Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk
Abstract
:1. Introduction
1.1. Credit Valuation Adjustment
1.2. Counterparty Credit Risk
1.3. CVA Value-at-Risk (VaR) and a Variant
1.4. Motivation
- As pointed out in [21], credit spread levels and changes provide signs of the characteristic fat-tailed behavior and, in both cases, recommend that both series are away from a distribution of the normal. This justifies why we chose the fatter tail logistic distribution in this work in contrast to the normal distribution.
- The logistic model has also recently been applied in the work [22] in another context and showed promising results.
- It is pointed out that, in Table 1 of [23], some empirical evidence for the use of a logistic distribution for modeling the distribution of a risk variable is provided.
- Note that experiencing all of the non-Gaussian distributions in modeling stock data [24] is not the major aim here since it is not feasible. As a matter of fact, our strategy is to adopt a fat-tailed distribution, namely, logistic distribution, that is good enough to accommodate the features of financial data with respect to computing the CVA VaR and CVA CVaR in higher dimensions.
1.5. Problems to Be Solved and Novelty
- Here, we first focused on IRS CVA and improved the existing EEs formulas given for IRS CVA using the logistic distribution. The existing relations are based on the normal distribution. In fact, here, the novelty is that we explored the model distribution for the exposure based on a proxy for the swap duration and the logistic distribution.
- Next, we employed the logistic distribution to propose a new formulation for CVA VaR. In fact, the existing CVA VaR formulation is based on the normal distribution and, here, we assumed that the CDS spread follows a logistic distribution and obtained a new formulation that is more consistent with real financial data having fatter tails.
- The final important factor that has been addressed in this piece of work is to assume that not only the credit spread but also the EPE follow logistic distributions having different parameters. This novelty of the work extends the computation of CVA VaR and CVaR in higher dimensions.
1.6. Organization
2. New EE Formulas
2.1. Logistic Distribution
2.2. PFE
2.3. Derivation of New EE Relations
3. Novel Risk Measure Formulas for CVA
3.1. CVA VaR under the Logistic Distribution
3.2. Extension to Higher Dependency Based on the Logistic Distribution
4. Applications
4.1. EEs for the Payer and Receiver Swap
4.2. Results for 1D CVA VaR and Advantages over the Existing Solver
4.3. Results for 2D CVA VaR
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Name | Mean | Variance | Median | Skewness | Kurtosis | q-Quantile |
---|---|---|---|---|---|---|
Normal | 0 | 3 | , | |||
Logistic | 0 | , |
T | 2.0 | 2.1 | 2.2 | 2.3 | 2.4 | 2.5 | 2.6 | 2.7 | 2.8 | 2.9 | 3.0 |
CVA charge | 3.44 | 3.60 | 3.77 | 3.93 | 4.09 | 4.25 | 4.41 | 4.57 | 4.73 | 4.89 | 5.05 |
CVA N, | 6.05 | 6.35 | 6.66 | 6.96 | 7.27 | 7.57 | 7.88 | 8.187 | 8.49 | 8.79 | 9.10 |
CVA L, | 7.26 | 7.60 | 7.95 | 8.29 | 8.63 | 8.97 | 9.31 | 9.6517 | 9.98 | 10.3211 | 10.65 |
CVA N, | 7.32 | 7.69 | 8.06 | 8.43 | 8.80 | 9.17 | 9.54 | 9.90 | 10.27 | 10.64 | 11.01 |
CVA L, | 9.49 | 9.94 | 10.39 | 10.84 | 11.29 | 11.73 | 12.18 | 12.62 | 13.05 | 13.49 | 13.93 |
CVA N, | 8.37 | 8.80 | 9.22 | 9.64 | 10.06 | 10.48 | 10.90 | 11.33 | 11.75 | 12.17 | 12.60 |
CVA L, | 11.55 | 12.10 | 12.65 | 13.19 | 13.74 | 14.28 | 14.82 | 15.35 | 15.88 | 16.42 | 16.95 |
CVA N, | 10.35 | 10.87 | 11.39 | 11.91 | 12.43 | 12.95 | 13.47 | 13.99 | 14.52 | 15.04 | 15.56 |
CVA L, | 16.09 | 16.86 | 17.63 | 18.39 | 19.14 | 19.90 | 20.65 | 21.39 | 22.14 | 22.88 | 23.62 |
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Song, Y.; Shateyi, S.; He, J.; Cui, X. Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk. Mathematics 2022, 10, 3828. https://doi.org/10.3390/math10203828
Song Y, Shateyi S, He J, Cui X. Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk. Mathematics. 2022; 10(20):3828. https://doi.org/10.3390/math10203828
Chicago/Turabian StyleSong, Yanlai, Stanford Shateyi, Jianying He, and Xueqing Cui. 2022. "Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk" Mathematics 10, no. 20: 3828. https://doi.org/10.3390/math10203828