# Credit Valuation Adjustment Compression by Genetic Optimization

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## Abstract

**:**

## 1. Introduction

#### Outline and Contributions

## 2. CVA Compression Modeling

#### 2.1. Credit Valuation Adjustment

#### 2.2. Fitness Criterion

#### 2.3. Genetic Optimization Algorithm

Algorithm 1: Pseudo-code of an optimization genetic algorithm |

## 3. Acceleration Techniques

- 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.

#### 3.1. MtM Store-and-Reuse Approach for Trade Incremental XVA Computations

- (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 $(0,{X}_{0})$. A priori, additional simulation level makes impractical the MtM store-and-reuse idea of swapping execution time against storage.
- (Common random numbers) ${\widehat{\mathrm{MtM}}}^{incr}$ should be based on the same paths of the drivers as ${\widehat{\mathrm{MtM}}}^{init}$. Otherwise, numerical noise (or variance) would arise during $\widehat{\mathrm{MtM}}$ aggregation.
- (Lagged market data) ${\widehat{\mathrm{MtM}}}^{incr}$ should be based on the same time, say 0, and initial condition ${X}_{0}$ (including, modulo calibration, market data), as ${\widehat{\mathrm{MtM}}}^{init}$. This condition ensures a consistent aggregation of ${\widehat{\mathrm{MtM}}}^{init}$ and ${\widehat{\mathrm{MtM}}}^{incr}$ into ${\widehat{\mathrm{MtM}}}^{augm}$.

- 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.
- The second implies storing the driver paths that were simulated for the purpose of obtaining ${\widehat{\mathrm{MtM}}}^{init}$; it also puts a bound on the accuracy of the estimation of ${\mathrm{MtM}}^{incr}$, 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.
- The third induces a lag between the market data (of the preceding night) that are used in the computation of ${\widehat{\mathrm{MtM}}}^{incr}$ and the exact ${\mathrm{MtM}}^{incr}$ 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.

#### 3.2. Parallelization of the Genetic Algorithm

## 4. Case Study

- 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?

#### 4.1. New Deal Parameterization

- Notional: From 10${}^{5}$ to ${10}^{7}$ by step of 10${}^{5}$ dollars.
- Maturity: From 1 to 20 years by step of 1 year, 30 years and 50 years.

- Currency: Euro, US dollar, GBP or Yen.
- Direction: A binary variable for payer or receiver.

#### 4.2. Design of the Genetic Algorithm

#### 4.3. Results in the Case of Payer Portfolio Without Penalization

#### 4.4. Results in the Case of Payer Portfolio With Penalization

#### 4.5. Results in the Case of a Hybrid Portfolio With Penalization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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

## References

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1 | See also, e.g., David Bachelier’s panel discussion Capital and Margin Optimisation, Quantminds International 2018 conference, Lisbon, 16 May 2018. |

2 | For details regarding the initial margin and the MVA, see (Crépey et al. 2019, sets. 5.2 and 6.4). |

3 | 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. |

4 | Term structure obtained by integrating the EPE profile against the CDS curve of the counterparty from time 0 to an increasing upper bound $t\le T$ (cf. Equation (3)). |

**Figure 5.**Market risk profile of the portfolio before and after optimization (payer portfolio without penalization).

**Figure 8.**Market risk profile of portfolio before and after optimization (payer portfolio with penalization).

**Figure 12.**Market risk profile of portfolio before and after optimization (hybrid portfolio with penalization).

**Table 1.**Evolution of optimal solutions after each iteration (payer portfolio without penalization).

Iter. | Mat. (yrs) | Not. (K€) | Rate (%) | Curr. | Pos. | $\mathbf{\Delta}$CVA (€) | $\frac{-\mathbf{\Delta}\mathbf{CVA}}{\mathbf{CVA}}$ (in %) | |DV01|(€) |
---|---|---|---|---|---|---|---|---|

0 | 10 | 4,800,000 | 1.6471 | GBP | Receive | −8019 | 23.0 | 4484 |

10 | 4,700,000 | 1.6471 | GBP | Receive | −7948 | 22.8 | 4390 | |

10 | 4,600,000 | 1.6471 | GBP | Receive | −7872 | 22.5 | 4297 | |

1 | 17 | 5,600,000 | 1.4623 | EUR | Receive | −17,249 | 49.4 | 8648 |

12 | 5,400,000 | 1.7036 | GBP | Receive | −9163 | 26.2 | 5957 | |

16 | 3,900,000 | 0.6377 | JPY | Receive | −8760 | 25.1 | 6137 | |

2 | 14 | 6,600,000 | 1.3416 | EUR | Receive | −21,680 | 62.1 | 8626 |

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 | |

3 | 14 | 6,600,000 | 1.3416 | EUR | Receive | −21,680 | 62.1 | 8626 |

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 | |

4 | 17 | 3,300,000 | 1.4623 | EUR | Receive | −27,300 | 78.2 | 5096 |

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 | |

5 | 17 | 3,300,000 | 1.4623 | EUR | Receive | −27,300 | 78.2 | 5096 |

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 |

Iter. | Mat. (yrs) | Not. (K€) | Rate (%) | Curr. | Pos. | $\mathbf{\Delta}$CVA (€) | $\frac{-\mathbf{\Delta}\mathbf{CVA}}{\mathbf{CVA}}$ (in %) | |DV01|(€) |
---|---|---|---|---|---|---|---|---|

0 | 10 | 4,500,000 | 1.6471 | GBP | Receive | −7790 | 22.3 | 4218 |

10 | 4,600,000 | 1.6471 | GBP | Receive | −7871 | 22.5 | 4311 | |

10 | 4,700,000 | 1.6471 | GBP | Receive | −7947 | 22.8 | 4405 | |

1 | 17 | 5,600,000 | 1.4731 | EUR | Receive | −16,892 | 48.4 | 8706 |

10 | 4,500,000 | 1.6471 | GBP | Receive | −7790 | 22.3 | 4217 | |

10 | 4,600,000 | 1.6471 | GBP | Receive | −7871 | 22.5 | 4311 | |

2 | 14 | 6,600,000 | 1.3336 | EUR | Receive | −21,888 | 62.7 | 8654 |

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 | |

3 | 14 | 6,600,000 | 1.3336 | EUR | Receive | −21,888 | 62.7 | 8654 |

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 | |

4 | 10 | 6,600,000 | 1.3336 | EUR | Receive | −21,888 | 62.7 | 8654 |

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 | |

5 | 11 | 2,900,000 | 1.3811 | EUR | Receive | −25,059 | 71.7 | 4039 |

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 |

Iter. | Mat. (yrs) | Not. (K€) | Rate (%) | Curr. | Pos. | $\mathbf{\Delta}$CVA (€) | $\frac{-\mathbf{\Delta}\mathbf{CVA}}{\mathbf{CVA}}$ (in %) | |DV01|(€) |
---|---|---|---|---|---|---|---|---|

0 | 1 | 6,000,000 | 0.025 | JPY | Receive | 14 | -0.2 | 609 |

1 | 6,100,000 | 0.025 | JPY | Receive | 14 | −0.2 | 619 | |

1 | 6,300,000 | 0.025 | JPY | Receive | 14 | −0.2 | 640 | |

1 | 8 | 1,500,000 | 0.8565 | EUR | Receive | −1905 | 29.7 | 1177 |

6 | 2,300,000 | 0.586 | EUR | Receive | −1166 | 18.2 | 1370 | |

9 | 700,000 | 1.608 | GBP | Receive | −820 | 12.8 | 595 | |

2 | 8 | 1,500,000 | 0.8565 | EUR | Receive | −1905 | 29.7 | 1177 |

6 | 2,300,000 | 0.586 | EUR | Receive | −1166 | 18.2 | 1370 | |

9 | 700,000 | 1.608 | GBP | Receive | −82 | 12.8 | 595 | |

3 | 9 | 1,900,000 | 0.9584 | EUR | Receive | −2284 | 35.6 | 1665 |

8 | 1,500,000 | 0.8565 | EUR | Receive | −1905 | 29.7 | 1177 | |

7 | 2,700,000 | 0.7225 | EUR | Receive | −1628 | 25.4 | 1865 | |

4 | 9 | 1,900,000 | 0.9584 | EUR | Receive | −2284 | 35.6 | 1665 |

8 | 1,500,000 | 0.8565 | EUR | Receive | −1905 | 29.7 | 1177 | |

7 | 2,700,000 | 0.7225 | EUR | Receive | −1628 | 25.4 | 1865 | |

5 | 9 | 1,900,000 | 0.9584 | EUR | Receive | −2284 | 35.6 | 1665 |

8 | 1,500,000 | 0.8565 | EUR | Receive | −1905 | 29.7 | 1177 | |

9 | 2,500,000 | 0.9584 | EUR | Receive | −1942 | 30.3 | 2192 |

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**MDPI and ACS Style**

Chataigner, M.; Crépey, S.
Credit Valuation Adjustment Compression by Genetic Optimization. *Risks* **2019**, *7*, 100.
https://doi.org/10.3390/risks7040100

**AMA Style**

Chataigner M, Crépey S.
Credit Valuation Adjustment Compression by Genetic Optimization. *Risks*. 2019; 7(4):100.
https://doi.org/10.3390/risks7040100

**Chicago/Turabian Style**

Chataigner, Marc, and Stéphane Crépey.
2019. "Credit Valuation Adjustment Compression by Genetic Optimization" *Risks* 7, no. 4: 100.
https://doi.org/10.3390/risks7040100