146:
703:
436:
809:
1553:
248:
166:
Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.
1474:
2088:
525:
495:
151:
discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree (i.e. i=0);
625:
1912:
594:
555:
2515:
1508:
945:
465:
761:
2045:
2025:
288:-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and
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1462:
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1468:
997:
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66:
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305:
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Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous
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768:
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837:
597:
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factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the
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903:
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35:
2500:
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1980:
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709:
1004:
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1995:
1938:
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2050:
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1061:
562:
273:
174:
2730:
2570:
2495:
2300:
2060:
1970:
1860:
1153:
729:
502:
472:
1041:
2700:
2665:
2580:
2550:
2320:
2315:
2138:
1975:
1640:
1578:
1517:
1229:
929:
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713:
609:
2720:
2525:
2176:
1933:
1850:
1819:
1712:
1692:
1682:
1538:
1533:
1436:
1362:
1214:
1136:
792:
724:—are very easily applied to the calibration. Relatedly, the model was originally described in
568:
281:
277:
55:
find all other rates in the time-step, where these are linked to the node immediately above (r
2386:
2123:
532:
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1804:
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in 1990. A personal account of the development of the model is provided in
Emanuel Derman's
186:
155:
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1963:
1907:
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1441:
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1234:
1224:
1219:
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558:
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449:
810:"Society of Actuaries Professional Actuarial Specialty Guide Asset-Liability Management"
148:(this node-spacing being consistent with p = 50%; Δt being the length of the time-step);
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230:
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1990:
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1307:
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159:
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2007:
2002:
1814:
1747:
1722:
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1302:
1249:
1179:
1105:
967:
946:"A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options"
886:
797:
265:
2250:
762:"Impact of Different Interest Rate Models on Bond Value Measures, G, Buetow et al"
1023:
2715:
2234:
2229:
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2214:
2017:
1958:
1953:
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1084:
1003:. Technical Note No. 23, Options, Futures, and Other Derivatives. Archived from
269:
190:
268:
the model parameters to fit both the current term structure of interest rates (
2725:
2265:
2209:
2093:
206:
866:
2219:
1100:
725:
194:
45:
41:
154:
repeat until the discounted value at the first node in the tree equals the
17:
1367:
1322:
285:
1047:
841:
2046:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1486:
904:"One on One Interview with Emanuel Derman (Financial Engineering News)"
244:
972:"Calibrating the Black–Derman–Toy model: some theoretical results"
708:
One reason that the model remains popular, is that the "standard"
52:
adjust the rate at the top-most node at the current time-step, i;
891:
Calibrating the Black–Derman-Toy model: some theoretical results
141:{\displaystyle \ln(r_{u}/r_{d})/2=\sigma _{i}{\sqrt {\Delta t}}}
1490:
1057:
1053:
1042:
R function for computing the Black–Derman–Toy short rate tree
233:, and Bill Toy. It was first developed for in-house use by
2026:
Autoregressive conditional heteroskedasticity (ARCH) model
1554:
Independent and identically distributed random variables
944:
Black, F.; Derman, E.; Toy, W. (January–February 1990).
838:"My Life as a Quant: Reflections on Physics and Finance"
203:
Lattice model (finance) § Interest rate derivatives
698:{\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}
606:
For constant (time independent) short rate volatility,
2031:
Autoregressive integrated moving average (ARIMA) model
1029:. Seminar Financial Engineering, University of Vienna.
26:
1475:
Securities
Industry and Financial Markets Association
636:
612:
571:
535:
505:
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452:
308:
69:
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2016:
1926:
1843:
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1278:
1172:
1114:
1024:"Implementation of the Black, Derman and Toy Model"
431:{\displaystyle d\ln(r)=\leftdt+\sigma _{t}\,dW_{t}}
697:
619:
588:
549:
519:
489:
459:
430:
140:
1913:Stochastic chains with memory of variable length
497:= value of the underlying asset at option expiry
893:, Applied Mathematical Finance 8, 27– 48 (2001)
205:. It is a one-factor model; that is, a single
1502:
1069:
8:
2041:Autoregressive–moving-average (ARMA) model
1509:
1495:
1487:
1463:Commercial Mortgage Securities Association
1076:
1062:
1054:
689:
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662:
635:
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128:
122:
107:
98:
89:
83:
68:
1469:International Capital Market Association
889:, Ken Seng Tan and Weidong Tian (2001).
756:
754:
467:= the instantaneous short rate at time t
1048:Excel BDT calculator and tree generator
750:
2347:Doob's martingale convergence theorems
237:in the 1980s and was published in the
31:Short-rate tree calibration under BDT:
2099:Constant elasticity of variance (CEV)
2089:Chan–Karolyi–Longstaff–Sanders (CKLS)
937:Mathematica in Education and Research
213:behaviour of the short rate with the
7:
1399:Commercial mortgage-backed security
2586:Skorokhod's representation theorem
2367:Law of large numbers (weak/strong)
1394:Collateralized mortgage obligation
998:"The Black, Derman, and Toy Model"
130:
25:
2556:Martingale representation theorem
928:Benninga, S.; Wiener, Z. (1998).
2601:Stochastic differential equation
2491:Doob's optional stopping theorem
2486:Doob–Meyer decomposition theorem
930:"Binomial Term Structure Models"
297:stochastic differential equation
63:being the node in question) via
2471:Convergence of random variables
2357:Fisher–Tippett–Gnedenko theorem
527:= instant short rate volatility
2069:Binomial options pricing model
1389:Collateralized debt obligation
1265:Reverse convertible securities
652:
646:
387:
381:
324:
318:
217:, and is still widely used.
104:
76:
1:
2536:Kolmogorov continuity theorem
2372:Law of the iterated logarithm
520:{\displaystyle \sigma _{t}\,}
490:{\displaystyle \theta _{t}\,}
2541:Kolmogorov extension theorem
2220:Generalized queueing network
1728:Interacting particle systems
1022:Klose, C.; Li C. Y. (2003).
979:Applied Mathematical Finance
970:; Tan, K.; Tian, W. (2001).
225:The model was introduced by
1673:Continuous-time random walk
1205:Contingent convertible bond
158:corresponding to the given
2811:
2681:Extreme value theory (EVT)
2481:Doob decomposition theorem
1773:Ornstein–Uhlenbeck process
1544:Chinese restaurant process
1245:Inverse floating rate note
981:: 8, 27–48. Archived from
953:Financial Analysts Journal
240:Financial Analysts Journal
2749:
2561:Optional stopping theorem
2362:Large deviation principle
2114:Heath–Jarrow–Morton (HJM)
2051:Moving-average (MA) model
2036:Autoregressive (AR) model
1861:Hidden Markov model (HMM)
1795:Schramm–Loewner evolution
1091:
620:{\displaystyle \sigma \,}
290:interest rate derivatives
199:interest rate derivatives
2476:Doléans-Dade exponential
2306:Progressively measurable
2104:Cox–Ingersoll–Ross (CIR)
1404:Mortgage-backed security
1173:Types of bonds by payout
1115:Types of bonds by issuer
863:"Black-Derman-Toy (BDT)"
728:language, and not using
589:{\displaystyle dW_{t}\,}
38:of an up move, p, to 50%
36:risk-neutral probability
2696:Mathematical statistics
2686:Large deviations theory
2516:Infinitesimal generator
2377:Maximal ergodic theorem
2296:Piecewise-deterministic
1898:Random dynamical system
1763:Markov additive process
955:: 24–32. Archived from
710:Root-finding algorithms
550:{\displaystyle W_{t}\,}
215:log-normal distribution
189:used in the pricing of
162:for the i-th time-step.
40:Step 1. For each input
2531:Karhunen–Loève theorem
2466:Cameron–Martin formula
2430:Burkholder–Davis–Gundy
1825:Variance gamma process
1338:Option-adjusted spread
1240:Inflation-indexed bond
699:
621:
590:
551:
521:
491:
461:
432:
179:Black–Derman–Toy model
142:
2780:Fixed income analysis
2661:Actuarial mathematics
2623:Uniform integrability
2618:Stratonovich integral
2546:Lévy–Prokhorov metric
2450:Marcinkiewicz–Zygmund
2337:Central limit theorem
1939:Gaussian random field
1768:McKean–Vlasov process
1688:Dyson Brownian motion
1549:Galton–Watson process
1384:Asset-backed security
1348:Weighted-average life
1185:Auction rate security
793:Fixed Income Analysis
700:
622:
591:
565:probability measure;
552:
522:
492:
462:
433:
143:
2736:Time series analysis
2691:Mathematical finance
2576:Reflection principle
1903:Regenerative process
1703:Fleming–Viot process
1518:Stochastic processes
1377:Securitized products
634:
610:
569:
533:
503:
473:
450:
306:
274:volatility structure
175:mathematical finance
67:
2731:Stochastic analysis
2571:Quadratic variation
2566:Prokhorov's theorem
2501:Feynman–Kac formula
1971:Markov random field
1619:Birth–death process
1154:Infrastructure bond
730:stochastic calculus
460:{\displaystyle r\,}
362:
260:Under BDT, using a
2701:Probability theory
2581:Skorokhod integral
2551:Malliavin calculus
2134:Korn-Kreer-Lenssen
2018:Time series models
1981:Pitman–Yor process
1230:Floating rate note
695:
617:
586:
547:
517:
487:
457:
428:
350:
278:interest rate caps
249:My Life as a Quant
160:spot interest rate
138:
2795:Options (finance)
2785:Short-rate models
2767:
2766:
2721:Signal processing
2440:Doob's upcrossing
2435:Doob's martingale
2399:Engelbert–Schmidt
2342:Donsker's theorem
2276:Feller-continuous
2144:Rendleman–Bartter
1934:Dirichlet process
1851:Branching process
1820:Telegraph process
1713:Geometric process
1693:Empirical process
1683:Diffusion process
1539:Branching process
1534:Bernoulli process
1484:
1483:
1437:Exchangeable bond
1363:Yield to maturity
1215:Exchangeable bond
1137:Subordinated debt
373:
171:
170:
136:
16:(Redirected from
2802:
2790:Financial models
2741:Machine learning
2628:Usual hypotheses
2511:Girsanov theorem
2496:Dynkin's formula
2261:Continuous paths
2169:Actuarial models
2109:Garman–Kohlhagen
2079:Black–Karasinski
2074:Black–Derman–Toy
2061:Financial models
1927:Fields and other
1856:Gaussian process
1805:Sigma-martingale
1609:Additive process
1511:
1504:
1497:
1488:
1427:Convertible bond
1270:Zero-coupon bond
1210:Convertible bond
1195:Commercial paper
1078:
1071:
1064:
1055:
1044:, Andrea Ruberto
1030:
1028:
1018:
1016:
1015:
1009:
1002:
989:
987:
976:
963:
961:
950:
940:
934:
914:
913:
911:
910:
900:
894:
884:
878:
877:
875:
874:
865:. Archived from
859:
853:
852:
850:
849:
840:. Archived from
834:
828:
827:
825:
823:
814:
806:
800:
789:
783:
782:
780:
779:
773:
767:. Archived from
766:
758:
704:
702:
701:
696:
694:
693:
667:
666:
627:, the model is:
626:
624:
623:
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592:
587:
584:
583:
556:
554:
553:
548:
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496:
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429:
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413:
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394:
390:
374:
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358:
349:
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343:
262:binomial lattice
187:short-rate model
147:
145:
144:
139:
137:
129:
127:
126:
111:
103:
102:
93:
88:
87:
34:Step 0. Set the
27:
21:
2810:
2809:
2805:
2804:
2803:
2801:
2800:
2799:
2770:
2769:
2768:
2763:
2745:
2706:Queueing theory
2649:
2591:Skorokhod space
2454:
2445:Kunita–Watanabe
2416:
2382:Sanov's theorem
2352:Ergodic theorem
2325:
2321:Time-reversible
2239:
2202:Queueing models
2196:
2192:Sparre–Anderson
2182:Cramér–Lundberg
2163:
2149:SABR volatility
2055:
2012:
1964:Boolean network
1922:
1908:Renewal process
1839:
1788:Non-homogeneous
1778:Poisson process
1668:Contact process
1631:Brownian motion
1601:Continuous time
1595:
1589:Maximal entropy
1520:
1515:
1485:
1480:
1451:
1442:Extendible bond
1432:Embedded option
1408:
1372:
1274:
1235:High-yield debt
1225:Fixed rate bond
1220:Extendible bond
1168:
1149:Government bond
1144:Distressed debt
1110:
1087:
1082:
1038:
1033:
1026:
1021:
1013:
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1007:
1000:
992:
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959:
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790:
786:
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771:
764:
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759:
752:
742:
714:Newton's method
685:
658:
632:
631:
608:
607:
575:
567:
566:
559:Brownian motion
536:
531:
530:
506:
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500:
476:
471:
470:
448:
447:
418:
404:
363:
335:
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304:
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258:
223:
185:) is a popular
118:
94:
79:
65:
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62:
58:
39:
23:
22:
15:
12:
11:
5:
2808:
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2798:
2797:
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2787:
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2772:
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2765:
2764:
2762:
2761:
2756:
2754:List of topics
2750:
2747:
2746:
2744:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2711:Renewal theory
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2676:Ergodic theory
2673:
2668:
2666:Control theory
2663:
2657:
2655:
2651:
2650:
2648:
2647:
2646:
2645:
2640:
2630:
2625:
2620:
2615:
2610:
2609:
2608:
2598:
2596:Snell envelope
2593:
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2578:
2573:
2568:
2563:
2558:
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2538:
2533:
2528:
2523:
2518:
2513:
2508:
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2442:
2437:
2432:
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2424:
2418:
2417:
2415:
2414:
2395:Borel–Cantelli
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2333:
2331:
2330:Limit theorems
2327:
2326:
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2323:
2318:
2313:
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2240:
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2028:
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2020:
2014:
2013:
2011:
2010:
2005:
2000:
1999:
1998:
1993:
1983:
1978:
1973:
1968:
1967:
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1961:
1951:
1949:Hopfield model
1946:
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1930:
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1921:
1920:
1915:
1910:
1905:
1900:
1895:
1894:
1893:
1888:
1883:
1878:
1868:
1866:Markov process
1863:
1858:
1853:
1847:
1845:
1841:
1840:
1838:
1837:
1835:Wiener sausage
1832:
1830:Wiener process
1827:
1822:
1817:
1812:
1810:Stable process
1807:
1802:
1800:Semimartingale
1797:
1792:
1791:
1790:
1785:
1775:
1770:
1765:
1760:
1755:
1750:
1745:
1743:Jump diffusion
1740:
1735:
1730:
1725:
1720:
1718:Hawkes process
1715:
1710:
1705:
1700:
1698:Feller process
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1663:Cauchy process
1660:
1659:
1658:
1653:
1648:
1643:
1638:
1628:
1627:
1626:
1616:
1614:Bessel process
1611:
1605:
1603:
1597:
1596:
1594:
1593:
1592:
1591:
1586:
1581:
1576:
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1556:
1551:
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1522:
1521:
1516:
1514:
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1506:
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1482:
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1478:
1472:
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1459:
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1449:
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1439:
1434:
1429:
1424:
1418:
1416:
1410:
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1401:
1396:
1391:
1386:
1380:
1378:
1374:
1373:
1371:
1370:
1365:
1360:
1355:
1350:
1345:
1343:Risk-free bond
1340:
1335:
1330:
1328:Mortgage yield
1325:
1320:
1315:
1310:
1305:
1300:
1295:
1290:
1284:
1282:
1280:Bond valuation
1276:
1275:
1273:
1272:
1267:
1262:
1257:
1255:Perpetual bond
1252:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1176:
1174:
1170:
1169:
1167:
1166:
1161:
1159:Municipal bond
1156:
1151:
1146:
1141:
1140:
1139:
1134:
1127:Corporate bond
1124:
1118:
1116:
1112:
1111:
1109:
1108:
1103:
1098:
1092:
1089:
1088:
1083:
1081:
1080:
1073:
1066:
1058:
1052:
1051:
1045:
1037:
1036:External links
1034:
1032:
1031:
1019:
990:
988:on 2012-04-22.
964:
962:on 2008-09-10.
941:
939:: vol.7 No. 3.
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314:
311:
257:
254:
231:Emanuel Derman
222:
219:
211:mean-reverting
169:
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2:
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2679:
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2674:
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2659:
2658:
2656:
2652:
2644:
2641:
2639:
2636:
2635:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2613:Stopping time
2611:
2607:
2604:
2603:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
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2537:
2534:
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2524:
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2441:
2438:
2436:
2433:
2431:
2428:
2427:
2425:
2423:
2419:
2412:
2408:
2404:
2403:Hewitt–Savage
2400:
2396:
2392:
2388:
2387:Zero–one laws
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2334:
2332:
2328:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2248:
2246:
2242:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2207:
2205:
2203:
2199:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2172:
2170:
2166:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2084:Black–Scholes
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2066:
2064:
2062:
2058:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2023:
2021:
2019:
2015:
2009:
2006:
2004:
2001:
1997:
1994:
1992:
1989:
1988:
1987:
1986:Point process
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1965:
1962:
1960:
1957:
1956:
1955:
1952:
1950:
1947:
1945:
1944:Gibbs measure
1942:
1940:
1937:
1935:
1932:
1931:
1929:
1925:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1873:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1848:
1846:
1842:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1789:
1786:
1784:
1781:
1780:
1779:
1776:
1774:
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1733:Itô diffusion
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1708:Gamma process
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1633:
1632:
1629:
1625:
1622:
1621:
1620:
1617:
1615:
1612:
1610:
1607:
1606:
1604:
1602:
1598:
1590:
1587:
1585:
1582:
1580:
1579:Self-avoiding
1577:
1575:
1572:
1571:
1570:
1567:
1565:
1564:Moran process
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1531:
1529:
1527:
1526:Discrete time
1523:
1519:
1512:
1507:
1505:
1500:
1498:
1493:
1492:
1489:
1476:
1473:
1470:
1467:
1464:
1461:
1460:
1458:
1454:
1448:
1447:Puttable bond
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1422:Callable bond
1420:
1419:
1417:
1415:
1411:
1405:
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1381:
1379:
1375:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1333:Nominal yield
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1308:Current yield
1306:
1304:
1303:Credit spread
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1285:
1283:
1281:
1277:
1271:
1268:
1266:
1263:
1261:
1260:Puttable bond
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1190:Callable bond
1188:
1186:
1183:
1181:
1178:
1177:
1175:
1171:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1138:
1135:
1133:
1130:
1129:
1128:
1125:
1123:
1120:
1119:
1117:
1113:
1107:
1104:
1102:
1099:
1097:
1094:
1093:
1090:
1086:
1079:
1074:
1072:
1067:
1065:
1060:
1059:
1056:
1049:
1046:
1043:
1040:
1039:
1035:
1025:
1020:
1010:on 2011-01-29
1006:
999:
995:
991:
984:
980:
973:
969:
965:
958:
954:
947:
942:
938:
931:
926:
925:
924:
923:
905:
899:
896:
892:
888:
883:
880:
869:on 2016-05-24
868:
864:
858:
855:
844:on 2010-03-28
843:
839:
833:
830:
818:
811:
805:
802:
799:
796:, p. 410, at
795:
794:
788:
785:
774:on 2011-10-07
770:
763:
757:
755:
751:
747:
746:
739:
737:
735:
731:
727:
723:
719:
718:secant method
715:
711:
690:
686:
682:
678:
675:
672:
669:
663:
659:
655:
649:
643:
640:
637:
630:
629:
628:
613:
599:
580:
576:
572:
564:
560:
557:= a standard
541:
537:
529:
511:
507:
499:
481:
477:
469:
453:
446:
443:
442:
441:
440:
423:
419:
415:
409:
405:
401:
398:
395:
391:
384:
378:
375:
368:
364:
359:
355:
351:
345:
340:
336:
331:
327:
321:
315:
312:
309:
302:
301:
300:
298:
293:
291:
287:
283:
279:
275:
271:
267:
263:
255:
253:
251:
250:
246:
242:
241:
236:
235:Goldman Sachs
232:
228:
227:Fischer Black
220:
218:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
167:
161:
157:
153:
150:
133:
123:
119:
115:
112:
108:
99:
95:
90:
84:
80:
73:
70:
54:
51:
50:
49:
47:
43:
37:
32:
29:
28:
19:
2671:Econometrics
2633:Wiener space
2521:Itô integral
2422:Inequalities
2311:Self-similar
2281:Gauss–Markov
2271:Exchangeable
2251:Càdlàg paths
2187:Risk process
2139:LIBOR market
2073:
2008:Random graph
2003:Random field
1815:Superprocess
1753:Lévy process
1748:Jump process
1723:Hunt process
1559:Markov chain
1456:Institutions
1414:Bond options
1358:Yield spread
1250:Lottery bond
1180:Accrual bond
1106:Fixed income
1050:, Serkan Gur
1012:. Retrieved
1005:the original
983:the original
978:
957:the original
952:
936:
921:
920:
907:. Retrieved
898:
887:Phelim Boyle
882:
871:. Retrieved
867:the original
857:
846:. Retrieved
842:the original
832:
820:. Retrieved
816:
804:
798:Google Books
791:
787:
776:. Retrieved
769:the original
744:
743:
707:
605:
598:differential
563:risk-neutral
294:
259:
247:
238:
224:
191:bond options
182:
178:
172:
165:
33:
30:
2716:Ruin theory
2654:Disciplines
2526:Itô's lemma
2301:Predictable
1976:Percolation
1959:Potts model
1954:Ising model
1918:White noise
1876:Differences
1738:Itô process
1678:Cox process
1574:Loop-erased
1569:Random walk
1353:Yield curve
1313:Dirty price
1288:Clean price
1164:Global bond
1132:Senior debt
1122:Agency bond
1085:Bond market
734:martingales
726:algorithmic
272:), and the
270:yield curve
46:iteratively
18:William Toy
2774:Categories
2726:Statistics
2506:Filtration
2407:Kolmogorov
2391:Blumenthal
2316:Stationary
2256:Continuous
2244:Properties
2129:Hull–White
1871:Martingale
1758:Local time
1646:Fractional
1624:pure birth
1014:2011-04-08
909:2021-06-09
873:2010-06-14
848:2010-04-26
778:2011-07-21
740:References
282:as implied
266:calibrates
207:stochastic
197:and other
156:zero-price
2638:Classical
1651:Geometric
1641:Excursion
1293:Convexity
1101:Debenture
968:Boyle, P.
722:bisection
712:—such as
679:σ
660:θ
644:
614:σ
508:σ
478:θ
406:σ
379:
365:σ
352:σ
337:θ
316:
280:(usually
195:swaptions
131:Δ
120:σ
74:
42:spot rate
2759:Category
2643:Abstract
2177:Bühlmann
1783:Compound
1368:Z-spread
1323:I-spread
1318:Duration
996:(2008).
994:Hull, J.
922:Articles
822:19 March
561:under a
360:′
286:Black-76
256:Formulae
2266:Ergodic
2154:Vašíček
1996:Poisson
1656:Meander
1477:(SIFMA)
817:soa.org
284:by the
221:History
2606:Tanaka
2291:Mixing
2286:Markov
2159:Wilkie
2124:Ho–Lee
2119:Heston
1891:Super-
1636:Bridge
1584:Biased
1471:(ICMA)
1465:(CMSA)
1298:Coupon
1200:Consol
444:where,
264:, one
245:memoir
201:; see
177:, the
2459:Tools
2235:M/M/c
2230:M/M/1
2225:M/G/1
2215:Fluid
1881:Local
1027:(PDF)
1008:(PDF)
1001:(PDF)
986:(PDF)
975:(PDF)
960:(PDF)
949:(PDF)
933:(PDF)
813:(PDF)
772:(PDF)
765:(PDF)
745:Notes
720:) or
716:(the
2411:Lévy
2210:Bulk
2094:Chen
1886:Sub-
1844:Both
1096:Bond
824:2024
596:its
276:for
1991:Cox
732:or
183:BDT
173:In
59:; r
2776::
2409:,
2405:,
2401:,
2397:,
2393:,
977:.
951:.
935:.
815:.
753:^
736:.
641:ln
376:ln
313:ln
299::
292:.
252:.
229:,
193:,
71:ln
48::
44:,
2413:)
2389:(
1510:e
1503:t
1496:v
1077:e
1070:t
1063:v
1017:.
912:.
876:.
851:.
826:.
781:.
691:t
687:W
683:d
676:+
673:t
670:d
664:t
656:=
653:)
650:r
647:(
638:d
600:.
581:t
577:W
573:d
542:t
538:W
512:t
482:t
454:r
424:t
420:W
416:d
410:t
402:+
399:t
396:d
392:]
388:)
385:r
382:(
369:t
356:t
346:+
341:t
332:[
328:=
325:)
322:r
319:(
310:d
181:(
134:t
124:i
116:=
113:2
109:/
105:)
100:d
96:r
91:/
85:u
81:r
77:(
61:d
57:u
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.