-wide) CVA, trade incremental XVA computations require two portfolio-wide calculations: one without the new trade and another one including it. However, it is possible to store the (including MtM) paths simulated for the initial portfolio and reuse them each time we want to compute a new trade incremental XVA. Then, each trade incremental XVA computation only requires the forward simulation of the mark-to-market process of the new deal. The corresponding MtM store-and-reuse approach to trade incremental XVA computations circumvents repeated valuations at the cost of disk memory. It exploits the trade additivity of clean valuation by recording the MtM paths of the initial portfolio on a disk. For every new deal, the augmented portfolio exposure is obtained by adding, along the paths of the risk factors, the mark-to-market of the initial portfolio and of the new deal. This augmented portfolio exposure is then plugged into the XVA engine. An optimally implemented MtM store-and-reuse approach brings down trade incremental XVA computations to the time of generating the clean price process of the trade itself, instead of the one of the augmented portfolio as a whole. Another advantage of this approach is its compliance with desk segregation: As far as clean valuation is concerned, the XVA desks just use the pricers of the clean desks. Hence, the MtM process plugged into the XVA computations is consistent with the one used for producing the market risk hedging sensitivities. However, such an approach comes at the costs of memory disk (obviously), but also data slippage as, for consistency, it requires anchoring all the trade incremental XVA computations at the market data and parameters corresponding to the generation of the initial portfolio exposure. In practice, an MtM process at the overall portfolio level can only be generated during night runs, between two market sessions. Moreover, we have to distinguish between ﬁrst-order (or ﬁrst-generation) XVAs, which are options on the MtM process, and higher-order (or second-generation) XVAs (see (Crépey et al. 2019)), which can be viewed as compound options of order two or more on the MtM process. Second-generation XVAs may also involve conditional risk measures, e.g., conditional value-at-risk for the dynamic initial margin calculations that are required for MVA computations, as opposed to conditional expectations only in the case of ﬁrst generation XVAs. A Monte Carlo simulation diffuses risk factors X (such as interest rates, credit spreads, etc.) along drivers Z (such as Brownian motions, Poisson processes, etc.), according to a model formulated as a Markovian system of stochastic differential equations, starting from some given initial condition X for all risk factors, suitably discretized in time and space. Modulo calibration, X can be identiﬁed with the time 0 market data. We denote by Y a suitable estimate of a process Y at all (outer) nodes of a Monte Carlo XVA engine. In particular, MtM is the fully discrete counterpart of the MtM process of the initial portfolio, namely the clean value of the portfolio at future exposure dates in a time grid and for different scenario paths. At ﬁrst sight, an MtM store-and-reuse approach is unsuitable for second-order XVAs, such as the MVA and the KVA (but also the CVA in the case of a CSA where the bank receives so-called initial margin), Indeed, in their case, the principle of swapping computations against storage would require to store not one portfolio exposure MtM, but a whole family of resimulated, future conditional portfolio exposures, (at least, over a certain time horizon), which seems hardly feasible in practice. However, even in the case of second-order XVA metrics, an MtM store and reuse approach can be For details regarding the initial margin and the MVA, see (Crépey et al. 2019, sets. 5.2 and 6.4). Risks 2019, 7, 100 9 of 21 implemented with the help of appropriate regression techniques (at the cost of an additional regression error; see (Crépey et al. 2019)). Formalizing the above discussion, the conditions for a straightforward and satisfactory application of the MtM store-and-reuse approach to a given XVA metric are as follows, referring by indices init, incr, and augm to the initial portfolio, the new deal, and the augmented portfolio: 1. (No nested resimulation of the portfolio exposure required) The formula for the corresponding (portfolio-wide, time-0) XVA metric should be estimatable without nested resimulation, only based on the portfolio exposure rooted at (0, X ). A priori, additional simulation level makes impractical the MtM store-and-reuse idea of swapping execution time against storage. incr init [ [ 2. (Common random numbers) MtM should be based on the same paths of the drivers as MtM . Otherwise, numerical noise (or variance) would arise during MtM aggregation. incr 3. (Lagged market data) MtM should be based on the same time, say 0, and initial condition init X (including, modulo calibration, market data), as MtM . This condition ensures a consistent init incr augm [ [ [ aggregation of MtM and MtM into MtM . These conditions have the following implications: 1. The ﬁrst seems to ban second-order generation XVAs, such as CVA in presence of initial margin, but these can in fact be included with the help of appropriate regression techniques. 2. The second implies storing the driver paths that were simulated for the purpose of obtaining init incr MtM ; it also puts a bound on the accuracy of the estimation of MtM , since the number of Monte Carlo paths is imposed by the initial run. Furthermore, the XVA desks may want to account for some wrong way risk dependency between the portfolio exposure and counterparty credit risk (see Section 2.1); approaches based on correlating the default intensity and the market exposure in Equation (5) are readily doable in the present framework, provided the trajectories of the drivers and/or risk factors are shared between the clean and XVA desks. 3. The third induces a lag between the market data (of the preceding night) that are used in the incr incr computation of MtM and the exact MtM process; when the lag on market data becomes unacceptably high (because of time ﬂow and/or volatility on the market), a full reevaluation of the portfolio exposure is required. Figure 2 depicts the embedding of an MtM store-and-reuse approach into the trade incremental XVA engine of a bank. Market Data Database Data from the night init Initial run: MtM , Clean desks binit Clean Pricers Incremental run: incr bincr MtM , Z init Incremental run : Z XVA desks augm augm [ b MtM , Z augm Aggregate XVA Engine DataBase [ XVA Incremental run: init init [ b MtM , Z Figure 2. MtM store-and-reuse implementation of a trade incremental XVA engine with drivers Z. Risks 2019, 7, 100 10 of 21 3.2. Parallelization of the Genetic Algorithm Most of the XVA compression computational time is spent in the evaluation of the incremental XVA metric involved in the ﬁtness criterion visible in Equation (7). The MtM store-and-reuse approach allows reducing the complexity of such trade incremental XVA computations to trade (as opposed to portfolio) size. However, to achieve XVA compression in real time, this is not enough; another key step is the parallelization of the genetic algorithm that is used for solving Equation (7). The genetic algorithm is a population based method, which implies to maintain a population of individuals (tentative new deals) through each iteration of the algorithm. The calculation of the objective function, for a given individual, does not depend on the ﬁtness value of the other individuals. Therefore we can vectorize the computation of the ﬁtness values within the population. Provided a suitable parallel architecture is available, a perfectly distributed genetic algorithm makes the execution time independent of the population size P (see Algorithm 1 and Figure 1). This makes an important difference with other metaheuristic optimization algorithm, such as simulated annealing or stochastic hill climbing, which only evaluate one or very few solutions per iteration, but need much more iterations to converge toward a good minimum (see (Adler 1993; Janaki Ram et al. 1996)). As discussed in (Pardalos et al. 1995), the above parallelization of the ﬁtness function evaluation, for a given population, should not be confused with a parallel genetic algorithm in the sense of an independent evolution of several smaller populations. In our context, where individuals only represent incremental trades, a parallelization of population ﬁtness evaluation is compatible with an MtM store-and-reuse approach for the trade incremental XVA computations. Combining the two techniques results in an XVA compression time independent of the sizes of the initial portfolio of the bank and of the population of the genetic algorithm used for the optimization, which represents an XVA compression computation time gain factor of the order of Number of trades in the initial portfolio population size. 4. Case Study In the remainder of the paper, we present CVA compression results on real swap portfolios , using an additional swap for the CVA compression. We aim at addressing questions such as: Which type of swap is suitable for achieving the compression of the CVA, in the context of a given initial portfolio? How does the compression distort the portfolio exposure, with or without penalization? To ease the implementation of the MtM store-and-reuse approach, we assume no CSA (cf. Section 3.1). 4.1. New Deal Parameterization A swap is parameterized by its notional, its maturity, its direction, and its currency. The quantitative parameters are encoded through grids of values: 5 7 5 Notional: From 10 to 10 by step of 10 dollars. Maturity: From 1 to 20 years by step of 1 year, 30 years and 50 years. The qualitative parameters are encoded as enumerations of values: Currency: Euro, US dollar, GBP or Yen. Direction: A binary variable for payer or receiver. The underlying interest rate and FX models are proprietary and cannot be disclosed in the paper. We use a deterministic credit spread model for the counterparty, calibrated to the CDS term structure of the latter. Risks 2019, 7, 100 11 of 21 Moreover, we impose the additional swap to be at par so that it can be entered at no cost, which is equally desirable from the bank and the client perspectives. The above parameterization deﬁnes a discrete search space A with 100 22 4 2 = 1.76 10 elements. 4.2. Design of the Genetic Algorithm We address the optimization problem in Equation (7) by a genetic algorithm as per Section 2.3. The new deal space A in Equation (7) is viewed as a space of chromosomes x, the genes (deal parameters) of which evolve randomly along the iterations of the algorithm as detailed in Section 2.3. In the theoretical literature on genetic algorithms, an individual is represented as a bit string. In practice, however, bit string representation of parameters does not give enough control on the mutation distribution. Namely, in bit string representation, mutations affect all bits uniformly, whereas we might want to mutate some parameters more frequently (the quantitative parameters, in particular, as the algorithm tends to quickly identify the relevant values of the qualitative parameters). Hence, we rather model our individuals x by a variable string, a choice also made in (Kondratyev and Giorgidze 2017). We choose rank proportionate selection to avoid ﬁtness scaling issues. More precisely, if we have a population P = f1, . . . , Pg of P individuals and the associated ﬁtnesses ( f ) , then the probability i i2P to select chromosome i is 2rank( f ) p = , P(P + 1) where rank is a function that ranks chromosomes according to their ﬁtness value (returning one for the highest value, in the context of a minimization problem). Regarding the crossover operator, we use a uniform crossover mask, i.e., the choice of gene inherited from one parent or another is drawn with a uniform probability. Regarding mutations, the probability to mutate a gene is proportional to the number of alleles (values) that it can take. The mutation operator then selects uniformly a new gene allele (value). In our experiments, the notional of the new swap can take 100 different values, its currency 4 values, its position 2 values, and its maturity 22 values. Hence, when a chromosome is selected for 100 22 mutation, the probability to mutate each of its genes is equal to for the notional, for the maturity, 128 128 4 2 for the currency, and for the position. Indeed, a more frequent mutation of notional and 128 128 maturity parameters are desirable. Diversity for currency and position is ensured at the initialization of the algorithm (with a large population) and maintained across the iterations thanks to the crossover operator. A prerequisite for a successful implementation is a reasonable speciﬁcation of the search space A. Hyperparameters strongly impact the behavior of the algorithm. In the case of XVA compression, which is time-consuming, searching good values for the hyperparameters by a grid search method would be too demanding computationally. In our numerics, the mutation rate r is set to 20% and the crossover rate r to 50%. In the genetic algorithms literature, the crossover rate is often close to one, but for problems with few genes (i.e., components of x, or parameters, only four in our case), it is recommended to select a smaller value. With parallelization in mind (see Section 3.2), we prefer to decrease the number of iterations even if it implies exploring more solutions. We set the genetic algorithm population size to P = 100 individuals and we limit the number of iterations to 5. Hence, we value the ﬁtness function on 600 tentative new swaps x. 4.3. Results in the Case of Payer Portfolio Without Penalization First, we consider a portfolio only composed of payer swaps. The expected exposure (EE) and the expected positive and negative exposures (EPE and ENE), i.e., EMtM and EMtM , are shown as a function of time t in Figure 3, which illustrates the asymmetric market risk proﬁle of the portfolio. Risks 2019, 7, 100 12 of 21 Figure 3. Market risk proﬁle of the portfolio (payer portfolio without penalization). Our ﬁrst point is to verify that the algorithm without penalization, i.e., for a = 0 in Equation (7), will select a receiver swap with a maturity comparable to those of the swap of the initial portfolio. Table 1 reports after each iteration the three best solutions (from top to bottom) ever found since the beginning of the algorithm (in terms of the ﬁtness criterion in Equation (7) with a = 0, i.e., DCVA). A negative incremental CVA means that the new swap decreases the counterparty risk of the bank. The initial portfolio CVA amounts to 34,929e. We also report the jDV01js of the augmented portfolios in order to be able to assess the impact of the penalization in our next experiment. Table 1. Evolution of optimal solutions after each iteration (payer portfolio without penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 10 4,800,000 1.6471 GBP Receive 8019 23.0 4484 0 10 4,700,000 1.6471 GBP Receive 7948 22.8 4390 10 4,600,000 1.6471 GBP Receive 7872 22.5 4297 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 1 12 5,400,000 1.7036 GBP Receive 9163 26.2 5957 16 3,900,000 0.6377 JPY Receive 8760 25.1 6137 14 6,600,000 1.3416 EUR Receive 21,680 62.1 8626 2 17 5,100,000 1.4623 EUR Receive 19,729 56.5 7875 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 14 6,600,000 1.3416 EUR Receive 21,680 62.1 8626 3 17 5,100,000 1.4623 EUR Receive 19,729 56.5 7875 17 5,600,000 1.4623 EUR Receive 17,249 49.4 8648 17 3,300,000 1.4623 EUR Receive 27,300 78.2 5096 4 12 6,100,000 1.2203 EUR Receive 25,382 72.7 6959 11 5,600,000 1.147 EUR Receive 23,009 65.9 5908 17 3,300,000 1.4623 EUR Receive 27,300 78.2 5096 5 12 6,100,000 1.2203 EUR Receive 25,382 72.7 6959 12 5,100,000 1.2203 EUR Receive 25,264 72.3 5818 As shown in Figure 4, a stabilization of the algorithm is observed after four iterations, on a new swap leading to a CVA gain of about 27,300e, i.e., about 78% of the initial portfolio CVA. The maturity and the notional are found the two most sensitive genes in the optimization. The maturity of the swap Risks 2019, 7, 100 13 of 21 is chosen by the algorithm so as to reduce the exposure peak: The decrease of the exposure in the ﬁrst eight years of the portfolio is visible in terms of EPE proﬁle in Figure 5 and of CVA proﬁle in Figure 6. Figure 4. Fitness value as a function of iteration number (payer portfolio without penalization). Figure 5. Market risk proﬁle of the portfolio before and after optimization (payer portfolio without penalization). Term structure obtained by integrating the EPE proﬁle against the CDS curve of the counterparty from time 0 to an increasing upper bound t T (cf. Equation (3)). Risks 2019, 7, 100 14 of 21 Figure 6. CVA proﬁle before and after optimization (payer portfolio without penalization). 4.4. Results in the Case of Payer Portfolio With Penalization We keep the same initial portfolio but we now penalize our objective function by the jDV01j of the new swap, setting the regularization parameter a to one in Equation (7). As shown below, this choice achieves a good balance between the two terms DCVA and aDV01 in Equation (7). In the present context of a payer portfolio, jDV01j control and CVA gain are two antagonistic targets. This may explain why the algorithm seems to struggle in ﬁnding a stable solution: indeed, the last iteration still decreases the ﬁtness signiﬁcantly (see Figure 7). Figure 7. Fitness as a function of iteration number (payer portfolio with penalization). During the execution (see Table 2), the algorithm ﬁrst optimizes the CVA and then (in iteration 5) reduces the jDV01j. This is due to the difference of order of magnitude between DCVA and jDV01j (recalling a = 1): DCVA is more important, hence the algorithm only takes care of the penalization once DCVA has been compressed. Risks 2019, 7, 100 15 of 21 Table 2. Evolution of optimal solutions after each iteration (payer portfolio with penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 10 4,500,000 1.6471 GBP Receive 7790 22.3 4218 0 10 4,600,000 1.6471 GBP Receive 7871 22.5 4311 10 4,700,000 1.6471 GBP Receive 7947 22.8 4405 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 1 10 4,500,000 1.6471 GBP Receive 7790 22.3 4217 10 4,600,000 1.6471 GBP Receive 7871 22.5 4311 14 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 2 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 17 6,100,000 1.4531 EUR Receive 15,038 43.1 9466 14 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 3 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 9 4,500,000 0.9584 EUR Receive 10,454 29.9 3945 10 6,600,000 1.3336 EUR Receive 21,888 62.7 8654 4 11 6,600,000 1.3999 EUR Receive 18,825 53.9 9207 17 5,600,000 1.4731 EUR Receive 16,892 48.4 8706 11 2,900,000 1.3811 EUR Receive 25,059 71.7 4039 5 18 1,500,000 1.48 EUR Receive 18,258 52.3 2442 17 1,500,000 1.4531 EUR Receive 16,553 47.4 2327 In the end, the gains in CVA are of the same order of magnitude as in the case without penalization (92% of the CVA gain without penalization), but for about 20% of jDV01j less than before. The second and third best solutions also achieve a great CVA gain, while diminishing the jDV01j by a factor three with respect to the nonpenalized case. By comparison with the unpenalized case (cf. Tables 1 and 2), the trades identiﬁed by the algorithm have a lower maturity or a smaller notional, hence a smaller jDV01j. Figures 8 and 9 show the corresponding market risk and CVA proﬁles before and after the optimization. Figure 8. Market risk profile of portfolio before and after optimization (payer portfolio with penalization). Risks 2019, 7, 100 16 of 21 Figure 9. CVA proﬁle before and after optimization (payer portfolio with penalization). 4.5. Results in the Case of a Hybrid Portfolio With Penalization Next, we challenge our algorithm with a more balanced initial portfolio, as shown in Figure 10 (to be compared with Figure 3). The initial CVA is now 6410e. We set the regularization parameter a in Equation (7) to 0.3, as opposed to 1 in the previous case, in view of the lower CVA of the initial portfolio. Figure 10. Market risk proﬁle of the portfolio (hybrid portfolio with penalization). As visible in Table 3 and Figure 11, the stabilization of the algorithm occurs after three iterations, showing that, for the hybrid portfolio, jDV01j penalization and DCVA play less antagonistic roles. This is obtained by a relatively small notional and a maturity limited to 9 years, versus 11 years in the previous case of a payer portfolio with penalization. The corresponding market risk and CVA proﬁles, before and after the optimization, are displayed in Figures 12 and 13. Figure 12 explains the choices operated by the algorithm: as we restrict our incremental strategy to one swap, the algorithm limits the EPE until the ﬁrst positive peak before 2026. A better strategy, but one outside our search space A, would be to add a second swap with entry date in 2028 and end date in 2037. Risks 2019, 7, 100 17 of 21 Table 3. Evolution of optimal solutions after each iteration (hybrid portfolio with penalization). DCVA Iter. Mat. (yrs) Not. (Ke) Rate (%) Curr. Pos. DCVA (e) (in %) jDV01j(e) CVA 1 6,000,000 0.025 JPY Receive 14 -0.2 609 0 1 6,100,000 0.025 JPY Receive 14 0.2 619 1 6,300,000 0.025 JPY Receive 14 0.2 640 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 1 6 2,300,000 0.586 EUR Receive 1166 18.2 1370 9 700,000 1.608 GBP Receive 820 12.8 595 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 2 6 2,300,000 0.586 EUR Receive 1166 18.2 1370 9 700,000 1.608 GBP Receive 82 12.8 595 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 3 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 7 2,700,000 0.7225 EUR Receive 1628 25.4 1865 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 4 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 7 2,700,000 0.7225 EUR Receive 1628 25.4 1865 9 1,900,000 0.9584 EUR Receive 2284 35.6 1665 5 8 1,500,000 0.8565 EUR Receive 1905 29.7 1177 9 2,500,000 0.9584 EUR Receive 1942 30.3 2192 Figure 11. Fitness value as a function of iteration number (hybrid portfolio with penalization). Risks 2019, 7, 100 18 of 21 Figure 12. Market risk proﬁle of portfolio before and after optimization (hybrid portfolio with penalization). Figure 13. CVA proﬁle before and after optimization (hybrid portfolio with penalization). 5. Conclusions There exists a trade-off between CVA compression and DV01 penalization, which have antagonistic inﬂuences on the incremental exposure. Provided the search space for incremental trades is adequately chosen and parameterized, genetic optimization can result in signiﬁcant CVA gains and, under DV01 penalization, this can be achieved without too much impact on the market risk of the bank position. On the portfolios considered in our case studies, with 10–20 trades, a basic XVA compression run on a standard PC without the acceleration techniques of Section 3 takes about 20 h. The time gain resulting from an MtM store-and-reuse implementation of the trade incremental XVA computations as per Section 3.1 primarily depends on the size of the initial portfolio, but also on the maturity, and complexity more generally (vanilla vs. callable or path-dependent, etc.), of the constituting trades. Likewise, the time gain resulting from a parallel implementation of the genetic algorithm as per Section 3.2 primarily depends on the population size P, but it can be deteriorated by grid latency, hardware limitation, or data ﬂow management features. In our simulations, an MtM store-and-reuse implementation of the trade incremental XVA computations reduces the XVA compression time to about 7 h. A further parallel implementation of the genetic optimization algorithm lowers the execution time to about 1 h. The case study of this paper is only a ﬁrst step toward more complex optimizations. One could thus enlarge the search space with, e.g., crosscurrency swaps. In this case, the market risk penalization Risks 2019, 7, 100 19 of 21 should be revisited to penalize other risk factors, beyond interest rate risk that is already accounted for by jDV01j. The penalization could also be reﬁned with a focus on forward mark-to-market, i.e., market risk in the future (our current jDV01j penalization only controls spot market risk). CVA compression strategies involving several additional trades could be implemented. A ﬁrst step toward such a multi-variate, multi-trade, compression would be an iterated application of single-trade XVA compressions, whereby, after each compression, the optimally augmented portfolio becomes the initial portfolio for the next compression. The beneﬁt of such an iterative approach would be the ability to work with a search space A (or a sequence of them) of constant size, as opposed to a global search space A that would need to grow exponentially with the number of new trades in the case of a single multi-trade compression cycle. Additional XVA metrics, and ultimately the all-inclusive XVA add-on in Equation (1), should be included in the compression (which, in particular, would allow one to identify possible XVA cuts across different netting sets). Author Contributions: Conceptualization, M.C. and S.C.; methodology, M.C.; software, M.C.; validation, M.C.; formal analysis, M.C. investigation, M.C. and S.C.; writing—original draft preparation, S.C.; writing—review and editing, S.C.; supervision, S.C.; project administration, S.C.; and funding acquisition, S.C. Funding: This research was conducted with the support of the Research Initiative “Modélisation des marchés actions, obligations et dérivés” ﬁnanced by HSBC France under the aegis of the Europlace Institute of Finance. The views and opinions expressed in this presentation are those of the author alone and do not necessarily reﬂect the views or policies of HSBC Investment Bank, its subsidiaries or afﬁliates. The PhD thesis of Marc Chataigner is co-funded by a public grant as part of investissement d’avenir project, reference ANR-11-LABX-0056-LLH LabEx LMH. Acknowledgments: The authors would like to thank Hugo Lebrun, Rémi Ligoureau, and Guillaume Macey (HSBC France) for their assistance in the conduct of this study. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Abbreviations The following abbreviations are used in this manuscript: CDS Credit default swap CVA Credit valuation adjustment DV01 Dollar value of an 01 EE Expected exposure EPE Expected positive exposure ENE Expected negative exposure FVA Funding valuation adjustment KVA Capital valuation adjustment MtM Mark-to-market MVA Margin valuation adjustment OIS Overnight indexed swap OTC Over-the-counter XVA Generic “X” valuation adjustment Appendix A. Single Point Crossover Let ( p , p ) be a pair of chromosomes chosen as parents and let (c , c ) denote the children. 1 2 1 2 We assume that each chromosome has four genes A, B, C, D, that p has gene versions f A , B , C , D g 1 1 1 1 1 and p has gene versionsf A , B , C , D g. For a single point crossover, we draw uniformly an integer i 2 2 2 2 2 such the ﬁrst i genes for c are inherited from p and the remaining genes are transferred from p to c , 1 1 2 1 and symmetrically so for c . 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Credit Valuation Adjustment Compression by Genetic Optimization

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