Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework
Abstract
:1. Introduction
2. Methodology
2.1. Modeling Interest Rates Using the Inverse-Call Transformation
2.1.1. Inverse-Call Transformation
2.1.2. Shadow Rates and Instantaneous Interest Rates
2.2. The ZCB Price under the Vasicek Model
2.3. The ZCB Price under the MeLo-1-V Model
2.3.1. The Monte-Carlo Price
2.3.2. The ZCB-Price Approximation Using the Gauss-Hermite Quadrature
2.4. Estimation of Unobservable State Variable and Model-Parameter Calibration
2.4.1. State-Space Model
2.4.2. Maximum-Likelihood Estimation (MLE)
2.4.3. Unscented Kalman Filter (UKF)
2.4.4. Particle Swarm Optimization (PSO)
Algorithm 2: Particle Swarm Optimization (PSO) [14] |
3. Results
3.1. The MeLo-1-V Model with Synthetically Generated Data
3.2. The MeLo-1-V Model with Actual Data
4. Discussion
4.1. Negative Interest Rates under the Vasicek
4.2. Comparison between the Vasicek Model and the MeLo-1-V Model
4.2.1. Scenarios Where the MeLo-1-V and Its Underlying Vasicek Are Roughly the Same
4.2.2. Scenarios Where the Vasicek Curve Seems to Be Unreliable
5. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Theorem 1
- with
- with and
- with and .
- Case with and
- Case with and
Appendix B. Proof of Theorem 2
References
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Maturity | Min | Max | Mean | Std | Skewness | Kurtosis | 1Q | 2Q | 3Q |
---|---|---|---|---|---|---|---|---|---|
3 months | −0.0244 | 4.5116 | 1.9620 | 1.4480 | 0.2013 | 1.5266 | 0.4811 | 2.0542 | 3.5388 |
6 months | 0.0148 | 4.6650 | 2.0797 | 1.4304 | 0.1895 | 1.5455 | 0.6716 | 2.0830 | 3.5441 |
9 months | 0.0304 | 4.8286 | 2.1967 | 1.3658 | 0.1776 | 1.6462 | 0.9727 | 2.1525 | 3.6033 |
1 year | 0.0708 | 4.8734 | 2.3127 | 1.2894 | 0.1357 | 1.7636 | 1.2945 | 2.2189 | 3.6097 |
3 years | 0.9586 | 4.7681 | 3.1776 | 0.7458 | −0.5211 | 2.7179 | 2.7851 | 3.2962 | 3.7141 |
5 years | 1.9980 | 4.8807 | 3.8579 | 0.4881 | −0.6204 | 3.0262 | 3.5364 | 3.9703 | 4.2129 |
10 years | 3.1180 | 5.7411 | 4.5384 | 0.4659 | 0.0373 | 2.2455 | 4.1477 | 4.5291 | 4.9369 |
15 years | 2.8365 | 5.5144 | 4.4929 | 0.5141 | −0.2366 | 2.2964 | 4.1005 | 4.4825 | 4.9547 |
Maturity | Min | Max | Mean | Std | Skewness | Kurtosis | 1Q | 2Q | 3Q |
---|---|---|---|---|---|---|---|---|---|
3 months | 0.0000 | 4.4627 | 1.9467 | 1.3896 | 0.1975 | 1.6029 | 0.6384 | 1.9025 | 3.3335 |
6 months | 0.0036 | 4.5071 | 2.1120 | 1.3346 | 0.1545 | 1.6378 | 0.8968 | 2.0844 | 3.4368 |
9 months | 0.0279 | 4.5477 | 2.2668 | 1.2809 | 0.1166 | 1.6740 | 1.1287 | 2.2525 | 3.5322 |
1 year | 0.0799 | 4.5848 | 2.4118 | 1.2284 | 0.0843 | 1.7093 | 1.3387 | 2.4076 | 3.6206 |
3 years | 1.1236 | 4.7882 | 3.2795 | 0.8759 | -0.0444 | 1.8868 | 2.5519 | 3.3023 | 4.1293 |
5 years | 2.1200 | 4.8881 | 3.7883 | 0.6423 | -0.0775 | 1.9469 | 3.2614 | 3.8088 | 4.4092 |
10 years | 3.3998 | 4.9491 | 4.3401 | 0.3556 | -0.0827 | 1.9647 | 4.0489 | 4.3513 | 4.6834 |
15 years | 3.8850 | 4.9103 | 4.5025 | 0.2371 | -0.0711 | 1.9498 | 4.3077 | 4.5090 | 4.7317 |
Maturity | Min | Max | Mean | Median | Std | MAE | RMSE |
---|---|---|---|---|---|---|---|
3 months | −0.8880 | 1.2776 | −0.0581 | −0.0541 | 0.3811 | 0.2940 | 0.3842 |
6 months | −0.9239 | 1.0959 | −0.0574 | −0.0806 | 0.3881 | 0.3156 | 0.3910 |
9 months | −0.8265 | 1.0415 | −0.0386 | −0.0398 | 0.3700 | 0.3004 | 0.3708 |
1 year | −1.1321 | 0.9634 | −0.0443 | −0.0411 | 0.3786 | 0.3065 | 0.3800 |
3 years | −0.9165 | 0.7445 | −0.0293 | −0.0219 | 0.3503 | 0.2882 | 0.3504 |
5 years | −0.8517 | 1.0144 | −0.0126 | −0.0187 | 0.3376 | 0.2690 | 0.3367 |
10 years | −0.7106 | 0.6723 | −0.0389 | −0.0142 | 0.2641 | 0.2086 | 0.2661 |
15 years | −0.5981 | 0.5265 | −0.0038 | −0.0141 | 0.2292 | 0.1846 | 0.2285 |
Maturity | Min | Max | Mean | Median | Std | MAE | RMSE |
---|---|---|---|---|---|---|---|
3 months | −0.5407 | 1.2452 | 0.0153 | 0.0385 | 0.1779 | 0.1390 | 0.1785 |
6 months | −0.4733 | 0.3394 | −0.0324 | −0.0039 | 0.1589 | 0.1276 | 0.1621 |
9 months | −1.3801 | 0.4284 | −0.0703 | −0.0873 | 0.1708 | 0.1478 | 0.1847 |
1 year | −1.6317 | 0.4660 | −0.0992 | −0.0950 | 0.2009 | 0.1631 | 0.2240 |
3 years | −0.8926 | 1.0397 | −0.1022 | −0.1908 | 0.4012 | 0.3551 | 0.4139 |
5 years | −0.8815 | 1.3998 | 0.0692 | −0.0352 | 0.5328 | 0.4552 | 0.5371 |
10 years | −0.8618 | 1.6063 | 0.1980 | 0.1416 | 0.5550 | 0.4633 | 0.5891 |
15 years | −1.0749 | 1.1497 | −0.0100 | −0.0766 | 0.4710 | 0.3902 | 0.4710 |
Maturity | Yield (Ex. 1) | Yield (Ex. 2) | Yield (Ex. 3) | Yield (Ex. 4) | Yield (Ex. 5) |
---|---|---|---|---|---|
1/12 | 5.7957 | 5.7911 | 0.2480 | −0.4961 | −4.8534 |
4/12 | 5.7718 | 5.7305 | 0.2165 | −0.4901 | −4.5738 |
6/12 | 5.6981 | 5.5345 | 0.1019 | −0.4854 | −4.1859 |
8/12 | 5.5801 | 5.2154 | −0.0915 | −0.4858 | −3.8324 |
1 | 5.4191 | 4.7768 | −0.3615 | −0.4910 | −3.5098 |
2 | 4.3687 | 1.8931 | −2.1652 | −0.5580 | −2.4706 |
3 | 2.7224 | −2.6475 | −5.0306 | −0.6928 | −1.7293 |
4 | 0.5467 | −8.6592 | −8.8389 | −0.8881 | −1.1891 |
5 | −2.0974 | −15.9733 | −13.4820 | −1.1368 | −0.7870 |
6 | −5.1548 | −24.4364 | −18.8616 | −1.4330 | −0.4815 |
7 | −8.5753 | −33.9090 | −24.8885 | −1.7709 | −0.2448 |
8 | −12.3132 | −44.2644 | −31.4815 | −2.1454 | −0.0580 |
9 | −16.3273 | −55.3877 | −38.5670 | −2.5518 | 0.0921 |
10 | −20.5799 | −67.1744 | −46.0782 | −2.9860 | 0.2146 |
12 | −29.6680 | −92.3696 | −62.1415 | −3.9228 | 0.4015 |
15 | −44.3380 | −133.0506 | −88.0919 | −5.4516 | 0.5906 |
20 | −70.1354 | −204.6084 | −133.7631 | −8.1688 | 0.7806 |
25 | −96.0515 | −276.5105 | −179.6724 | −10.9203 | 0.8947 |
30 | −120.9957 | −345.7247 | −223.8763 | −13.5814 | 0.9708 |
35 | −144.3949 | −410.6574 | −265.3527 | −16.0857 | 1.0252 |
50 | −203.6505 | −575.1068 | −370.4153 | −22.4504 | 1.1230 |
Case 1 | Case 2 | |||||
---|---|---|---|---|---|---|
Maturity | Yield (Vasicek) | Yield (MeLo-1-V) | Diff. | Yield (Vasicek) | Yield (MeLo-1-V) | Diff. |
1/12 | 2.5047 | 2.5047 | 0.0000 | 0.0246 | 0.5361 | −0.5115 |
4/12 | 2.5142 | 2.5142 | 0.0000 | 0.0729 | 0.6468 | −0.5739 |
6/12 | 2.5282 | 2.5282 | 0.0000 | 0.1429 | 0.7898 | −0.6469 |
8/12 | 2.5421 | 2.5421 | 0.0000 | 0.2103 | 0.9140 | −0.7037 |
1 | 2.5559 | 2.5558 | 0.0001 | 0.2750 | 1.0254 | −0.7504 |
2 | 2.6094 | 2.6091 | 0.0003 | 0.5101 | 1.3910 | −0.8809 |
3 | 2.6608 | 2.6601 | 0.0007 | 0.7113 | 1.6754 | −0.9641 |
4 | 2.7101 | 2.7089 | 0.0012 | 0.8839 | 1.9065 | −1.0226 |
5 | 2.7574 | 2.7556 | 0.0018 | 1.0322 | 2.0977 | −1.0655 |
6 | 2.8029 | 2.8003 | 0.0026 | 1.1598 | 2.2572 | −1.0974 |
7 | 2.8466 | 2.8430 | 0.0036 | 1.2698 | 2.3908 | −1.1210 |
8 | 2.8885 | 2.8840 | 0.0045 | 1.3648 | 2.5026 | −1.1378 |
9 | 2.9289 | 2.9231 | 0.0058 | 1.4471 | 2.5962 | −1.1491 |
10 | 2.9677 | 2.9606 | 0.0071 | 1.5185 | 2.6740 | −1.1555 |
12 | 3.0411 | 3.0308 | 0.0103 | 1.6345 | 2.7912 | −1.1567 |
15 | 3.1413 | 3.1254 | 0.0159 | 1.7592 | 2.8940 | −1.1348 |
20 | 3.2860 | 3.2581 | 0.0279 | 1.8854 | 2.9399 | −1.0545 |
25 | 3.4075 | 3.3649 | 0.0426 | 1.9560 | 2.8964 | −0.9404 |
30 | 3.5104 | 3.4507 | 0.0597 | 1.9983 | 2.8073 | −0.8090 |
35 | 3.5982 | 3.5192 | 0.0790 | 2.0254 | 2.6967 | −0.6713 |
50 | 3.7952 | 3.6475 | 0.1477 | 2.0676 | 2.3444 | −0.2768 |
Example 1 | Example 3 | Example 5 | ||||
---|---|---|---|---|---|---|
Maturity | Yield (Vasicek) | Yield (MeLo-1-V) | Yield (Vasicek) | Yield (MeLo-1-V) | Yield (Vasicek) | Yield (MeLo-1-V) |
1/12 | 5.7957 | 5.8618 | 0.2480 | 1.7537 | −4.8534 | 0.0128 |
4/12 | 5.7718 | 6.1852 | 0.2165 | 2.8204 | −4.5738 | 0.0204 |
6/12 | 5.6981 | 6.6505 | 0.1019 | 3.8359 | −4.1859 | 0.0403 |
8/12 | 5.5801 | 7.0312 | −0.0915 | 4.5639 | −3.8324 | 0.0728 |
1 | 5.4191 | 7.3388 | −0.3615 | 5.1264 | −3.5098 | 0.1170 |
2 | 4.3687 | 8.0579 | −2.1652 | 6.4461 | −2.4706 | 0.3570 |
3 | 2.7224 | 8.2548 | −5.0306 | 6.9329 | −1.7293 | 0.6075 |
4 | 0.5467 | 8.1533 | −8.8389 | 6.9861 | −1.1891 | 0.8208 |
5 | −2.0974 | 7.8840 | −13.4820 | 6.8093 | −0.7870 | 0.9906 |
6 | −5.1548 | 7.5285 | −18.8616 | 6.5191 | −0.4815 | 1.1222 |
7 | −8.5753 | 7.1376 | −24.8885 | 6.1822 | −0.2448 | 1.2232 |
8 | −12.3132 | 6.7417 | −31.4815 | 5.8356 | −0.0580 | 1.3007 |
9 | −16.3273 | 6.3579 | −38.5670 | 5.4988 | 0.0921 | 1.3601 |
10 | −20.5799 | 5.9952 | −46.0782 | 5.1811 | 0.2146 | 1.4058 |
12 | −29.6680 | 5.3454 | −62.1415 | 4.6152 | 0.4015 | 1.4675 |
15 | −44.3380 | 4.5553 | −88.0919 | 3.9324 | 0.5906 | 1.5139 |
20 | −70.1354 | 3.6158 | −133.7631 | 3.1265 | 0.7806 | 1.5297 |
25 | −96.0515 | 2.9812 | −179.6724 | 2.5835 | 0.8947 | 1.5112 |
30 | −120.9957 | 2.5303 | −223.8763 | 2.1976 | 0.9708 | 1.4782 |
35 | −144.3949 | 2.1955 | −265.3527 | 1.9105 | 1.0252 | 1.4389 |
50 | −203.6505 | 1.5688 | −370.4153 | 1.3710 | 1.1230 | 1.3141 |
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Jun, J.-Y.; Rakotondratsimba, Y. Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework. Mathematics 2024, 12, 2690. https://doi.org/10.3390/math12172690
Jun J-Y, Rakotondratsimba Y. Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework. Mathematics. 2024; 12(17):2690. https://doi.org/10.3390/math12172690
Chicago/Turabian StyleJun, Jae-Yun, and Yves Rakotondratsimba. 2024. "Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework" Mathematics 12, no. 17: 2690. https://doi.org/10.3390/math12172690