An Asymptotic Solution for Call Options on Zero-Coupon Bonds
Abstract
:1. Introduction
2. Previous Literature: Interest Rate Models
Therefore, while today’s models are indubitably more effective (at least in certain respects) than the early ones, and, therefore, one can certainly speak of an ‘evolution’ of interest-rate models, this does not necessary imply that certain choices, abandoned in the past, might not have ultimately led a more fruitful description of interest-rate derivatives products if they had been pursued more actively.[14] (p. 672)
As for the log-normal assumption for the bond price, this was partly satisfactory, because it did not constrain the price to be smaller than face value, thereby assigning a non-zero probability to negative interest rates…What gave greater discomfort, however, was the so-called pull-to-par problem.[14] (p. 683)
Although this is not necessary for what follows, we may suppose that it is a stochastic process–defined on a complete probability space –from which some realization is observed.[15] (p. 82)
3. Derivation of Bond Options on Zero-Coupon Bonds
3.1. A Process for Zero-Coupon Bonds
3.2. Model Definition
4. Solution
Discussion of Results
5. Properties of Greeks
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Derivation of Asymptotic Solution
References
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Volatility 15% | ||||||
---|---|---|---|---|---|---|
Option Maturity 0.25 Years | Option Maturity 0.5 Years | Option Maturity 1 Year | ||||
Bond Price | PTP | Black–Scholes | PTP | Black–Scholes | PTP | Black–Scholes |
72 | 0.1635 | 0.2140 | 0.6151 | 0.7087 | 1.6037 | 1.7065 |
76 | 0.8201 | 0.8820 | 1.6655 | 1.7368 | 3.0303 | 3.0701 |
80 | 2.3971 | 2.4419 | 3.4379 | 3.4801 | 4.9730 | 4.9730 |
84 | 4.9643 | 5.0273 | 5.8893 | 5.9606 | 7.3626 | 7.4004 |
88 | 8.3351 | 8.3938 | 8.9581 | 9.0560 | 10.1888 | 10.2867 |
Volatility 20% | ||||||
72 | 0.5158 | 0.5835 | 1.3647 | 1.4599 | 2.9080 | 2.9788 |
76 | 1.4694 | 1.5393 | 2.6761 | 2.7496 | 4.5372 | 4.5614 |
80 | 3.1787 | 3.2384 | 4.5485 | 4.6047 | 6.5580 | 6.5580 |
84 | 5.6463 | 5.7183 | 6.9416 | 7.0165 | 8.9257 | 8.9488 |
88 | 8.7695 | 8.8449 | 9.8153 | 9.9148 | 11.6266 | 11.6935 |
Volatility 25% | ||||||
72 | 0.9935 | 1.0738 | 2.2248 | 2.3227 | 4.2850 | 4.3357 |
76 | 2.1610 | 2.2414 | 3.7128 | 3.7936 | 6.0542 | 6.0704 |
80 | 3.9599 | 4.0345 | 5.6578 | 5.7280 | 8.1394 | 8.1394 |
84 | 6.3730 | 6.4564 | 8.0254 | 8.1087 | 10.5079 | 10.5233 |
88 | 9.3322 | 9.4207 | 10.7848 | 10.8881 | 13.1461 | 13.1937 |
Bond Maturity | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bond Price | Volatility | Option Maturity | 5 Years | 5 Years | 10 Years | 10 Years | 20 Years | 20 Years | 30 Years | 30 Years |
O(1) | O(1/T_B) | O(1) | O(1/T_B) | O(1) | O(1/T_B) | O(1) | O(1/T_B) | |||
72 | 15% | 0.25 | 57.42% | 42.58% | 72.95% | 27.05% | 84.36% | 15.64% | 89.00% | 11.00% |
72 | 20% | 0.50 | 79.32% | 20.68% | 88.47% | 11.53% | 93.88% | 6.12% | 95.84% | 4.16% |
72 | 25% | 1.0 | 82.91% | 17.09% | 90.66% | 9.34% | 95.10% | 4.90% | 96.68% | 3.32% |
80 | 15% | 0.25 | 97.61% | 2.39% | 98.79% | 1.21% | 99.39% | 0.61% | 99.59% | 0.41% |
80 | 20% | 0.50 | 95.35% | 4.65% | 97.62% | 2.38% | 98.79% | 1.21% | 99.19% | 0.81% |
80 | 25% | 1.00 | 91.16% | 8.84% | 95.37% | 4.63% | 97.63% | 2.37% | 98.41% | 1.59% |
88 | 15% | 0.25 | 97.90% | 2.10% | 98.94% | 1.06% | 99.47% | 0.53% | 99.64% | 0.36% |
88 | 20% | 0.50 | 96.14% | 3.86% | 98.03% | 1.97% | 99.01% | 0.99% | 99.34% | 0.66% |
88 | 25% | 1.00 | 93.25% | 6.75% | 96.51% | 3.49% | 98.22% | 1.78% | 98.81% | 1.19% |
Volatility 15% | ||||||
---|---|---|---|---|---|---|
Option Maturity 0.25 Years | Option Maturity 0.5 Years | Option Maturity 1 Year | ||||
Bond Price | PTP | Black–Scholes | PTP | Black–Scholes | PTP | Black–Scholes |
72 | 0.0807 | 0.0884 | 0.1794 | 0.1796 | 0.2890 | 0.2762 |
76 | 0.2674 | 0.2644 | 0.3518 | 0.3420 | 0.4238 | 0.4076 |
80 | 0.5214 | 0.5216 | 0.5304 | 0.5305 | 0.5431 | 0.5431 |
84 | 0.7558 | 0.7595 | 0.6945 | 0.7042 | 0.6527 | 0.6677 |
88 | 0.9127 | 0.9074 | 0.8328 | 0.8353 | 0.7583 | 0.7715 |
Volatility 20% | ||||||
72 | 0.1567 | 0.1608 | 0.2586 | 0.2557 | 0.3554 | 0.3439 |
76 | 0.3288 | 0.3262 | 0.3986 | 0.3919 | 0.4581 | 0.4477 |
80 | 0.5246 | 0.5249 | 0.5350 | 0.5352 | 0.5497 | 0.5497 |
84 | 0.7057 | 0.7090 | 0.6607 | 0.6676 | 0.6342 | 0.6439 |
88 | 0.8470 | 0.8451 | 0.7726 | 0.7771 | 0.7151 | 0.7263 |
Volatility 25% | ||||||
72 | 0.2182 | 0.2205 | 0.3140 | 0.3108 | 0.4003 | 0.3911 |
76 | 0.3695 | 0.3677 | 0.4301 | 0.4256 | 0.4832 | 0.4760 |
80 | 0.5285 | 0.5289 | 0.5405 | 0.5408 | 0.5576 | 0.5576 |
84 | 0.6757 | 0.6783 | 0.6425 | 0.6475 | 0.6264 | 0.6331 |
88 | 0.7986 | 0.7981 | 0.7351 | 0.7395 | 0.6919 | 0.7006 |
Volatility 15% | ||||||
---|---|---|---|---|---|---|
Option Maturity 0.25 Years | Option Maturity 0.5 Years | Option Maturity 1 Year | ||||
Bond Price | PTP | Black–Scholes | PTP | Black–Scholes | PTP | Black–Scholes |
72 | 0.0311 | 0.0297 | 0.0375 | 0.0343 | 0.0335 | 0.0310 |
76 | 0.0601 | 0.0574 | 0.0464 | 0.0456 | 0.0328 | 0.0341 |
80 | 0.0607 | 0.0664 | 0.0405 | 0.0469 | 0.0261 | 0.0331 |
84 | 0.0516 | 0.0494 | 0.0394 | 0.0388 | 0.0277 | 0.0288 |
88 | 0.0266 | 0.0251 | 0.0289 | 0.0266 | 0.0245 | 0.0229 |
Volatility 20% | ||||||
72 | 0.0356 | 0.0339 | 0.0333 | 0.0316 | 0.0263 | 0.0256 |
76 | 0.0485 | 0.0474 | 0.0356 | 0.0357 | 0.0247 | 0.0260 |
80 | 0.0468 | 0.0498 | 0.0317 | 0.0351 | 0.0209 | 0.0247 |
84 | 0.0417 | 0.0408 | 0.0304 | 0.0306 | 0.0210 | 0.0222 |
88 | 0.0284 | 0.0271 | 0.0251 | 0.0240 | 0.0193 | 0.0189 |
Volatility 25% | ||||||
72 | 0.0342 | 0.0329 | 0.0286 | 0.0278 | 0.0214 | 0.0213 |
76 | 0.0402 | 0.0397 | 0.0288 | 0.0292 | 0.0198 | 0.0210 |
80 | 0.0380 | 0.0398 | 0.0259 | 0.0281 | 0.0173 | 0.0197 |
84 | 0.0345 | 0.0341 | 0.0247 | 0.0250 | 0.0169 | 0.0179 |
88 | 0.0265 | 0.0256 | 0.0214 | 0.0209 | 0.0157 | 0.0158 |
Volatility 15% | ||||||
---|---|---|---|---|---|---|
Option Maturity 0.25 Years | Option Maturity 0.5 Years | Option Maturity 1 Year | ||||
Bond Price | PTP | Black–Scholes | PTP | Black–Scholes | PTP | Black–Scholes |
72 | −1.5080 | −1.7614 | −1.9599 | −2.0625 | −1.9409 | −1.8967 |
76 | −3.7232 | −3.8260 | −3.0916 | −3.0815 | −2.4562 | −2.3523 |
80 | −4.9471 | −4.9767 | −3.6125 | −3.5700 | −2.6910 | −2.5720 |
84 | −4.1172 | −4.2153 | −3.3577 | −3.3438 | −2.6396 | −2.5313 |
88 | −2.2974 | −2.5464 | −2.5534 | −2.6368 | −2.3466 | −2.2848 |
Volatility 20% | ||||||
72 | −3.3608 | −3.5703 | −3.3246 | −3.3592 | −2.8699 | −2.7585 |
76 | −5.5126 | −5.5952 | −4.3018 | −4.2647 | −3.3007 | −3.1532 |
80 | −6.5250 | −6.5644 | −4.7436 | −4.6870 | −3.5121 | −3.3538 |
84 | −5.9483 | −6.0294 | −4.6004 | −4.5597 | −3.5103 | −3.3563 |
88 | −4.3202 | −4.5199 | −3.9859 | −4.0052 | −3.3198 | −3.1907 |
Volatility 25% | ||||||
72 | −5.2361 | −5.4112 | −4.6030 | −4.5959 | −3.7367 | −3.5750 |
76 | −7.2097 | −7.2880 | −5.4659 | −5.4090 | −4.1214 | −3.9336 |
80 | −8.1003 | −8.1493 | −5.8709 | −5.8000 | −4.3274 | −4.1301 |
84 | −7.7012 | −7.7795 | −5.8065 | −5.7458 | −4.3632 | −4.1667 |
88 | −6.3302 | −6.4976 | −5.3428 | −5.3222 | −4.2452 | −4.0632 |
Volatility 15% | ||||||
---|---|---|---|---|---|---|
Option Maturity 0.25 Years | Option Maturity 0.5 Years | Option Maturity 1 Year | ||||
Bond Price | PTP | Black–Scholes | PTP | Black–Scholes | PTP | Black–Scholes |
72 | 5.3239 | 5.7688 | 13.2655 | 13.3423 | 24.8610 | 24.0773 |
76 | 12.3032 | 12.4331 | 19.7749 | 19.7351 | 29.9482 | 29.5034 |
80 | 15.6359 | 15.9343 | 22.2208 | 22.5015 | 31.7286 | 31.7286 |
84 | 12.9213 | 13.0714 | 20.5244 | 20.5186 | 30.9206 | 30.5045 |
88 | 6.8675 | 7.2971 | 15.3781 | 15.4301 | 27.3902 | 26.6244 |
Volatility 20% | ||||||
72 | 8.5109 | 8.7884 | 16.3433 | 16.3721 | 27.0008 | 26.4963 |
76 | 13.5092 | 13.6974 | 20.5392 | 20.6476 | 30.2736 | 30.0587 |
80 | 15.6286 | 15.9266 | 22.1999 | 22.4796 | 31.6670 | 31.6670 |
84 | 14.1929 | 14.4005 | 21.4398 | 21.5719 | 31.5097 | 31.3062 |
88 | 10.2028 | 10.4806 | 18.5209 | 18.5645 | 29.7817 | 29.2965 |
Volatility 25% | ||||||
72 | 10.4378 | 10.6743 | 17.9003 | 17.9824 | 27.9662 | 27.6470 |
76 | 14.0918 | 14.3189 | 20.8875 | 21.0651 | 30.3834 | 30.2648 |
80 | 15.6185 | 15.9158 | 22.1708 | 22.4493 | 31.5816 | 31.5816 |
84 | 14.8086 | 15.0538 | 21.8605 | 22.0577 | 31.7404 | 31.6274 |
88 | 12.1349 | 12.3871 | 20.0960 | 20.2053 | 30.8716 | 30.5674 |
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Tomas, M.J., III; Yu, J. An Asymptotic Solution for Call Options on Zero-Coupon Bonds. Mathematics 2021, 9, 1940. https://doi.org/10.3390/math9161940
Tomas MJ III, Yu J. An Asymptotic Solution for Call Options on Zero-Coupon Bonds. Mathematics. 2021; 9(16):1940. https://doi.org/10.3390/math9161940
Chicago/Turabian StyleTomas, Michael J., III, and Jun Yu. 2021. "An Asymptotic Solution for Call Options on Zero-Coupon Bonds" Mathematics 9, no. 16: 1940. https://doi.org/10.3390/math9161940
APA StyleTomas, M. J., III, & Yu, J. (2021). An Asymptotic Solution for Call Options on Zero-Coupon Bonds. Mathematics, 9(16), 1940. https://doi.org/10.3390/math9161940