2126:
125:
900:
821:
1435:
155:
1259:
1862:
855:
1392:
1372:
1776:
2167:
809:
1693:
195:
As the model generates a symmetric ("bell shaped") distribution of rates in the future, negative rates are possible. Further, it does not incorporate
1703:
1377:
815:
1387:
1745:
212:
1460:
1642:
1932:
1922:
1445:
422:
1832:
1796:
1749:
2100:
1837:
745:
947:
848:
740:
1902:
1480:
1450:
350:
256:
1753:
1737:
2201:
1947:
1652:
872:
374:
1852:
1817:
1786:
1781:
1420:
1217:
1134:
200:
1791:
1119:
2186:
2160:
1415:
1222:
735:
611:
174:
1141:
60:
1877:
1757:
2105:
1882:
1718:
1617:
1602:
1014:
930:
841:
1892:
1528:
1887:
1490:
1074:
1019:
935:
551:
1822:
1812:
1455:
1425:
2191:
1827:
992:
890:
591:
1538:
1114:
895:
2196:
2153:
1907:
1708:
1622:
1607:
997:
415:
40:
1741:
1627:
1049:
1129:
1104:
750:
196:
170:
1847:
1430:
965:
297:
2042:
2032:
1723:
1505:
1244:
1109:
920:
177:
1327:
1984:
1912:
1171:
684:
586:
394:
181:
2007:
1989:
1969:
1964:
1683:
1515:
1495:
1342:
1285:
1124:
1034:
730:
694:
531:
490:
20:
1475:
208:
2082:
2037:
2027:
1768:
1713:
1688:
1657:
1637:
1397:
1382:
1249:
408:
248:
279:
2077:
1917:
1842:
1647:
1407:
1317:
1207:
500:
382:
2047:
2012:
1927:
1897:
1667:
1662:
1485:
1322:
987:
925:
864:
576:
323:
133:
1728:
2067:
1872:
1523:
1280:
1197:
1166:
1059:
1039:
1029:
885:
880:
783:
709:
561:
483:
346:
252:
2137:
1733:
2087:
1974:
1857:
1227:
1202:
1151:
1079:
1002:
955:
773:
644:
616:
556:
541:
328:
28:
2133:
2052:
1952:
1937:
1698:
1632:
1310:
1254:
1237:
982:
788:
778:
581:
571:
566:
495:
378:
301:
1867:
1099:
157:
from market prices, meaning that it can exactly return the price of bonds comprising the
2057:
2022:
1942:
1548:
1295:
1212:
1181:
1176:
1156:
1146:
1089:
1084:
1064:
1044:
1009:
977:
960:
689:
674:
639:
626:
601:
505:
473:
442:
366:
185:
2180:
1959:
1500:
1337:
1332:
1290:
1232:
1054:
970:
910:
793:
768:
679:
664:
654:
606:
546:
536:
371:
342:
44:
2017:
1979:
1533:
1465:
1354:
1349:
1161:
1094:
1069:
905:
704:
649:
596:
526:
452:
1597:
2062:
1581:
1576:
1571:
1561:
1364:
1305:
1300:
1264:
1024:
915:
760:
699:
659:
634:
510:
478:
468:
431:
219:
analogue to the Ho–Lee model, although is less widely used than the latter two.
189:
162:
158:
32:
2072:
1612:
1556:
1440:
362:
1566:
447:
267:
216:
204:
36:
211:(mean reverting with lognormal variant available) are often preferred. The
714:
669:
363:
Valuation and
Hedging of Interest Rates Derivatives with the Ho-Lee Model
166:
51:
2125:
1393:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
833:
332:
319:
Term structure movements and pricing interest rate contingent claims
130:
The model can be calibrated to market data by implying the form of
292:
837:
404:
400:
245:
Fixed Income
Securities: Valuation, Risk, and Risk Management
1373:
Autoregressive conditional heteroskedasticity (ARCH) model
47:. It was developed in 1986 by Thomas Ho and Sang Bin Lee.
901:
Independent and identically distributed random variables
2141:
1378:
Autoregressive integrated moving average (ARIMA) model
120:{\displaystyle dr_{t}=\theta _{t}\,dt+\sigma \,dW_{t}}
822:
Securities
Industry and Financial Markets Association
136:
63:
2000:
1805:
1767:
1676:
1590:
1547:
1514:
1406:
1363:
1273:
1190:
946:
871:
802:
759:
723:
625:
519:
461:
149:
119:
1260:Stochastic chains with memory of variable length
161:. This calibration, and subsequent valuation of
2161:
849:
416:
8:
199:. For both of these reasons, models such as
192:" bond option formulae are also available.
50:Under this model, the short rate follows a
2168:
2154:
1388:Autoregressive–moving-average (ARMA) model
856:
842:
834:
810:Commercial Mortgage Securities Association
423:
409:
401:
239:
237:
141:
135:
111:
103:
90:
84:
71:
62:
16:Short-rate model in financial mathematics
816:International Capital Market Association
339:Options, futures, and other derivatives
233:
1694:Doob's martingale convergence theorems
1446:Constant elasticity of variance (CEV)
1436:Chan–Karolyi–Longstaff–Sanders (CKLS)
7:
2122:
2120:
395:Binomial Tree – Excel implementation
746:Commercial mortgage-backed security
2140:. You can help Knowledge (XXG) by
1933:Skorokhod's representation theorem
1714:Law of large numbers (weak/strong)
741:Collateralized mortgage obligation
365:, Markus Leippold and Zvi Wiener,
14:
1903:Martingale representation theorem
2124:
1948:Stochastic differential equation
1838:Doob's optional stopping theorem
1833:Doob–Meyer decomposition theorem
1818:Convergence of random variables
1704:Fisher–Tippett–Gnedenko theorem
173:, is typically performed via a
1416:Binomial options pricing model
736:Collateralized debt obligation
612:Reverse convertible securities
213:Kalotay–Williams–Fabozzi model
31:widely used in the pricing of
1:
1883:Kolmogorov continuity theorem
1719:Law of the iterated logarithm
372:Term Structure Lattice Models
304:, Financial Modelling Agency.
1888:Kolmogorov extension theorem
1567:Generalized queueing network
1075:Interacting particle systems
1020:Continuous-time random walk
552:Contingent convertible bond
150:{\displaystyle \theta _{t}}
2218:
2119:
2028:Extreme value theory (EVT)
1828:Doob decomposition theorem
1120:Ornstein–Uhlenbeck process
891:Chinese restaurant process
592:Inverse floating rate note
2096:
1908:Optional stopping theorem
1709:Large deviation principle
1461:Heath–Jarrow–Morton (HJM)
1398:Moving-average (MA) model
1383:Autoregressive (AR) model
1208:Hidden Markov model (HMM)
1142:Schramm–Loewner evolution
438:
294:Interest Rate Derivatives
171:interest rate derivatives
43:, and in modeling future
41:interest rate derivatives
1823:Doléans-Dade exponential
1653:Progressively measurable
1451:Cox–Ingersoll–Ross (CIR)
751:Mortgage-backed security
520:Types of bonds by payout
462:Types of bonds by issuer
243:Pietro Veronesi (2010).
207:and mean reverting) and
2202:Economic theories stubs
2043:Mathematical statistics
2033:Large deviations theory
1863:Infinitesimal generator
1724:Maximal ergodic theorem
1643:Piecewise-deterministic
1245:Random dynamical system
1110:Markov additive process
1878:Karhunen–Loève theorem
1813:Cameron–Martin formula
1777:Burkholder–Davis–Gundy
1172:Variance gamma process
685:Option-adjusted spread
587:Inflation-indexed bond
151:
121:
2187:Fixed income analysis
2136:related article is a
2008:Actuarial mathematics
1970:Uniform integrability
1965:Stratonovich integral
1893:Lévy–Prokhorov metric
1797:Marcinkiewicz–Zygmund
1684:Central limit theorem
1286:Gaussian random field
1115:McKean–Vlasov process
1035:Dyson Brownian motion
896:Galton–Watson process
731:Asset-backed security
695:Weighted-average life
532:Auction rate security
317:T.S.Y. Ho, S.B. Lee,
291:Graeme West, (2010).
152:
122:
21:financial mathematics
2083:Time series analysis
2038:Mathematical finance
1923:Reflection principle
1250:Regenerative process
1050:Fleming–Viot process
865:Stochastic processes
724:Securitized products
134:
61:
2078:Stochastic analysis
1918:Quadratic variation
1913:Prokhorov's theorem
1848:Feynman–Kac formula
1318:Markov random field
966:Birth–death process
501:Infrastructure bond
383:Columbia University
268:Thomas S.Y. Ho Ph.D
178:lattice based model
2048:Probability theory
1928:Skorokhod integral
1898:Malliavin calculus
1481:Korn-Kreer-Lenssen
1365:Time series models
1328:Pitman–Yor process
577:Floating rate note
377:2012-01-23 at the
324:Journal of Finance
312:Primary references
300:2012-04-17 at the
282:, shanghai.nyu.edu
270:, thcdecisions.com
147:
117:
2192:Short-rate models
2149:
2148:
2114:
2113:
2068:Signal processing
1787:Doob's upcrossing
1782:Doob's martingale
1746:Engelbert–Schmidt
1689:Donsker's theorem
1623:Feller-continuous
1491:Rendleman–Bartter
1281:Dirichlet process
1198:Branching process
1167:Telegraph process
1060:Geometric process
1040:Empirical process
1030:Diffusion process
886:Branching process
881:Bernoulli process
831:
830:
784:Exchangeable bond
710:Yield to maturity
562:Exchangeable bond
484:Subordinated debt
2209:
2197:Financial models
2170:
2163:
2156:
2128:
2121:
2088:Machine learning
1975:Usual hypotheses
1858:Girsanov theorem
1843:Dynkin's formula
1608:Continuous paths
1516:Actuarial models
1456:Garman–Kohlhagen
1426:Black–Karasinski
1421:Black–Derman–Toy
1408:Financial models
1274:Fields and other
1203:Gaussian process
1152:Sigma-martingale
956:Additive process
858:
851:
844:
835:
774:Convertible bond
617:Zero-coupon bond
557:Convertible bond
542:Commercial paper
425:
418:
411:
402:
381:, Martin Haugh,
305:
289:
283:
277:
271:
265:
259:
241:
201:Black–Derman–Toy
156:
154:
153:
148:
146:
145:
126:
124:
123:
118:
116:
115:
89:
88:
76:
75:
29:short-rate model
2217:
2216:
2212:
2211:
2210:
2208:
2207:
2206:
2177:
2176:
2175:
2174:
2134:economic theory
2117:
2115:
2110:
2092:
2053:Queueing theory
1996:
1938:Skorokhod space
1801:
1792:Kunita–Watanabe
1763:
1729:Sanov's theorem
1699:Ergodic theorem
1672:
1668:Time-reversible
1586:
1549:Queueing models
1543:
1539:Sparre–Anderson
1529:Cramér–Lundberg
1510:
1496:SABR volatility
1402:
1359:
1311:Boolean network
1269:
1255:Renewal process
1186:
1135:Non-homogeneous
1125:Poisson process
1015:Contact process
978:Brownian motion
948:Continuous time
942:
936:Maximal entropy
867:
862:
832:
827:
798:
789:Extendible bond
779:Embedded option
755:
719:
621:
582:High-yield debt
572:Fixed rate bond
567:Extendible bond
515:
496:Government bond
491:Distressed debt
457:
434:
429:
379:Wayback Machine
359:
341:, 5th edition,
333:10.2307/2328161
309:
308:
302:Wayback Machine
290:
286:
278:
274:
266:
262:
242:
235:
225:
137:
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107:
80:
67:
59:
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17:
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11:
5:
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2172:
2165:
2158:
2150:
2147:
2146:
2129:
2112:
2111:
2109:
2108:
2103:
2101:List of topics
2097:
2094:
2093:
2091:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2058:Renewal theory
2055:
2050:
2045:
2040:
2035:
2030:
2025:
2023:Ergodic theory
2020:
2015:
2013:Control theory
2010:
2004:
2002:
1998:
1997:
1995:
1994:
1993:
1992:
1987:
1977:
1972:
1967:
1962:
1957:
1956:
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1945:
1943:Snell envelope
1940:
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1930:
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1761:
1742:Borel–Cantelli
1731:
1726:
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1701:
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1691:
1686:
1680:
1678:
1677:Limit theorems
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1298:
1296:Hopfield model
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1262:
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1215:
1213:Markov process
1210:
1205:
1200:
1194:
1192:
1188:
1187:
1185:
1184:
1182:Wiener sausage
1179:
1177:Wiener process
1174:
1169:
1164:
1159:
1157:Stable process
1154:
1149:
1147:Semimartingale
1144:
1139:
1138:
1137:
1132:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1090:Jump diffusion
1087:
1082:
1077:
1072:
1067:
1065:Hawkes process
1062:
1057:
1052:
1047:
1045:Feller process
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1010:Cauchy process
1007:
1006:
1005:
1000:
995:
990:
985:
975:
974:
973:
963:
961:Bessel process
958:
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712:
707:
702:
697:
692:
690:Risk-free bond
687:
682:
677:
675:Mortgage yield
672:
667:
662:
657:
652:
647:
642:
637:
631:
629:
627:Bond valuation
623:
622:
620:
619:
614:
609:
604:
602:Perpetual bond
599:
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589:
584:
579:
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559:
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506:Municipal bond
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474:Corporate bond
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430:
428:
427:
420:
413:
405:
399:
398:
397:, thomasho.com
386:
385:
369:
367:Wharton School
358:
357:External links
355:
354:
353:
337:John C. Hull,
335:
307:
306:
284:
272:
260:
232:
231:
224:
221:
197:mean reversion
184:valuations of
144:
140:
128:
127:
114:
110:
106:
102:
99:
96:
93:
87:
83:
79:
74:
70:
66:
52:normal process
45:interest rates
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13:
10:
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3:
2:
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2130:
2127:
2123:
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2107:
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2102:
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2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2005:
2003:
1999:
1991:
1988:
1986:
1983:
1982:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1960:Stopping time
1958:
1954:
1951:
1950:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
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1911:
1909:
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1901:
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1770:
1766:
1759:
1755:
1751:
1750:Hewitt–Savage
1747:
1743:
1739:
1735:
1734:Zero–one laws
1732:
1730:
1727:
1725:
1722:
1720:
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1560:
1558:
1555:
1554:
1552:
1550:
1546:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1521:
1519:
1517:
1513:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1431:Black–Scholes
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1413:
1411:
1409:
1405:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1370:
1368:
1366:
1362:
1356:
1353:
1351:
1348:
1344:
1341:
1339:
1336:
1335:
1334:
1333:Point process
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1312:
1309:
1307:
1304:
1303:
1302:
1299:
1297:
1294:
1292:
1291:Gibbs measure
1289:
1287:
1284:
1282:
1279:
1278:
1276:
1272:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1220:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1195:
1193:
1189:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1136:
1133:
1131:
1128:
1127:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1080:Itô diffusion
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1055:Gamma process
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
980:
979:
976:
972:
969:
968:
967:
964:
962:
959:
957:
954:
953:
951:
949:
945:
937:
934:
932:
929:
927:
926:Self-avoiding
924:
922:
919:
918:
917:
914:
912:
911:Moran process
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
878:
876:
874:
873:Discrete time
870:
866:
859:
854:
852:
847:
845:
840:
839:
836:
823:
820:
817:
814:
811:
808:
807:
805:
801:
795:
794:Puttable bond
792:
790:
787:
785:
782:
780:
777:
775:
772:
770:
769:Callable bond
767:
766:
764:
762:
758:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
728:
726:
722:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
680:Nominal yield
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
655:Current yield
653:
651:
650:Credit spread
648:
646:
643:
641:
638:
636:
633:
632:
630:
628:
624:
618:
615:
613:
610:
608:
607:Puttable bond
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
537:Callable bond
535:
533:
530:
528:
525:
524:
522:
518:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
485:
482:
480:
477:
476:
475:
472:
470:
467:
466:
464:
460:
454:
451:
449:
446:
444:
441:
440:
437:
433:
426:
421:
419:
414:
412:
407:
406:
403:
396:
393:
392:
391:
390:
384:
380:
376:
373:
370:
368:
364:
361:
360:
356:
352:
351:0-13-009056-5
348:
344:
343:Prentice Hall
340:
336:
334:
330:
326:
325:
320:
316:
315:
314:
313:
303:
299:
296:
295:
288:
285:
281:
276:
273:
269:
264:
261:
258:
257:0-470-10910-6
254:
250:
246:
240:
238:
234:
230:
229:
222:
220:
218:
214:
210:
206:
202:
198:
193:
191:
187:
183:
179:
176:
172:
168:
164:
160:
142:
138:
112:
108:
104:
100:
97:
94:
91:
85:
81:
77:
72:
68:
64:
57:
56:
55:
53:
48:
46:
42:
38:
34:
30:
26:
22:
2142:expanding it
2131:
2116:
2018:Econometrics
1980:Wiener space
1868:Itô integral
1769:Inequalities
1658:Self-similar
1628:Gauss–Markov
1618:Exchangeable
1598:Càdlàg paths
1534:Risk process
1486:LIBOR market
1470:
1355:Random graph
1350:Random field
1162:Superprocess
1100:Lévy process
1095:Jump process
1070:Hunt process
906:Markov chain
803:Institutions
761:Bond options
705:Yield spread
597:Lottery bond
527:Accrual bond
453:Fixed income
389:Online tools
388:
387:
338:
322:
318:
311:
310:
293:
287:
280:Sang Bin Lee
275:
263:
244:
227:
226:
194:
163:bond options
129:
49:
33:bond options
25:Ho-Lee model
24:
18:
2063:Ruin theory
2001:Disciplines
1873:Itô's lemma
1648:Predictable
1323:Percolation
1306:Potts model
1301:Ising model
1265:White noise
1223:Differences
1085:Itô process
1025:Cox process
921:Loop-erased
916:Random walk
700:Yield curve
660:Dirty price
635:Clean price
511:Global bond
479:Senior debt
469:Agency bond
432:Bond market
182:Closed form
159:yield curve
2181:Categories
2073:Statistics
1853:Filtration
1754:Kolmogorov
1738:Blumenthal
1663:Stationary
1603:Continuous
1591:Properties
1476:Hull–White
1218:Martingale
1105:Local time
993:Fractional
971:pure birth
327:41, 1986.
223:References
209:Hull–White
190:Black-like
169:and other
39:and other
1985:Classical
998:Geometric
988:Excursion
640:Convexity
448:Debenture
217:lognormal
205:lognormal
167:swaptions
139:θ
101:σ
82:θ
37:swaptions
2106:Category
1990:Abstract
1524:Bühlmann
1130:Compound
715:Z-spread
670:I-spread
665:Duration
375:Archived
298:Archived
175:binomial
1613:Ergodic
1501:Vašíček
1343:Poisson
1003:Meander
824:(SIFMA)
188:, and "
1953:Tanaka
1638:Mixing
1633:Markov
1506:Wilkie
1471:Ho–Lee
1466:Heston
1238:Super-
983:Bridge
931:Biased
818:(ICMA)
812:(CMSA)
645:Coupon
547:Consol
349:
255:
23:, the
2132:This
1806:Tools
1582:M/M/c
1577:M/M/1
1572:M/G/1
1562:Fluid
1228:Local
249:Wiley
228:Notes
215:is a
186:bonds
27:is a
2138:stub
1758:Lévy
1557:Bulk
1441:Chen
1233:Sub-
1191:Both
443:Bond
347:ISBN
253:ISBN
1338:Cox
329:doi
19:In
2183::
1756:,
1752:,
1748:,
1744:,
1740:,
345:,
321:,
251:.
247:.
236:^
180:.
165:,
54::
35:,
2169:e
2162:t
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2144:.
1760:)
1736:(
857:e
850:t
843:v
424:e
417:t
410:v
331::
203:(
143:t
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109:W
105:d
98:+
95:t
92:d
86:t
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73:t
69:r
65:d
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