|
2 | 2 | "cells": [ |
3 | 3 | { |
4 | 4 | "cell_type": "markdown", |
5 | | - "id": "8c99041d", |
| 5 | + "id": "35184849", |
6 | 6 | "metadata": {}, |
7 | 7 | "source": [ |
8 | 8 | "$$\n", |
|
13 | 13 | }, |
14 | 14 | { |
15 | 15 | "cell_type": "markdown", |
16 | | - "id": "34888168", |
| 16 | + "id": "5eff41c4", |
17 | 17 | "metadata": {}, |
18 | 18 | "source": [ |
19 | 19 | "\n", |
|
22 | 22 | }, |
23 | 23 | { |
24 | 24 | "cell_type": "markdown", |
25 | | - "id": "7994cc74", |
| 25 | + "id": "c7794ae5", |
26 | 26 | "metadata": {}, |
27 | 27 | "source": [ |
28 | 28 | "# Affine Models of Asset Prices" |
29 | 29 | ] |
30 | 30 | }, |
31 | 31 | { |
32 | 32 | "cell_type": "markdown", |
33 | | - "id": "a6d38d89", |
| 33 | + "id": "af26c6c0", |
34 | 34 | "metadata": {}, |
35 | 35 | "source": [ |
36 | 36 | "## Overview\n", |
|
87 | 87 | { |
88 | 88 | "cell_type": "code", |
89 | 89 | "execution_count": null, |
90 | | - "id": "005f4235", |
| 90 | + "id": "ac6e3757", |
91 | 91 | "metadata": { |
92 | 92 | "hide-output": false |
93 | 93 | }, |
|
101 | 101 | }, |
102 | 102 | { |
103 | 103 | "cell_type": "markdown", |
104 | | - "id": "014d7b73", |
| 104 | + "id": "0fd86787", |
105 | 105 | "metadata": {}, |
106 | 106 | "source": [ |
107 | 107 | "## The model" |
108 | 108 | ] |
109 | 109 | }, |
110 | 110 | { |
111 | 111 | "cell_type": "markdown", |
112 | | - "id": "e2226ffd", |
| 112 | + "id": "876ed648", |
113 | 113 | "metadata": {}, |
114 | 114 | "source": [ |
115 | 115 | "### State dynamics and short rate\n", |
|
169 | 169 | }, |
170 | 170 | { |
171 | 171 | "cell_type": "markdown", |
172 | | - "id": "8336c9ba", |
| 172 | + "id": "18699682", |
173 | 173 | "metadata": {}, |
174 | 174 | "source": [ |
175 | 175 | "### Properties of the SDF\n", |
|
197 | 197 | }, |
198 | 198 | { |
199 | 199 | "cell_type": "markdown", |
200 | | - "id": "56b5196f", |
| 200 | + "id": "04b8425b", |
201 | 201 | "metadata": {}, |
202 | 202 | "source": [ |
203 | 203 | "## Pricing risky assets" |
204 | 204 | ] |
205 | 205 | }, |
206 | 206 | { |
207 | 207 | "cell_type": "markdown", |
208 | | - "id": "4deabcaa", |
| 208 | + "id": "bccd375e", |
209 | 209 | "metadata": {}, |
210 | 210 | "source": [ |
211 | 211 | "### Lognormal returns\n", |
|
237 | 237 | }, |
238 | 238 | { |
239 | 239 | "cell_type": "markdown", |
240 | | - "id": "eafa8f5d", |
| 240 | + "id": "8b2e7eb9", |
241 | 241 | "metadata": {}, |
242 | 242 | "source": [ |
243 | 243 | "### Expected excess returns\n", |
|
267 | 267 | }, |
268 | 268 | { |
269 | 269 | "cell_type": "markdown", |
270 | | - "id": "a29a1313", |
| 270 | + "id": "aaf22d4f", |
271 | 271 | "metadata": {}, |
272 | 272 | "source": [ |
273 | 273 | "## Affine term structure of yields\n", |
|
278 | 278 | }, |
279 | 279 | { |
280 | 280 | "cell_type": "markdown", |
281 | | - "id": "f2ffc534", |
| 281 | + "id": "27b4d5b6", |
282 | 282 | "metadata": {}, |
283 | 283 | "source": [ |
284 | 284 | "### Bond prices\n", |
|
309 | 309 | }, |
310 | 310 | { |
311 | 311 | "cell_type": "markdown", |
312 | | - "id": "4f297348", |
| 312 | + "id": "d684e495", |
313 | 313 | "metadata": {}, |
314 | 314 | "source": [ |
315 | 315 | "### Exponential affine prices\n", |
|
342 | 342 | }, |
343 | 343 | { |
344 | 344 | "cell_type": "markdown", |
345 | | - "id": "d671b120", |
| 345 | + "id": "461c6fb2", |
346 | 346 | "metadata": {}, |
347 | 347 | "source": [ |
348 | 348 | "### Yields\n", |
|
370 | 370 | }, |
371 | 371 | { |
372 | 372 | "cell_type": "markdown", |
373 | | - "id": "751dfac6", |
| 373 | + "id": "9de3c548", |
374 | 374 | "metadata": {}, |
375 | 375 | "source": [ |
376 | 376 | "## Python implementation\n", |
|
382 | 382 | { |
383 | 383 | "cell_type": "code", |
384 | 384 | "execution_count": null, |
385 | | - "id": "9ecd357d", |
| 385 | + "id": "8b556719", |
386 | 386 | "metadata": { |
387 | 387 | "hide-output": false |
388 | 388 | }, |
|
450 | 450 | }, |
451 | 451 | { |
452 | 452 | "cell_type": "markdown", |
453 | | - "id": "8acf7018", |
| 453 | + "id": "315963ce", |
454 | 454 | "metadata": {}, |
455 | 455 | "source": [ |
456 | 456 | "### A one-factor Gaussian example\n", |
|
461 | 461 | { |
462 | 462 | "cell_type": "code", |
463 | 463 | "execution_count": null, |
464 | | - "id": "0733a32a", |
| 464 | + "id": "7d545621", |
465 | 465 | "metadata": { |
466 | 466 | "hide-output": false |
467 | 467 | }, |
|
494 | 494 | }, |
495 | 495 | { |
496 | 496 | "cell_type": "markdown", |
497 | | - "id": "f47c150b", |
| 497 | + "id": "4a35ba66", |
498 | 498 | "metadata": {}, |
499 | 499 | "source": [ |
500 | 500 | "### Yield curve shapes\n", |
|
505 | 505 | { |
506 | 506 | "cell_type": "code", |
507 | 507 | "execution_count": null, |
508 | | - "id": "110bd778", |
| 508 | + "id": "32d830bc", |
509 | 509 | "metadata": { |
510 | 510 | "hide-output": false |
511 | 511 | }, |
|
559 | 559 | }, |
560 | 560 | { |
561 | 561 | "cell_type": "markdown", |
562 | | - "id": "10b7da62", |
| 562 | + "id": "bc31fbf8", |
563 | 563 | "metadata": {}, |
564 | 564 | "source": [ |
565 | 565 | "The model generates upward-sloping, flat, and inverted yield curves as the short\n", |
|
568 | 568 | }, |
569 | 569 | { |
570 | 570 | "cell_type": "markdown", |
571 | | - "id": "aa4a4058", |
| 571 | + "id": "86e90f63", |
572 | 572 | "metadata": {}, |
573 | 573 | "source": [ |
574 | 574 | "### Short rate dynamics" |
|
577 | 577 | { |
578 | 578 | "cell_type": "code", |
579 | 579 | "execution_count": null, |
580 | | - "id": "30bb2ff3", |
| 580 | + "id": "e828c891", |
581 | 581 | "metadata": { |
582 | 582 | "hide-output": false |
583 | 583 | }, |
|
607 | 607 | }, |
608 | 608 | { |
609 | 609 | "cell_type": "markdown", |
610 | | - "id": "3e379be9", |
| 610 | + "id": "fde48eac", |
611 | 611 | "metadata": {}, |
612 | 612 | "source": [ |
613 | 613 | "### A two-factor model\n", |
|
621 | 621 | { |
622 | 622 | "cell_type": "code", |
623 | 623 | "execution_count": null, |
624 | | - "id": "9ab7bf59", |
| 624 | + "id": "d9aad405", |
625 | 625 | "metadata": { |
626 | 626 | "hide-output": false |
627 | 627 | }, |
|
658 | 658 | { |
659 | 659 | "cell_type": "code", |
660 | 660 | "execution_count": null, |
661 | | - "id": "4d4c35b0", |
| 661 | + "id": "e2b832b2", |
662 | 662 | "metadata": { |
663 | 663 | "hide-output": false |
664 | 664 | }, |
|
723 | 723 | }, |
724 | 724 | { |
725 | 725 | "cell_type": "markdown", |
726 | | - "id": "21963a4f", |
| 726 | + "id": "1a5856e2", |
727 | 727 | "metadata": {}, |
728 | 728 | "source": [ |
729 | 729 | "## Risk premiums\n", |
|
745 | 745 | { |
746 | 746 | "cell_type": "code", |
747 | 747 | "execution_count": null, |
748 | | - "id": "101b2eed", |
| 748 | + "id": "947414ac", |
749 | 749 | "metadata": { |
750 | 750 | "hide-output": false |
751 | 751 | }, |
|
821 | 821 | }, |
822 | 822 | { |
823 | 823 | "cell_type": "markdown", |
824 | | - "id": "79e29c37", |
| 824 | + "id": "f131072b", |
825 | 825 | "metadata": {}, |
826 | 826 | "source": [ |
827 | 827 | "## Risk-neutral probabilities\n", |
|
856 | 856 | }, |
857 | 857 | { |
858 | 858 | "cell_type": "markdown", |
859 | | - "id": "4154ba28", |
| 859 | + "id": "8217e6b5", |
860 | 860 | "metadata": {}, |
861 | 861 | "source": [ |
862 | 862 | "### The risk-neutral VAR\n", |
|
883 | 883 | }, |
884 | 884 | { |
885 | 885 | "cell_type": "markdown", |
886 | | - "id": "a462f694", |
| 886 | + "id": "e9c3e4e2", |
887 | 887 | "metadata": {}, |
888 | 888 | "source": [ |
889 | 889 | "### Verification via risk-neutral pricing\n", |
|
903 | 903 | { |
904 | 904 | "cell_type": "code", |
905 | 905 | "execution_count": null, |
906 | | - "id": "1b372983", |
| 906 | + "id": "6444d391", |
907 | 907 | "metadata": { |
908 | 908 | "hide-output": false |
909 | 909 | }, |
|
942 | 942 | }, |
943 | 943 | { |
944 | 944 | "cell_type": "markdown", |
945 | | - "id": "8d28d182", |
| 945 | + "id": "189320f8", |
946 | 946 | "metadata": {}, |
947 | 947 | "source": [ |
948 | 948 | "The analytical and Monte Carlo bond prices agree closely, validating the\n", |
|
951 | 951 | }, |
952 | 952 | { |
953 | 953 | "cell_type": "markdown", |
954 | | - "id": "d91183be", |
| 954 | + "id": "823fd88e", |
955 | 955 | "metadata": {}, |
956 | 956 | "source": [ |
957 | 957 | "## Distorted beliefs\n", |
|
963 | 963 | }, |
964 | 964 | { |
965 | 965 | "cell_type": "markdown", |
966 | | - "id": "60bf9f4f", |
| 966 | + "id": "60d209bc", |
967 | 967 | "metadata": {}, |
968 | 968 | "source": [ |
969 | 969 | "### The subjective measure\n", |
|
1009 | 1009 | }, |
1010 | 1010 | { |
1011 | 1011 | "cell_type": "markdown", |
1012 | | - "id": "fe3e3779", |
| 1012 | + "id": "cbfc70ba", |
1013 | 1013 | "metadata": {}, |
1014 | 1014 | "source": [ |
1015 | 1015 | "### Pricing under distorted beliefs\n", |
|
1036 | 1036 | }, |
1037 | 1037 | { |
1038 | 1038 | "cell_type": "markdown", |
1039 | | - "id": "f9127709", |
| 1039 | + "id": "30cd1124", |
1040 | 1040 | "metadata": {}, |
1041 | 1041 | "source": [ |
1042 | 1042 | "### Numerical illustration" |
|
1045 | 1045 | { |
1046 | 1046 | "cell_type": "code", |
1047 | 1047 | "execution_count": null, |
1048 | | - "id": "8c452049", |
| 1048 | + "id": "68770e0d", |
1049 | 1049 | "metadata": { |
1050 | 1050 | "hide-output": false |
1051 | 1051 | }, |
|
1089 | 1089 | { |
1090 | 1090 | "cell_type": "code", |
1091 | 1091 | "execution_count": null, |
1092 | | - "id": "a86fe6b0", |
| 1092 | + "id": "d79f17b5", |
1093 | 1093 | "metadata": { |
1094 | 1094 | "hide-output": false |
1095 | 1095 | }, |
|
1155 | 1155 | }, |
1156 | 1156 | { |
1157 | 1157 | "cell_type": "markdown", |
1158 | | - "id": "2c6a437a", |
| 1158 | + "id": "099ca676", |
1159 | 1159 | "metadata": {}, |
1160 | 1160 | "source": [ |
1161 | 1161 | "When expert beliefs are overly persistent ($ \\hat\\phi $ has larger eigenvalues than\n", |
|
1171 | 1171 | }, |
1172 | 1172 | { |
1173 | 1173 | "cell_type": "markdown", |
1174 | | - "id": "aa0117ce", |
| 1174 | + "id": "e473fbfc", |
1175 | 1175 | "metadata": {}, |
1176 | 1176 | "source": [ |
1177 | 1177 | "## The bond price recursion\n", |
|
1206 | 1206 | }, |
1207 | 1207 | { |
1208 | 1208 | "cell_type": "markdown", |
1209 | | - "id": "9822eec0", |
| 1209 | + "id": "9376537b", |
1210 | 1210 | "metadata": {}, |
1211 | 1211 | "source": [ |
1212 | 1212 | "## Concluding remarks\n", |
|
1239 | 1239 | } |
1240 | 1240 | ], |
1241 | 1241 | "metadata": { |
1242 | | - "date": 1772859322.5392563, |
| 1242 | + "date": 1772867702.0173764, |
1243 | 1243 | "filename": "affine_risk_prices.md", |
1244 | 1244 | "kernelspec": { |
1245 | 1245 | "display_name": "Python", |
|
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