20:
753:
prices for instance, interest rates cannot rise indefinitely. This is because at very high levels they would hamper economic activity, prompting a decrease in interest rates. Similarly, interest rates do not usually decrease below 0. As a result, interest rates move in a limited range, showing a
1921:
1123:
1503:
1343:
926:
The main disadvantage is that, under
Vasicek's model, it is theoretically possible for the interest rate to become negative, an undesirable feature under pre-crisis assumptions. This shortcoming was fixed in the
1234:
165:
1409:
1741:
1654:
2674:
685:
2595:
1747:
500:
3209:
965:
917:
797:
3033:
848:
708:
583:
563:
447:
427:
253:
203:
280:
3636:
2629:
233:
3166:
3146:
1555:
1535:
728:
643:
623:
603:
543:
520:
405:
385:
365:
343:
323:
300:
3550:
2583:
3467:
181:
under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system. The
3477:
3151:
2589:
3161:
1420:
3519:
3234:
3416:
3706:
3696:
3219:
2196:
2140:
2121:
2102:
3606:
3570:
1245:
3523:
3874:
3611:
2519:
2721:
2622:
2514:
3676:
3254:
3224:
1943:
948:
928:
3527:
3511:
3721:
3426:
2646:
2169:
76:
3626:
3591:
3560:
3555:
3194:
2991:
2908:
932:
3565:
2893:
1932:
1129:
734:
3905:
3189:
2996:
2509:
2385:
1965:
209:
of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters
2915:
1138:
3651:
3531:
2002:
Mahdavi
Damghani B. (2013). "The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model".
85:
3879:
3656:
3492:
3391:
3376:
2788:
2704:
2615:
3666:
3302:
3661:
2160:
1354:
3264:
1665:
749:, an essential characteristic of the interest rate that sets it apart from other financial prices. Thus, as opposed to
2848:
2793:
2709:
2325:
944:
60:
3596:
3586:
3229:
3199:
936:
56:
3910:
3601:
2766:
2664:
2365:
3312:
2888:
2669:
1563:
429:: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher
3915:
3681:
3482:
3396:
3381:
2771:
2189:
3515:
3401:
2823:
2903:
2878:
2524:
746:
651:
52:
3621:
3204:
2739:
23:
A trajectory of the short rate and the corresponding yield curves at T=0 (purple) and two later points in time
3900:
3816:
3806:
3497:
3279:
3018:
2883:
2694:
1916:{\displaystyle A(t,T)=\left(b-{\frac {\sigma ^{2}}{2a^{2}}}\right)\left-{\frac {\sigma ^{2}}{4a}}B^{2}(t,T)}
3101:
605:
amounts to increasing the speed at which the system will stabilize statistically around the long term mean
3758:
3686:
2945:
2458:
2360:
1974:
1118:{\displaystyle r_{t}=r_{0}e^{-at}+b\left(1-e^{-at}\right)+\sigma e^{-at}\int _{0}^{t}e^{as}\,dW_{s}.\,\!}
3781:
3763:
3743:
3738:
3457:
3289:
3269:
3116:
3059:
2898:
2808:
2504:
2468:
2305:
2264:
459:
3249:
1938:
940:
737:. Making the long term mean stochastic to another SDE is a simplified version of the cointelation SDE.
3856:
3811:
3801:
3542:
3487:
3462:
3431:
3411:
3171:
3156:
3023:
2182:
206:
28:
3851:
3691:
3616:
3421:
3181:
3091:
2981:
2274:
1979:
3821:
3786:
3701:
3671:
3441:
3436:
3259:
3096:
2761:
2699:
2638:
2350:
2068:
2042:
182:
36:
3502:
2156:
880:
760:
3841:
3646:
3297:
3054:
2971:
2940:
2833:
2813:
2803:
2659:
2654:
2557:
2483:
2335:
2257:
2136:
2117:
2098:
2060:
817:
282:, completely characterize the dynamics, and can be quickly characterized as follows, assuming
72:
3507:
3244:
693:
568:
548:
432:
412:
238:
188:
3861:
3748:
3631:
3001:
2976:
2925:
2853:
2776:
2729:
2547:
2418:
2390:
2330:
2315:
2052:
2011:
1984:
1514:
863:
44:
258:
3826:
3726:
3711:
3472:
3406:
3084:
3028:
3011:
2756:
2562:
2552:
2355:
2345:
2340:
2269:
3641:
2873:
212:
2165:
585:
increases the amount of randomness entering the system, but at the same time increasing
3831:
3796:
3716:
3322:
3069:
2986:
2955:
2950:
2930:
2920:
2863:
2858:
2838:
2818:
2783:
2751:
2734:
2463:
2448:
2413:
2400:
2375:
2279:
2247:
2216:
2087:
2030:
1540:
1520:
713:
628:
608:
588:
528:
505:
390:
370:
350:
328:
308:
285:
178:
3894:
3733:
3111:
3106:
3064:
3006:
2828:
2744:
2684:
2567:
2542:
2453:
2438:
2428:
2380:
2320:
2310:
2094:
2072:
1988:
40:
2031:"Forecasting interest rates through Vasicek and CIR models: A partitioning approach"
923:, generating a tendency for the interest rate to move upwards (toward equilibrium).
3791:
3753:
3307:
3239:
3128:
3123:
2935:
2868:
2843:
2679:
2478:
2423:
2370:
2300:
2226:
811:
750:
3371:
951:. In recent research both models were used for data partitioning and forecasting.
814:
value towards which the interest rate reverts. Indeed, in the absence of shocks (
3836:
3355:
3350:
3345:
3335:
3138:
3079:
3074:
3038:
2798:
2689:
2534:
2473:
2433:
2408:
2284:
2252:
2242:
2205:
1132:
stochastic process we get that state variable is distributed normally with mean
48:
3846:
3386:
3330:
3214:
2064:
1963:
Vasicek, O. (1977). "An equilibrium characterization of the term structure".
1498:{\displaystyle \lim _{t\to \infty }\mathrm {Var} ={\frac {\sigma ^{2}}{2a}}.}
55:, and has also been adapted for credit markets. It was introduced in 1977 by
3340:
2221:
522:
will regroup around the long term mean with such variance after a long time.
2114:
Interest Rate Models – Theory and
Practice with Smile, Inflation and Credit
799:
represents the expected instantaneous change in the interest rate at time
387:
characterizes the velocity at which such trajectories will regroup around
19:
2488:
2443:
808:
47:
as it describes interest rate movements as driven by only one source of
3167:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
2607:
2015:
2056:
2029:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (July 2020).
1338:{\displaystyle \mathrm {Var} ={\frac {\sigma ^{2}}{2a}}(1-e^{-2at}).}
862:, governing the speed of adjustment, needs to be positive to ensure
2047:
939:, among many others. The Vasicek model was further extended in the
18:
2611:
2178:
2174:
2166:
Yield Curve
Estimation and Prediction with the Vasicek Model
3147:
Autoregressive conditional heteroskedasticity (ARCH) model
2675:
Independent and identically distributed random variables
959:
We solve the stochastic differential equation to obtain
645:. This is clear when looking at the long term variance,
943:. The Vasicek model is also a canonical example of the
3152:
Autoregressive integrated moving average (ARIMA) model
1229:{\displaystyle \mathrm {E} =r_{0}e^{-at}+b(1-e^{-at})}
2596:
Securities
Industry and Financial Markets Association
1750:
1668:
1566:
1543:
1523:
1423:
1357:
1248:
1141:
968:
883:
820:
763:
716:
696:
654:
631:
611:
591:
571:
551:
531:
508:
462:
435:
415:
393:
373:
353:
331:
325:: "long term mean level". All future trajectories of
311:
288:
261:
241:
215:
191:
160:{\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}}
88:
453:
The following derived quantity is also of interest,
3774:
3579:
3541:
3450:
3364:
3321:
3288:
3180:
3137:
3047:
2964:
2720:
2645:
2576:
2533:
2497:
2399:
2293:
2235:
502:: "long term variance". All future trajectories of
2086:
1915:
1735:
1648:
1549:
1529:
1497:
1404:{\displaystyle \lim _{t\to \infty }\mathrm {E} =b}
1403:
1337:
1228:
1117:
911:
842:
791:
722:
702:
679:
637:
617:
597:
577:
557:
537:
514:
494:
441:
421:
399:
379:
359:
345:will evolve around a mean level b in the long run;
337:
317:
294:
274:
247:
227:
197:
159:
1736:{\displaystyle B(t,T)={\frac {1-e^{-a(T-t)}}{a}}}
1114:
3034:Stochastic chains with memory of variable length
1425:
1359:
625:with a corridor of variance determined also by
866:around the long term value. For example, when
2623:
2190:
1557:is exponential affine in the interest rate:
1517:may be priced in the Vasicek model. The time
745:Vasicek's model was the first one to capture
8:
1649:{\displaystyle P(t,T)=e^{A(t,T)-B(t,T)r(t)}}
1537:value of a discount bond with maturity date
51:. The model can be used in the valuation of
1128:Using similar techniques as applied to the
850:), the interest rate remains constant when
3162:Autoregressive–moving-average (ARMA) model
2630:
2616:
2608:
2584:Commercial Mortgage Securities Association
2197:
2183:
2175:
2116:(2nd ed. 2006 ed.). Springer Verlag.
2046:
1978:
1892:
1872:
1866:
1803:
1789:
1783:
1749:
1703:
1690:
1667:
1592:
1565:
1542:
1522:
1476:
1470:
1458:
1440:
1428:
1422:
1386:
1374:
1362:
1356:
1314:
1285:
1279:
1267:
1249:
1247:
1211:
1180:
1170:
1154:
1142:
1140:
1113:
1104:
1096:
1087:
1077:
1072:
1056:
1029:
996:
986:
973:
967:
900:
882:
828:
819:
780:
762:
715:
695:
680:{\displaystyle {\frac {\sigma ^{2}}{2a}}}
661:
655:
653:
630:
610:
590:
570:
550:
530:
507:
475:
468:
463:
461:
434:
414:
392:
372:
352:
330:
310:
287:
266:
260:
240:
214:
190:
151:
143:
130:
121:
96:
87:
2590:International Capital Market Association
754:tendency to revert to a long run value.
1955:
3468:Doob's martingale convergence theorems
2112:Damiano Brigo, Fabio Mercurio (2001).
2089:Options, Futures and Other Derivatives
565:tend to oppose each other: increasing
255:, together with the initial condition
3220:Constant elasticity of variance (CEV)
3210:Chan–Karolyi–Longstaff–Sanders (CKLS)
1513:Under the no-arbitrage assumption, a
735:Ornstein–Uhlenbeck stochastic process
7:
16:Mathematical model of interest rates
2520:Commercial mortgage-backed security
2131:Jessica James, Nick Webber (2000).
3707:Skorokhod's representation theorem
3488:Law of large numbers (weak/strong)
2515:Collateralized mortgage obligation
1447:
1444:
1441:
1435:
1375:
1369:
1256:
1253:
1250:
1143:
495:{\displaystyle {\sigma ^{2}}/(2a)}
14:
3677:Martingale representation theorem
3722:Stochastic differential equation
3612:Doob's optional stopping theorem
3607:Doob–Meyer decomposition theorem
2170:Middle East Technical University
77:stochastic differential equation
3592:Convergence of random variables
3478:Fisher–Tippett–Gnedenko theorem
3190:Binomial options pricing model
2510:Collateralized debt obligation
2386:Reverse convertible securities
1966:Journal of Financial Economics
1910:
1898:
1855:
1843:
1837:
1825:
1766:
1754:
1722:
1710:
1684:
1672:
1641:
1635:
1629:
1617:
1608:
1596:
1582:
1570:
1464:
1451:
1432:
1392:
1379:
1366:
1329:
1301:
1273:
1260:
1223:
1198:
1160:
1147:
919:becomes positive for positive
906:
887:
786:
767:
489:
480:
127:
108:
1:
3657:Kolmogorov continuity theorem
3493:Law of the iterated logarithm
931:, exponential Vasicek model,
71:The model specifies that the
43:. It is a type of one-factor
3662:Kolmogorov extension theorem
3341:Generalized queueing network
2849:Interacting particle systems
2161:Wisconsin School of Business
1989:10.1016/0304-405X(77)90016-2
955:Asymptotic mean and variance
59:, and can be also seen as a
39:describing the evolution of
2794:Continuous-time random walk
2326:Contingent convertible bond
945:affine term structure model
73:instantaneous interest rate
61:stochastic investment model
3932:
3802:Extreme value theory (EVT)
3602:Doob decomposition theorem
2894:Ornstein–Uhlenbeck process
2665:Chinese restaurant process
2366:Inverse floating rate note
2093:. Upper Saddle River, NJ:
1933:Ornstein–Uhlenbeck process
912:{\displaystyle a(b-r_{t})}
792:{\displaystyle a(b-r_{t})}
3870:
3682:Optional stopping theorem
3483:Large deviation principle
3235:Heath–Jarrow–Morton (HJM)
3172:Moving-average (MA) model
3157:Autoregressive (AR) model
2982:Hidden Markov model (HMM)
2916:Schramm–Loewner evolution
2212:
53:interest rate derivatives
3597:Doléans-Dade exponential
3427:Progressively measurable
3225:Cox–Ingersoll–Ross (CIR)
2525:Mortgage-backed security
2294:Types of bonds by payout
2236:Types of bonds by issuer
1944:Cox–Ingersoll–Ross model
949:Cox–Ingersoll–Ross model
929:Cox–Ingersoll–Ross model
843:{\displaystyle dW_{t}=0}
367:: "speed of reversion".
3817:Mathematical statistics
3807:Large deviations theory
3637:Infinitesimal generator
3498:Maximal ergodic theorem
3417:Piecewise-deterministic
3019:Random dynamical system
2884:Markov additive process
2133:Interest Rate Modelling
703:{\displaystyle \sigma }
578:{\displaystyle \sigma }
558:{\displaystyle \sigma }
449:implies more randomness
442:{\displaystyle \sigma }
422:{\displaystyle \sigma }
248:{\displaystyle \sigma }
198:{\displaystyle \sigma }
3652:Karhunen–Loève theorem
3587:Cameron–Martin formula
3551:Burkholder–Davis–Gundy
2946:Variance gamma process
2459:Option-adjusted spread
2361:Inflation-indexed bond
2085:Hull, John C. (2003).
2035:Journal of Forecasting
1917:
1737:
1650:
1551:
1531:
1499:
1405:
1348:Consequently, we have
1339:
1230:
1119:
937:Black–Karasinski model
933:Black–Derman–Toy model
913:
844:
793:
724:
704:
681:
639:
619:
599:
579:
559:
539:
516:
496:
443:
423:
401:
381:
361:
339:
319:
296:
276:
249:
229:
199:
161:
24:
3906:Fixed income analysis
3782:Actuarial mathematics
3744:Uniform integrability
3739:Stratonovich integral
3667:Lévy–Prokhorov metric
3571:Marcinkiewicz–Zygmund
3458:Central limit theorem
3060:Gaussian random field
2889:McKean–Vlasov process
2809:Dyson Brownian motion
2670:Galton–Watson process
2505:Asset-backed security
2469:Weighted-average life
2306:Auction rate security
1918:
1738:
1651:
1552:
1532:
1500:
1406:
1340:
1231:
1120:
914:
845:
794:
725:
705:
690:which increases with
682:
640:
620:
600:
580:
560:
540:
517:
497:
444:
424:
402:
382:
362:
340:
320:
297:
277:
275:{\displaystyle r_{0}}
250:
230:
200:
162:
22:
3857:Time series analysis
3812:Mathematical finance
3697:Reflection principle
3024:Regenerative process
2824:Fleming–Viot process
2639:Stochastic processes
2498:Securitized products
1748:
1666:
1564:
1541:
1521:
1421:
1355:
1246:
1139:
966:
881:
818:
761:
714:
694:
652:
629:
609:
589:
569:
549:
529:
506:
460:
433:
413:
391:
371:
351:
329:
309:
302:to be non-negative:
286:
259:
239:
213:
189:
86:
3852:Stochastic analysis
3692:Quadratic variation
3687:Prokhorov's theorem
3622:Feynman–Kac formula
3092:Markov random field
2740:Birth–death process
2275:Infrastructure bond
1082:
710:but decreases with
228:{\displaystyle b,a}
3822:Probability theory
3702:Skorokhod integral
3672:Malliavin calculus
3255:Korn-Kreer-Lenssen
3139:Time series models
3102:Pitman–Yor process
2351:Floating rate note
2016:10.1002/wilm.10252
1913:
1733:
1646:
1547:
1527:
1495:
1439:
1401:
1373:
1335:
1226:
1130:Ornstein–Uhlenbeck
1115:
1068:
909:
840:
789:
720:
700:
677:
635:
615:
595:
575:
555:
535:
512:
492:
439:
419:
397:
377:
357:
335:
315:
292:
272:
245:
225:
195:
183:standard deviation
157:
37:mathematical model
25:
3911:Short-rate models
3888:
3887:
3842:Signal processing
3561:Doob's upcrossing
3556:Doob's martingale
3520:Engelbert–Schmidt
3463:Donsker's theorem
3397:Feller-continuous
3265:Rendleman–Bartter
3055:Dirichlet process
2972:Branching process
2941:Telegraph process
2834:Geometric process
2814:Empirical process
2804:Diffusion process
2660:Branching process
2655:Bernoulli process
2605:
2604:
2558:Exchangeable bond
2484:Yield to maturity
2336:Exchangeable bond
2258:Subordinated debt
2157:The Vasicek Model
2142:978-0-471-97523-6
2123:978-3-540-22149-4
2104:978-0-13-009056-0
1886:
1810:
1731:
1550:{\displaystyle T}
1530:{\displaystyle t}
1490:
1424:
1358:
1299:
947:, along with the
877:, the drift term
757:The drift factor
733:This model is an
723:{\displaystyle a}
675:
638:{\displaystyle a}
618:{\displaystyle b}
598:{\displaystyle a}
538:{\displaystyle a}
515:{\displaystyle r}
400:{\displaystyle b}
380:{\displaystyle a}
360:{\displaystyle a}
338:{\displaystyle r}
318:{\displaystyle b}
295:{\displaystyle a}
205:, determines the
3923:
3916:Financial models
3862:Machine learning
3749:Usual hypotheses
3632:Girsanov theorem
3617:Dynkin's formula
3382:Continuous paths
3290:Actuarial models
3230:Garman–Kohlhagen
3200:Black–Karasinski
3195:Black–Derman–Toy
3182:Financial models
3048:Fields and other
2977:Gaussian process
2926:Sigma-martingale
2730:Additive process
2632:
2625:
2618:
2609:
2548:Convertible bond
2391:Zero-coupon bond
2331:Convertible bond
2316:Commercial paper
2199:
2192:
2185:
2176:
2159:, Bjørn Eraker,
2146:
2127:
2108:
2092:
2077:
2076:
2057:10.1002/for.2642
2050:
2026:
2020:
2019:
2004:Wilmott Magazine
1999:
1993:
1992:
1982:
1960:
1939:Hull–White model
1922:
1920:
1919:
1914:
1897:
1896:
1887:
1885:
1877:
1876:
1867:
1862:
1858:
1816:
1812:
1811:
1809:
1808:
1807:
1794:
1793:
1784:
1742:
1740:
1739:
1734:
1732:
1727:
1726:
1725:
1691:
1655:
1653:
1652:
1647:
1645:
1644:
1556:
1554:
1553:
1548:
1536:
1534:
1533:
1528:
1504:
1502:
1501:
1496:
1491:
1489:
1481:
1480:
1471:
1463:
1462:
1450:
1438:
1410:
1408:
1407:
1402:
1391:
1390:
1378:
1372:
1344:
1342:
1341:
1336:
1328:
1327:
1300:
1298:
1290:
1289:
1280:
1272:
1271:
1259:
1235:
1233:
1232:
1227:
1222:
1221:
1191:
1190:
1175:
1174:
1159:
1158:
1146:
1124:
1122:
1121:
1116:
1109:
1108:
1095:
1094:
1081:
1076:
1067:
1066:
1045:
1041:
1040:
1039:
1007:
1006:
991:
990:
978:
977:
941:Hull–White model
918:
916:
915:
910:
905:
904:
858:. The parameter
849:
847:
846:
841:
833:
832:
803:. The parameter
798:
796:
795:
790:
785:
784:
729:
727:
726:
721:
709:
707:
706:
701:
686:
684:
683:
678:
676:
674:
666:
665:
656:
644:
642:
641:
636:
624:
622:
621:
616:
604:
602:
601:
596:
584:
582:
581:
576:
564:
562:
561:
556:
544:
542:
541:
536:
521:
519:
518:
513:
501:
499:
498:
493:
479:
474:
473:
472:
448:
446:
445:
440:
428:
426:
425:
420:
406:
404:
403:
398:
386:
384:
383:
378:
366:
364:
363:
358:
344:
342:
341:
336:
324:
322:
321:
316:
301:
299:
298:
293:
281:
279:
278:
273:
271:
270:
254:
252:
251:
246:
234:
232:
231:
226:
204:
202:
201:
196:
166:
164:
163:
158:
156:
155:
126:
125:
101:
100:
45:short-rate model
3931:
3930:
3926:
3925:
3924:
3922:
3921:
3920:
3891:
3890:
3889:
3884:
3866:
3827:Queueing theory
3770:
3712:Skorokhod space
3575:
3566:Kunita–Watanabe
3537:
3503:Sanov's theorem
3473:Ergodic theorem
3446:
3442:Time-reversible
3360:
3323:Queueing models
3317:
3313:Sparre–Anderson
3303:Cramér–Lundberg
3284:
3270:SABR volatility
3176:
3133:
3085:Boolean network
3043:
3029:Renewal process
2960:
2909:Non-homogeneous
2899:Poisson process
2789:Contact process
2752:Brownian motion
2722:Continuous time
2716:
2710:Maximal entropy
2641:
2636:
2606:
2601:
2572:
2563:Extendible bond
2553:Embedded option
2529:
2493:
2395:
2356:High-yield debt
2346:Fixed rate bond
2341:Extendible bond
2289:
2270:Government bond
2265:Distressed debt
2231:
2208:
2203:
2153:
2143:
2130:
2124:
2111:
2105:
2084:
2081:
2080:
2028:
2027:
2023:
2001:
2000:
1996:
1962:
1961:
1957:
1952:
1929:
1888:
1878:
1868:
1821:
1817:
1799:
1795:
1785:
1776:
1772:
1746:
1745:
1699:
1692:
1664:
1663:
1588:
1562:
1561:
1539:
1538:
1519:
1518:
1511:
1482:
1472:
1454:
1419:
1418:
1382:
1353:
1352:
1310:
1291:
1281:
1263:
1244:
1243:
1207:
1176:
1166:
1150:
1137:
1136:
1100:
1083:
1052:
1025:
1018:
1014:
992:
982:
969:
964:
963:
957:
896:
879:
878:
871:
855:
824:
816:
815:
807:represents the
776:
759:
758:
743:
712:
711:
692:
691:
667:
657:
650:
649:
627:
626:
607:
606:
587:
586:
567:
566:
547:
546:
527:
526:
504:
503:
464:
458:
457:
431:
430:
411:
410:
389:
388:
369:
368:
349:
348:
327:
326:
307:
306:
284:
283:
262:
257:
256:
237:
236:
211:
210:
187:
186:
175:
147:
117:
92:
84:
83:
69:
57:OldĹ™ich VašĂÄŤek
17:
12:
11:
5:
3929:
3927:
3919:
3918:
3913:
3908:
3903:
3901:Interest rates
3893:
3892:
3886:
3885:
3883:
3882:
3877:
3875:List of topics
3871:
3868:
3867:
3865:
3864:
3859:
3854:
3849:
3844:
3839:
3834:
3832:Renewal theory
3829:
3824:
3819:
3814:
3809:
3804:
3799:
3797:Ergodic theory
3794:
3789:
3787:Control theory
3784:
3778:
3776:
3772:
3771:
3769:
3768:
3767:
3766:
3761:
3751:
3746:
3741:
3736:
3731:
3730:
3729:
3719:
3717:Snell envelope
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3583:
3581:
3577:
3576:
3574:
3573:
3568:
3563:
3558:
3553:
3547:
3545:
3539:
3538:
3536:
3535:
3516:Borel–Cantelli
3505:
3500:
3495:
3490:
3485:
3480:
3475:
3470:
3465:
3460:
3454:
3452:
3451:Limit theorems
3448:
3447:
3445:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3368:
3366:
3362:
3361:
3359:
3358:
3353:
3348:
3343:
3338:
3333:
3327:
3325:
3319:
3318:
3316:
3315:
3310:
3305:
3300:
3294:
3292:
3286:
3285:
3283:
3282:
3277:
3272:
3267:
3262:
3257:
3252:
3247:
3242:
3237:
3232:
3227:
3222:
3217:
3212:
3207:
3202:
3197:
3192:
3186:
3184:
3178:
3177:
3175:
3174:
3169:
3164:
3159:
3154:
3149:
3143:
3141:
3135:
3134:
3132:
3131:
3126:
3121:
3120:
3119:
3114:
3104:
3099:
3094:
3089:
3088:
3087:
3082:
3072:
3070:Hopfield model
3067:
3062:
3057:
3051:
3049:
3045:
3044:
3042:
3041:
3036:
3031:
3026:
3021:
3016:
3015:
3014:
3009:
3004:
2999:
2989:
2987:Markov process
2984:
2979:
2974:
2968:
2966:
2962:
2961:
2959:
2958:
2956:Wiener sausage
2953:
2951:Wiener process
2948:
2943:
2938:
2933:
2931:Stable process
2928:
2923:
2921:Semimartingale
2918:
2913:
2912:
2911:
2906:
2896:
2891:
2886:
2881:
2876:
2871:
2866:
2864:Jump diffusion
2861:
2856:
2851:
2846:
2841:
2839:Hawkes process
2836:
2831:
2826:
2821:
2819:Feller process
2816:
2811:
2806:
2801:
2796:
2791:
2786:
2784:Cauchy process
2781:
2780:
2779:
2774:
2769:
2764:
2759:
2749:
2748:
2747:
2737:
2735:Bessel process
2732:
2726:
2724:
2718:
2717:
2715:
2714:
2713:
2712:
2707:
2702:
2697:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2651:
2649:
2643:
2642:
2637:
2635:
2634:
2627:
2620:
2612:
2603:
2602:
2600:
2599:
2593:
2587:
2580:
2578:
2574:
2573:
2571:
2570:
2565:
2560:
2555:
2550:
2545:
2539:
2537:
2531:
2530:
2528:
2527:
2522:
2517:
2512:
2507:
2501:
2499:
2495:
2494:
2492:
2491:
2486:
2481:
2476:
2471:
2466:
2464:Risk-free bond
2461:
2456:
2451:
2449:Mortgage yield
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2405:
2403:
2401:Bond valuation
2397:
2396:
2394:
2393:
2388:
2383:
2378:
2376:Perpetual bond
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2297:
2295:
2291:
2290:
2288:
2287:
2282:
2280:Municipal bond
2277:
2272:
2267:
2262:
2261:
2260:
2255:
2248:Corporate bond
2245:
2239:
2237:
2233:
2232:
2230:
2229:
2224:
2219:
2213:
2210:
2209:
2204:
2202:
2201:
2194:
2187:
2179:
2173:
2172:
2168:, D. Bayazit,
2163:
2152:
2151:External links
2149:
2148:
2147:
2141:
2128:
2122:
2109:
2103:
2079:
2078:
2041:(4): 569–579.
2021:
1994:
1980:10.1.1.164.447
1973:(2): 177–188.
1954:
1953:
1951:
1948:
1947:
1946:
1941:
1936:
1928:
1925:
1924:
1923:
1912:
1909:
1906:
1903:
1900:
1895:
1891:
1884:
1881:
1875:
1871:
1865:
1861:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1820:
1815:
1806:
1802:
1798:
1792:
1788:
1782:
1779:
1775:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1743:
1730:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1702:
1698:
1695:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1657:
1656:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1591:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1546:
1526:
1510:
1507:
1506:
1505:
1494:
1488:
1485:
1479:
1475:
1469:
1466:
1461:
1457:
1453:
1449:
1446:
1443:
1437:
1434:
1431:
1427:
1412:
1411:
1400:
1397:
1394:
1389:
1385:
1381:
1377:
1371:
1368:
1365:
1361:
1346:
1345:
1334:
1331:
1326:
1323:
1320:
1317:
1313:
1309:
1306:
1303:
1297:
1294:
1288:
1284:
1278:
1275:
1270:
1266:
1262:
1258:
1255:
1252:
1237:
1236:
1225:
1220:
1217:
1214:
1210:
1206:
1203:
1200:
1197:
1194:
1189:
1186:
1183:
1179:
1173:
1169:
1165:
1162:
1157:
1153:
1149:
1145:
1126:
1125:
1112:
1107:
1103:
1099:
1093:
1090:
1086:
1080:
1075:
1071:
1065:
1062:
1059:
1055:
1051:
1048:
1044:
1038:
1035:
1032:
1028:
1024:
1021:
1017:
1013:
1010:
1005:
1002:
999:
995:
989:
985:
981:
976:
972:
956:
953:
908:
903:
899:
895:
892:
889:
886:
869:
853:
839:
836:
831:
827:
823:
788:
783:
779:
775:
772:
769:
766:
747:mean reversion
742:
739:
719:
699:
688:
687:
673:
670:
664:
660:
634:
614:
594:
574:
554:
534:
524:
523:
511:
491:
488:
485:
482:
478:
471:
467:
451:
450:
438:
418:
408:
396:
376:
356:
346:
334:
314:
291:
269:
265:
244:
224:
221:
218:
194:
179:Wiener process
173:
168:
167:
154:
150:
146:
142:
139:
136:
133:
129:
124:
120:
116:
113:
110:
107:
104:
99:
95:
91:
68:
65:
41:interest rates
15:
13:
10:
9:
6:
4:
3:
2:
3928:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3898:
3896:
3881:
3878:
3876:
3873:
3872:
3869:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3779:
3777:
3773:
3765:
3762:
3760:
3757:
3756:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3734:Stopping time
3732:
3728:
3725:
3724:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3584:
3582:
3578:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3548:
3546:
3544:
3540:
3533:
3529:
3525:
3524:Hewitt–Savage
3521:
3517:
3513:
3509:
3508:Zero–one laws
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3455:
3453:
3449:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3369:
3367:
3363:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3328:
3326:
3324:
3320:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3295:
3293:
3291:
3287:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3205:Black–Scholes
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3185:
3183:
3179:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3142:
3140:
3136:
3130:
3127:
3125:
3122:
3118:
3115:
3113:
3110:
3109:
3108:
3107:Point process
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3086:
3083:
3081:
3078:
3077:
3076:
3073:
3071:
3068:
3066:
3065:Gibbs measure
3063:
3061:
3058:
3056:
3053:
3052:
3050:
3046:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2994:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2969:
2967:
2963:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2910:
2907:
2905:
2902:
2901:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2854:ItĂ´ diffusion
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2829:Gamma process
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2754:
2753:
2750:
2746:
2743:
2742:
2741:
2738:
2736:
2733:
2731:
2728:
2727:
2725:
2723:
2719:
2711:
2708:
2706:
2703:
2701:
2700:Self-avoiding
2698:
2696:
2693:
2692:
2691:
2688:
2686:
2685:Moran process
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2652:
2650:
2648:
2647:Discrete time
2644:
2640:
2633:
2628:
2626:
2621:
2619:
2614:
2613:
2610:
2597:
2594:
2591:
2588:
2585:
2582:
2581:
2579:
2575:
2569:
2568:Puttable bond
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2543:Callable bond
2541:
2540:
2538:
2536:
2532:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2502:
2500:
2496:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2454:Nominal yield
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2429:Current yield
2427:
2425:
2424:Credit spread
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2404:
2402:
2398:
2392:
2389:
2387:
2384:
2382:
2381:Puttable bond
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2311:Callable bond
2309:
2307:
2304:
2302:
2299:
2298:
2296:
2292:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2259:
2256:
2254:
2251:
2250:
2249:
2246:
2244:
2241:
2240:
2238:
2234:
2228:
2225:
2223:
2220:
2218:
2215:
2214:
2211:
2207:
2200:
2195:
2193:
2188:
2186:
2181:
2180:
2177:
2171:
2167:
2164:
2162:
2158:
2155:
2154:
2150:
2144:
2138:
2134:
2129:
2125:
2119:
2115:
2110:
2106:
2100:
2096:
2095:Prentice Hall
2091:
2090:
2083:
2082:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2044:
2040:
2036:
2032:
2025:
2022:
2017:
2013:
2010:(67): 50–61.
2009:
2005:
1998:
1995:
1990:
1986:
1981:
1976:
1972:
1968:
1967:
1959:
1956:
1949:
1945:
1942:
1940:
1937:
1934:
1931:
1930:
1926:
1907:
1904:
1901:
1893:
1889:
1882:
1879:
1873:
1869:
1863:
1859:
1852:
1849:
1846:
1840:
1834:
1831:
1828:
1822:
1818:
1813:
1804:
1800:
1796:
1790:
1786:
1780:
1777:
1773:
1769:
1763:
1760:
1757:
1751:
1744:
1728:
1719:
1716:
1713:
1707:
1704:
1700:
1696:
1693:
1687:
1681:
1678:
1675:
1669:
1662:
1661:
1660:
1638:
1632:
1626:
1623:
1620:
1614:
1611:
1605:
1602:
1599:
1593:
1589:
1585:
1579:
1576:
1573:
1567:
1560:
1559:
1558:
1544:
1524:
1516:
1515:discount bond
1508:
1492:
1486:
1483:
1477:
1473:
1467:
1459:
1455:
1429:
1417:
1416:
1415:
1398:
1395:
1387:
1383:
1363:
1351:
1350:
1349:
1332:
1324:
1321:
1318:
1315:
1311:
1307:
1304:
1295:
1292:
1286:
1282:
1276:
1268:
1264:
1242:
1241:
1240:
1239:and variance
1218:
1215:
1212:
1208:
1204:
1201:
1195:
1192:
1187:
1184:
1181:
1177:
1171:
1167:
1163:
1155:
1151:
1135:
1134:
1133:
1131:
1110:
1105:
1101:
1097:
1091:
1088:
1084:
1078:
1073:
1069:
1063:
1060:
1057:
1053:
1049:
1046:
1042:
1036:
1033:
1030:
1026:
1022:
1019:
1015:
1011:
1008:
1003:
1000:
997:
993:
987:
983:
979:
974:
970:
962:
961:
960:
954:
952:
950:
946:
942:
938:
934:
930:
924:
922:
901:
897:
893:
890:
884:
876:
872:
865:
861:
857:
837:
834:
829:
825:
821:
813:
810:
806:
802:
781:
777:
773:
770:
764:
755:
752:
748:
740:
738:
736:
731:
717:
697:
671:
668:
662:
658:
648:
647:
646:
632:
612:
592:
572:
552:
532:
509:
486:
483:
476:
469:
465:
456:
455:
454:
436:
416:
409:
394:
374:
354:
347:
332:
312:
305:
304:
303:
289:
267:
263:
242:
222:
219:
216:
208:
192:
184:
180:
176:
152:
148:
144:
140:
137:
134:
131:
122:
118:
114:
111:
105:
102:
97:
93:
89:
82:
81:
80:
78:
74:
66:
64:
62:
58:
54:
50:
46:
42:
38:
34:
33:Vasicek model
30:
21:
3792:Econometrics
3754:Wiener space
3642:ItĂ´ integral
3543:Inequalities
3432:Self-similar
3402:Gauss–Markov
3392:Exchangeable
3372:CĂ dlĂ g paths
3308:Risk process
3274:
3260:LIBOR market
3129:Random graph
3124:Random field
2936:Superprocess
2874:LĂ©vy process
2869:Jump process
2844:Hunt process
2680:Markov chain
2577:Institutions
2535:Bond options
2479:Yield spread
2371:Lottery bond
2301:Accrual bond
2227:Fixed income
2132:
2113:
2088:
2038:
2034:
2024:
2007:
2003:
1997:
1970:
1964:
1958:
1658:
1512:
1509:Bond pricing
1413:
1347:
1238:
1127:
958:
925:
920:
874:
867:
859:
851:
804:
800:
756:
744:
732:
689:
525:
452:
171:
169:
75:follows the
70:
32:
26:
3837:Ruin theory
3775:Disciplines
3647:ItĂ´'s lemma
3422:Predictable
3097:Percolation
3080:Potts model
3075:Ising model
3039:White noise
2997:Differences
2859:ItĂ´ process
2799:Cox process
2695:Loop-erased
2690:Random walk
2474:Yield curve
2434:Dirty price
2409:Clean price
2285:Global bond
2253:Senior debt
2243:Agency bond
2206:Bond market
812:equilibrium
185:parameter,
49:market risk
3895:Categories
3847:Statistics
3627:Filtration
3528:Kolmogorov
3512:Blumenthal
3437:Stationary
3377:Continuous
3365:Properties
3250:Hull–White
2992:Martingale
2879:Local time
2767:Fractional
2745:pure birth
2048:1901.02246
1950:References
741:Discussion
207:volatility
3759:Classical
2772:Geometric
2762:Excursion
2414:Convexity
2222:Debenture
2135:. Wiley.
2073:126507446
2065:0277-6693
1975:CiteSeerX
1870:σ
1864:−
1850:−
1841:−
1787:σ
1781:−
1717:−
1705:−
1697:−
1612:−
1474:σ
1436:∞
1433:→
1370:∞
1367:→
1316:−
1308:−
1283:σ
1213:−
1205:−
1182:−
1070:∫
1058:−
1050:σ
1031:−
1023:−
998:−
894:−
873:is below
864:stability
774:−
698:σ
659:σ
573:σ
553:σ
466:σ
437:σ
417:σ
243:σ
193:σ
141:σ
115:−
3880:Category
3764:Abstract
3298:BĂĽhlmann
2904:Compound
2489:Z-spread
2444:I-spread
2439:Duration
1927:See also
809:long-run
407:in time;
3387:Ergodic
3275:VašĂÄŤek
3117:Poisson
2777:Meander
2598:(SIFMA)
67:Details
29:finance
3727:Tanaka
3412:Mixing
3407:Markov
3280:Wilkie
3245:Ho–Lee
3240:Heston
3012:Super-
2757:Bridge
2705:Biased
2592:(ICMA)
2586:(CMSA)
2419:Coupon
2321:Consol
2139:
2120:
2101:
2071:
2063:
1977:
1659:where
170:where
31:, the
3580:Tools
3356:M/M/c
3351:M/M/1
3346:M/G/1
3336:Fluid
3002:Local
2069:S2CID
2043:arXiv
751:stock
177:is a
35:is a
3532:LĂ©vy
3331:Bulk
3215:Chen
3007:Sub-
2965:Both
2217:Bond
2137:ISBN
2118:ISBN
2099:ISBN
2061:ISSN
2008:2013
1414:and
935:and
545:and
235:and
3112:Cox
2053:doi
2012:doi
1985:doi
1426:lim
1360:lim
856:= b
27:In
3897::
3530:,
3526:,
3522:,
3518:,
3514:,
2097:.
2067:.
2059:.
2051:.
2039:39
2037:.
2033:.
2006:.
1983:.
1969:.
730:.
79::
63:.
3534:)
3510:(
2631:e
2624:t
2617:v
2198:e
2191:t
2184:v
2145:.
2126:.
2107:.
2075:.
2055::
2045::
2018:.
2014::
1991:.
1987::
1971:5
1935:.
1911:)
1908:T
1905:,
1902:t
1899:(
1894:2
1890:B
1883:a
1880:4
1874:2
1860:]
1856:)
1853:t
1847:T
1844:(
1838:)
1835:T
1832:,
1829:t
1826:(
1823:B
1819:[
1814:)
1805:2
1801:a
1797:2
1791:2
1778:b
1774:(
1770:=
1767:)
1764:T
1761:,
1758:t
1755:(
1752:A
1729:a
1723:)
1720:t
1714:T
1711:(
1708:a
1701:e
1694:1
1688:=
1685:)
1682:T
1679:,
1676:t
1673:(
1670:B
1642:)
1639:t
1636:(
1633:r
1630:)
1627:T
1624:,
1621:t
1618:(
1615:B
1609:)
1606:T
1603:,
1600:t
1597:(
1594:A
1590:e
1586:=
1583:)
1580:T
1577:,
1574:t
1571:(
1568:P
1545:T
1525:t
1493:.
1487:a
1484:2
1478:2
1468:=
1465:]
1460:t
1456:r
1452:[
1448:r
1445:a
1442:V
1430:t
1399:b
1396:=
1393:]
1388:t
1384:r
1380:[
1376:E
1364:t
1333:.
1330:)
1325:t
1322:a
1319:2
1312:e
1305:1
1302:(
1296:a
1293:2
1287:2
1277:=
1274:]
1269:t
1265:r
1261:[
1257:r
1254:a
1251:V
1224:)
1219:t
1216:a
1209:e
1202:1
1199:(
1196:b
1193:+
1188:t
1185:a
1178:e
1172:0
1168:r
1164:=
1161:]
1156:t
1152:r
1148:[
1144:E
1111:.
1106:s
1102:W
1098:d
1092:s
1089:a
1085:e
1079:t
1074:0
1064:t
1061:a
1054:e
1047:+
1043:)
1037:t
1034:a
1027:e
1020:1
1016:(
1012:b
1009:+
1004:t
1001:a
994:e
988:0
984:r
980:=
975:t
971:r
921:a
907:)
902:t
898:r
891:b
888:(
885:a
875:b
870:t
868:r
860:a
854:t
852:r
838:0
835:=
830:t
826:W
822:d
805:b
801:t
787:)
782:t
778:r
771:b
768:(
765:a
718:a
672:a
669:2
663:2
633:a
613:b
593:a
533:a
510:r
490:)
487:a
484:2
481:(
477:/
470:2
395:b
375:a
355:a
333:r
313:b
290:a
268:0
264:r
223:a
220:,
217:b
174:t
172:W
153:t
149:W
145:d
138:+
135:t
132:d
128:)
123:t
119:r
112:b
109:(
106:a
103:=
98:t
94:r
90:d
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