31:
3677:. A more recent extension for handling cluster volatility, negative interest rates and different distributions is the so-called "CIR #" by Orlando, Mininni and Bufalo (2018, 2019, 2020, 2021, 2023) and a simpler extension focussing on negative interest rates was proposed by Di Francesco and Kamm (2021, 2022), which are referred to as the CIR- and CIR-- models.
91:
3064:
3661:
for bond prices. Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities. The most general approach is in
Maghsoodi (1996). A more tractable approach is in Brigo and
1835:
1542:
2746:
1147:
2000:
2892:
3437:
2909:
672:) also becomes very small, which dampens the effect of the random shock on the rate. Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards
2174:
3587:
1692:
236:
2530:
2242:
1415:
2557:
878:
3237:
3643:
1207:
1874:
4885:
2290:
928:
799:
2050:
1637:
2344:
670:
523:
2094:
1590:
988:
606:
1882:
1347:
4806:
2413:
2773:
3248:
717:
5420:
3095:
1259:
977:
3059:{\displaystyle {\frac {r_{t+\Delta t}-r_{t}}{{\sqrt {r}}_{t}}}={\frac {ab\Delta t}{{\sqrt {r}}_{t}}}-a{\sqrt {r}}_{t}\Delta t+\sigma \,{\sqrt {\Delta t}}\varepsilon _{t},}
1680:
1404:
3738:
Yuliya
Mishura, Andrey Pilipenko & Anton Yurchenko-Tytarenko(10 Jan 2024): Low-dimensional Cox-Ingersoll-Ross process, Stochastics, DOI:10.1080/17442508.2023.2300291
1301:
438:
396:
331:
5244:
4253:
2374:
636:
266:
123:
5847:
4840:
5377:
5357:
2550:
1367:
563:
543:
482:
462:
375:
355:
310:
290:
5761:
4794:
5678:
4339:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (2020). "Forecasting interest rates through
Vasicek and CIR models: A partitioning approach".
5688:
5362:
4800:
3867:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (1 January 2019). "A new approach to forecast market interest rates through the CIR model".
5372:
5730:
2102:
1830:{\displaystyle {\partial p \over {\partial t}}+{\partial \over {\partial r}}={1 \over {2}}\sigma ^{2}{\partial ^{2} \over {\partial r^{2}}}(rp)}
5445:
5627:
3715:
3445:
5917:
5907:
5430:
4407:
4291:
3851:
5817:
5781:
139:
5734:
6085:
5822:
4730:
885:
4932:
4833:
4725:
3662:
Mercurio (2001b) where an external time-dependent shift is added to the model for consistency with an input term structure of rates.
1537:{\displaystyle f(r_{\infty };a,b,\sigma )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}r_{\infty }^{\alpha -1}e^{-\beta r_{\infty }},}
5887:
5465:
4206:
5738:
5722:
2420:
2185:
5932:
5637:
4857:
130:
2292:, this density describes a gamma distribution. Therefore, the asymptotic distribution of the CIR model is a gamma distribution.
5837:
5802:
5771:
5766:
5405:
5202:
5119:
2741:{\displaystyle \operatorname {Var} =r_{0}{\frac {\sigma ^{2}}{a}}(e^{-at}-e^{-2at})+{\frac {b\sigma ^{2}}{2a}}(1-e^{-at})^{2}.}
5776:
5104:
720:
79:
6116:
5400:
5207:
4720:
4596:
5126:
808:
5862:
5742:
6090:
5867:
5703:
5602:
5587:
4999:
4915:
4826:
5877:
5513:
3910:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (19 August 2019). "Interest rates calibration with a CIR model".
5872:
5475:
5059:
5004:
4920:
4536:
3654:
5807:
5797:
5440:
5410:
3147:
3834:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (2018). "A New
Approach to CIR Short-Term Rates Modelling".
3595:
3141:, a bond may be priced using this interest rate process. The bond price is exponential affine in the interest rate:
1158:
6126:
6121:
5812:
4977:
4875:
4576:
5523:
5099:
4880:
6131:
5892:
5693:
5607:
5592:
4982:
4400:
1843:
5726:
5612:
5034:
2250:
891:
5114:
5089:
4735:
2306:
441:
71:
59:
5832:
5415:
4950:
1142:{\displaystyle f(r_{t+T};r_{t},a,b,\sigma )=c\,e^{-u-v}\left({\frac {v}{u}}\right)^{q/2}I_{q}(2{\sqrt {uv}}),}
750:
2008:
1995:{\displaystyle a(b-r)p_{\infty }={1 \over {2}}\sigma ^{2}\left(p_{\infty }+r{dp_{\infty } \over {dr}}\right)}
1595:
6111:
6027:
6017:
5708:
5490:
5229:
5094:
4905:
2315:
641:
494:
5312:
2887:{\displaystyle r_{t+\Delta t}-r_{t}=a(b-r_{t})\,\Delta t+\sigma \,{\sqrt {r_{t}\Delta t}}\varepsilon _{t},}
2055:
1551:
571:
5969:
5897:
5156:
4669:
4571:
3658:
3432:{\displaystyle A(t,T)=\left({\frac {2he^{(a+h)(T-t)/2}}{2h+(a+h)(e^{h(T-t)}-1)}}\right)^{2ab/\sigma ^{2}}}
2757:
1683:
1306:
4053:"Time series forecasting with the CIR# model: from hectic markets sentiments to regular seasonal tourism"
3749:
2379:
5992:
5974:
5954:
5949:
5668:
5500:
5480:
5327:
5270:
5109:
5019:
4715:
4679:
4516:
4475:
4247:
4012:"Interest rates forecasting: Between Hull and White and the CIR#—How to make a single-factor model work"
3666:
3116:
67:
5460:
3686:
682:
3073:
6067:
6022:
6012:
5753:
5698:
5673:
5642:
5622:
5382:
5367:
5234:
4393:
1212:
39:
4216:
Cox, J.C., J.E. Ingersoll and S.A. Ross (1985). "A Theory of the Term
Structure of Interest Rates".
4052:
933:
6062:
5902:
5827:
5632:
5392:
5302:
5192:
4485:
3138:
4303:"A deterministic-shift extension of analytically tractable and time-homogeneous short rate models"
3789:"A deterministic–shift extension of analytically–tractable and time–homogeneous short–rate models"
6032:
5997:
5912:
5882:
5652:
5647:
5470:
5307:
4972:
4910:
4849:
4561:
4374:
4348:
4322:
4235:
4125:
4105:
3992:
3966:
3935:
3892:
3816:
3105:
1658:
1407:
1382:
488:
5713:
1264:
401:
4261:
Maghsoodi, Y. (1996). "Solution of the extended CIR Term
Structure and Bond Option Valuation".
380:
315:
6052:
5857:
5508:
5265:
5182:
5151:
5044:
5024:
5014:
4870:
4865:
4768:
4694:
4546:
4468:
4366:
4287:
4202:
4074:
4033:
3984:
3927:
3884:
3847:
3808:
3769:
5718:
5455:
4302:
6072:
5959:
5842:
5212:
5187:
5136:
5064:
4987:
4940:
4758:
4629:
4601:
4541:
4526:
4358:
4314:
4270:
4227:
4166:
4156:
4115:
4064:
4023:
3976:
3919:
3876:
3839:
3800:
3761:
51:
2352:
614:
244:
101:
6037:
5937:
5922:
5683:
5617:
5295:
5239:
5222:
4967:
4773:
4763:
4566:
4556:
4551:
4480:
3838:. Contributions to Management Science. Springer International Publishing. pp. 35–43.
5852:
5084:
6042:
6007:
5927:
5533:
5280:
5197:
5166:
5161:
5141:
5131:
5074:
5069:
5049:
5029:
4994:
4962:
4945:
4674:
4659:
4624:
4611:
4586:
4490:
4458:
4427:
4274:
4191:
3954:
3765:
3123:
2535:
1352:
548:
528:
467:
447:
360:
340:
295:
275:
269:
126:
6105:
5944:
5485:
5322:
5317:
5275:
5217:
5039:
4955:
4895:
4778:
4753:
4664:
4649:
4639:
4591:
4531:
4521:
4378:
4198:
4145:"On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework"
4129:
3996:
3939:
3896:
3691:
75:
47:
4326:
3955:"Forecasting interest rates through Vasicek and CIR models: A partitioning approach"
3820:
6002:
5964:
5518:
5450:
5339:
5334:
5146:
5079:
5054:
4890:
4689:
4634:
4581:
4511:
4437:
4218:
745:
The distribution of future values of a CIR process can be computed in closed form:
728:
673:
5582:
6047:
5566:
5561:
5556:
5546:
5349:
5290:
5285:
5249:
5009:
4900:
4745:
4684:
4644:
4619:
4495:
4463:
4453:
4416:
3843:
63:
55:
4333:
4120:
4093:
525:, avoids the possibility of negative interest rates for all positive values of
6057:
5597:
5541:
5425:
3696:
3674:
3670:
4370:
4078:
4037:
3988:
3931:
3923:
3888:
3880:
3812:
3773:
5551:
4432:
4069:
3716:"A Theory of the Term Structure of Interest Rates - The Econometric Society"
3665:
A significant extension of the CIR model to the case of stochastic mean and
334:
4284:
Interest Rate Models — Theory and
Practice with Smile, Inflation and Credit
2169:{\displaystyle {\alpha -1 \over {r}}-\beta ={d \over {dr}}\log p_{\infty }}
30:
17:
54:) as it describes interest rate movements as driven by only one source of
4699:
4654:
4161:
4144:
4171:
3582:{\displaystyle B(t,T)={\frac {2(e^{h(T-t)}-1)}{2h+(a+h)(e^{h(T-t)}-1)}}}
90:
5378:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4818:
4318:
4239:
3804:
3750:"Solution of the Extended Cir Term Structure and Bond Option Valuation"
724:
719:, the Feller square-root process can be obtained from the square of an
464:, with speed of adjustment governed by the strictly positive parameter
4362:
4028:
4011:
3980:
3953:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (July 2020).
3788:
3097:
is n.i.i.d. (0,1). This equation can be used for a linear regression.
231:{\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma {\sqrt {r_{t}}}\,dW_{t},}
4231:
4353:
4110:
3971:
1379:
Due to mean reversion, as time becomes large, the distribution of
89:
29:
4822:
4389:
4385:
565:. An interest rate of zero is also precluded if the condition
27:
Stochastic model for the evolution of financial interest rates
2525:{\displaystyle \operatorname {E} =r_{0}e^{-at}+b(1-e^{-at})}
2237:{\displaystyle p_{\infty }\propto r^{\alpha -1}e^{-\beta r}}
727:
and possesses a stationary distribution. It is used in the
5358:
Autoregressive conditional heteroskedasticity (ARCH) model
4334:
Open Source library implementing the CIR process in python
4094:"How to handle negative interest rates in a CIR framework"
440:, is exactly the same as in the Vasicek model. It ensures
1349:
is a modified Bessel function of the first kind of order
4886:
Independent and identically distributed random variables
98:
The CIR model describes the instantaneous interest rate
3119:
of the CIR process can be achieved using two variants:
5363:
Autoregressive integrated moving average (ARIMA) model
873:{\displaystyle c={\frac {2a}{(1-e^{-aT})\sigma ^{2}}}}
4807:
Securities
Industry and Financial Markets Association
3598:
3448:
3251:
3150:
3076:
2912:
2776:
2560:
2538:
2423:
2415:; otherwise it can occasionally touch the zero point,
2382:
2355:
2318:
2253:
2188:
2105:
2058:
2011:
1885:
1846:
1695:
1661:
1598:
1554:
1418:
1385:
1355:
1309:
1267:
1215:
1161:
991:
936:
894:
811:
753:
685:
644:
617:
574:
551:
531:
497:
470:
450:
404:
383:
363:
343:
318:
298:
278:
247:
142:
104:
5985:
5790:
5752:
5661:
5575:
5532:
5499:
5391:
5348:
5258:
5175:
4931:
4856:
4787:
4744:
4708:
4610:
4504:
4446:
4092:Di Francesco, Marco; Kamm, Kevin (4 October 2021).
357:corresponds to the speed of adjustment to the mean
4190:
3637:
3581:
3431:
3231:
3089:
3058:
2886:
2740:
2544:
2524:
2407:
2368:
2338:
2284:
2236:
2168:
2088:
2044:
1994:
1868:
1829:
1674:
1631:
1584:
1536:
1398:
1361:
1341:
1295:
1253:
1201:
1141:
971:
922:
872:
793:
711:
664:
630:
600:
557:
537:
517:
476:
456:
432:
390:
369:
349:
325:
304:
284:
260:
230:
117:
4057:Technological and Economic Development of Economy
4051:Orlando, Giuseppe; Bufalo, Michele (2023-07-14).
4010:Orlando, Giuseppe; Bufalo, Michele (2021-05-26).
3228:
5245:Stochastic chains with memory of variable length
2764:The continuous SDE can be discretized as follows
930:degrees of freedom and non-centrality parameter
444:of the interest rate towards the long run value
3232:{\displaystyle P(t,T)=A(t,T)e^{-B(t,T)r_{t}}\!}
979:. Formally the probability density function is:
3787:Brigo, Damiano; Mercurio, Fabio (2001-07-01).
3638:{\displaystyle h={\sqrt {a^{2}+2\sigma ^{2}}}}
1202:{\displaystyle q={\frac {2ab}{\sigma ^{2}}}-1}
272:(modelling the random market risk factor) and
4834:
4401:
2096:and rearranging terms leads to the equation:
8:
4252:: CS1 maint: multiple names: authors list (
1840:Our interest is in the particular case when
638:) is close to zero, the standard deviation (
58:. The model can be used in the valuation of
1869:{\displaystyle \partial _{t}p\rightarrow 0}
5373:Autoregressive–moving-average (ARMA) model
4841:
4827:
4819:
4795:Commercial Mortgage Securities Association
4408:
4394:
4386:
4286:(2nd ed. 2006 ed.). Springer Verlag.
4149:International Journal of Financial Studies
2285:{\displaystyle p_{\infty }\in (0,\infty ]}
1876:, which leads to the simplified equation:
1643:
923:{\displaystyle {\frac {4ab}{\sigma ^{2}}}}
4352:
4170:
4160:
4143:Di Francesco, Marco; Kamm, Kevin (2022).
4119:
4109:
4068:
4027:
3970:
3627:
3611:
3605:
3597:
3546:
3483:
3470:
3447:
3421:
3412:
3402:
3365:
3322:
3291:
3278:
3250:
3220:
3194:
3149:
3081:
3075:
3047:
3033:
3032:
3014:
3007:
2992:
2985:
2968:
2957:
2950:
2942:
2920:
2913:
2911:
2875:
2857:
2851:
2850:
2837:
2828:
2803:
2781:
2775:
2729:
2713:
2683:
2673:
2652:
2633:
2615:
2609:
2603:
2587:
2574:
2559:
2537:
2507:
2476:
2466:
2450:
2437:
2422:
2399:
2381:
2360:
2354:
2328:
2322:
2317:
2258:
2252:
2222:
2206:
2193:
2187:
2160:
2140:
2135:
2119:
2106:
2104:
2080:
2071:
2057:
2036:
2027:
2010:
1977:
1970:
1960:
1948:
1933:
1922:
1917:
1908:
1884:
1851:
1845:
1806:
1798:
1792:
1786:
1780:
1769:
1764:
1724:
1719:
1706:
1696:
1694:
1666:
1660:
1623:
1614:
1597:
1576:
1567:
1553:
1523:
1512:
1496:
1491:
1465:
1459:
1429:
1417:
1390:
1384:
1354:
1326:
1314:
1308:
1281:
1266:
1239:
1229:
1214:
1185:
1168:
1160:
1123:
1111:
1097:
1093:
1079:
1059:
1054:
1021:
1002:
990:
957:
947:
935:
912:
895:
893:
861:
842:
818:
810:
773:
758:
752:
708:
702:
684:
654:
648:
643:
622:
616:
597:
591:
573:
550:
530:
507:
501:
496:
469:
449:
421:
403:
387:
382:
362:
342:
322:
317:
297:
277:
252:
246:
219:
211:
203:
197:
184:
175:
150:
141:
109:
103:
4801:International Capital Market Association
4301:Brigo, Damiano; Fabio Mercurio (2001b).
794:{\displaystyle r_{t+T}={\frac {Y}{2c}},}
3707:
3653:The CIR model uses a special case of a
2045:{\displaystyle \alpha =2ab/\sigma ^{2}}
1632:{\displaystyle \alpha =2ab/\sigma ^{2}}
611:is met. More generally, when the rate (
5679:Doob's martingale convergence theorems
4282:Damiano Brigo; Fabio Mercurio (2001).
4245:
4193:Options, Futures and Other Derivatives
2376:the process will never touch zero, if
2339:{\displaystyle \sigma {\sqrt {r_{t}}}}
1655:To derive the asymptotic distribution
665:{\displaystyle \sigma {\sqrt {r_{t}}}}
518:{\displaystyle \sigma {\sqrt {r_{t}}}}
50:. It is a type of "one factor model" (
5431:Constant elasticity of variance (CEV)
5421:Chan–Karolyi–Longstaff–Sanders (CKLS)
2089:{\displaystyle \beta =2a/\sigma ^{2}}
1648:Derivation of asymptotic distribution
1585:{\displaystyle \beta =2a/\sigma ^{2}}
601:{\displaystyle 2ab\geq \sigma ^{2}\,}
7:
3836:New Methods in Fixed Income Modeling
1342:{\displaystyle I_{q}(2{\sqrt {uv}})}
886:non-central chi-squared distribution
4731:Commercial mortgage-backed security
2408:{\displaystyle 2ab\geq \sigma ^{2}}
1682:for the CIR model, we must use the
34:Three trajectories of CIR processes
5918:Skorokhod's representation theorem
5699:Law of large numbers (weak/strong)
4726:Collateralized mortgage obligation
4275:10.1111/j.1467-9965.1996.tb00113.x
3766:10.1111/j.1467-9965.1996.tb00113.x
3748:Maghsoodi, Yoosef (January 1996).
3035:
3020:
2977:
2927:
2863:
2838:
2788:
2424:
2276:
2259:
2194:
2161:
1971:
1949:
1909:
1848:
1799:
1789:
1725:
1721:
1707:
1699:
1667:
1524:
1492:
1472:
1430:
1391:
25:
5888:Martingale representation theorem
1410:with the probability density of:
712:{\displaystyle 4ab=\sigma ^{2}\,}
398:to volatility. The drift factor,
5933:Stochastic differential equation
5823:Doob's optional stopping theorem
5818:Doob–Meyer decomposition theorem
3869:Studies in Economics and Finance
3090:{\displaystyle \varepsilon _{t}}
731:to model stochastic volatility.
131:stochastic differential equation
5803:Convergence of random variables
5689:Fisher–Tippett–Gnedenko theorem
1254:{\displaystyle u=cr_{t}e^{-aT}}
62:. It was introduced in 1985 by
5401:Binomial options pricing model
4721:Collateralized debt obligation
4597:Reverse convertible securities
3573:
3562:
3550:
3539:
3536:
3524:
3510:
3499:
3487:
3476:
3464:
3452:
3392:
3381:
3369:
3358:
3355:
3343:
3319:
3307:
3304:
3292:
3267:
3255:
3213:
3201:
3187:
3175:
3166:
3154:
2834:
2815:
2726:
2700:
2667:
2626:
2593:
2567:
2519:
2494:
2456:
2430:
2279:
2267:
1901:
1889:
1860:
1824:
1815:
1758:
1752:
1740:
1734:
1481:
1475:
1453:
1422:
1336:
1320:
1133:
1117:
1045:
995:
972:{\displaystyle 2cr_{t}e^{-aT}}
854:
829:
427:
408:
181:
162:
44:Cox–Ingersoll–Ross (CIR) model
1:
5868:Kolmogorov continuity theorem
5704:Law of the iterated logarithm
5873:Kolmogorov extension theorem
5552:Generalized queueing network
5060:Interacting particle systems
2312:Level dependent volatility (
5005:Continuous-time random walk
4537:Contingent convertible bond
3912:The Journal of Risk Finance
3844:10.1007/978-3-319-95285-7_2
3655:basic affine jump diffusion
2179:Integrating shows us that:
1675:{\displaystyle p_{\infty }}
1399:{\displaystyle r_{\infty }}
129:square-root process, whose
46:describes the evolution of
6148:
6013:Extreme value theory (EVT)
5813:Doob decomposition theorem
5105:Ornstein–Uhlenbeck process
4876:Chinese restaurant process
4577:Inverse floating rate note
4197:. Upper Saddle River, NJ:
4121:10.1007/s40324-021-00267-w
3720:www.econometricsociety.org
1296:{\displaystyle v=cr_{t+T}}
721:Ornstein–Uhlenbeck process
433:{\displaystyle a(b-r_{t})}
80:Ornstein–Uhlenbeck_process
6081:
5893:Optional stopping theorem
5694:Large deviation principle
5446:Heath–Jarrow–Morton (HJM)
5383:Moving-average (MA) model
5368:Autoregressive (AR) model
5193:Hidden Markov model (HMM)
5127:Schramm–Loewner evolution
4423:
4307:Finance & Stochastics
391:{\displaystyle \sigma \,}
326:{\displaystyle \sigma \,}
60:interest rate derivatives
5808:Doléans-Dade exponential
5638:Progressively measurable
5436:Cox–Ingersoll–Ross (CIR)
4736:Mortgage-backed security
4505:Types of bonds by payout
4447:Types of bonds by issuer
3924:10.1108/JRF-05-2019-0080
3881:10.1108/SEF-03-2019-0116
3657:, which still permits a
6028:Mathematical statistics
6018:Large deviations theory
5848:Infinitesimal generator
5709:Maximal ergodic theorem
5628:Piecewise-deterministic
5230:Random dynamical system
5095:Markov additive process
4070:10.3846/tede.2023.19294
3793:Finance and Stochastics
3673:(1996) and is known as
3139:no-arbitrage assumption
2532:, so long term mean is
1374:Asymptotic distribution
74:as an extension of the
5863:Karhunen–Loève theorem
5798:Cameron–Martin formula
5762:Burkholder–Davis–Gundy
5157:Variance gamma process
4670:Option-adjusted spread
4572:Inflation-indexed bond
4341:Journal of Forecasting
4189:Hull, John C. (2003).
4016:Journal of Forecasting
3959:Journal of Forecasting
3659:closed-form expression
3639:
3583:
3433:
3233:
3091:
3060:
2900:which is equivalent to
2888:
2758:Ordinary least squares
2742:
2546:
2526:
2409:
2370:
2340:
2286:
2238:
2170:
2090:
2046:
1996:
1870:
1831:
1684:Fokker-Planck equation
1676:
1633:
1586:
1538:
1400:
1363:
1343:
1297:
1255:
1203:
1143:
973:
924:
874:
795:
713:
666:
632:
602:
559:
539:
519:
478:
458:
434:
392:
371:
351:
327:
306:
286:
262:
232:
119:
95:
35:
6117:Fixed income analysis
5993:Actuarial mathematics
5955:Uniform integrability
5950:Stratonovich integral
5878:Lévy–Prokhorov metric
5782:Marcinkiewicz–Zygmund
5669:Central limit theorem
5271:Gaussian random field
5100:McKean–Vlasov process
5020:Dyson Brownian motion
4881:Galton–Watson process
4716:Asset-backed security
4680:Weighted-average life
4517:Auction rate security
3667:stochastic volatility
3640:
3584:
3434:
3234:
3117:Stochastic simulation
3102:Martingale estimation
3092:
3061:
2889:
2743:
2547:
2527:
2410:
2371:
2369:{\displaystyle r_{0}}
2341:
2287:
2239:
2171:
2091:
2047:
1997:
1871:
1832:
1677:
1634:
1587:
1539:
1401:
1364:
1344:
1298:
1256:
1204:
1144:
974:
925:
875:
796:
714:
667:
633:
631:{\displaystyle r_{t}}
603:
560:
540:
520:
479:
459:
435:
393:
372:
352:
328:
307:
287:
263:
261:{\displaystyle W_{t}}
233:
120:
118:{\displaystyle r_{t}}
93:
68:Jonathan E. Ingersoll
33:
6068:Time series analysis
6023:Mathematical finance
5908:Reflection principle
5235:Regenerative process
5035:Fleming–Viot process
4850:Stochastic processes
4709:Securitized products
4263:Mathematical Finance
4162:10.3390/ijfs10020038
3754:Mathematical Finance
3596:
3446:
3249:
3148:
3074:
2910:
2774:
2558:
2536:
2421:
2380:
2353:
2316:
2251:
2186:
2103:
2056:
2009:
1883:
1844:
1693:
1659:
1596:
1552:
1416:
1383:
1353:
1307:
1265:
1213:
1159:
989:
934:
892:
809:
751:
683:
642:
615:
572:
549:
529:
495:
468:
448:
402:
381:
361:
341:
316:
296:
276:
245:
140:
102:
40:mathematical finance
6063:Stochastic analysis
5903:Quadratic variation
5898:Prokhorov's theorem
5833:Feynman–Kac formula
5303:Markov random field
4951:Birth–death process
4486:Infrastructure bond
2349:For given positive
1507:
740:Future distribution
6033:Probability theory
5913:Skorokhod integral
5883:Malliavin calculus
5466:Korn-Kreer-Lenssen
5350:Time series models
5313:Pitman–Yor process
4562:Floating rate note
4319:10.1007/PL00013541
4183:Further References
3805:10.1007/PL00013541
3635:
3579:
3429:
3229:
3106:Maximum likelihood
3087:
3056:
2884:
2738:
2542:
2522:
2405:
2366:
2336:
2282:
2234:
2166:
2086:
2042:
1992:
1866:
1827:
1672:
1629:
1582:
1534:
1487:
1408:gamma distribution
1396:
1359:
1339:
1293:
1251:
1199:
1139:
969:
920:
870:
791:
709:
662:
628:
598:
555:
535:
515:
489:standard deviation
474:
454:
430:
388:
367:
347:
323:
302:
282:
258:
228:
115:
96:
36:
6127:Short-rate models
6122:Stochastic models
6099:
6098:
6053:Signal processing
5772:Doob's upcrossing
5767:Doob's martingale
5731:Engelbert–Schmidt
5674:Donsker's theorem
5608:Feller-continuous
5476:Rendleman–Bartter
5266:Dirichlet process
5183:Branching process
5152:Telegraph process
5045:Geometric process
5025:Empirical process
5015:Diffusion process
4871:Branching process
4866:Bernoulli process
4816:
4815:
4769:Exchangeable bond
4695:Yield to maturity
4547:Exchangeable bond
4469:Subordinated debt
4293:978-3-540-22149-4
3853:978-3-319-95284-0
3633:
3577:
3396:
3041:
3012:
2998:
2990:
2963:
2955:
2869:
2698:
2624:
2545:{\displaystyle b}
2334:
2297:
2296:
2148:
2124:
1985:
1927:
1813:
1774:
1732:
1714:
1485:
1362:{\displaystyle q}
1334:
1191:
1131:
1087:
918:
868:
786:
660:
558:{\displaystyle b}
538:{\displaystyle a}
513:
477:{\displaystyle a}
457:{\displaystyle b}
370:{\displaystyle b}
350:{\displaystyle a}
305:{\displaystyle b}
285:{\displaystyle a}
209:
16:(Redirected from
6139:
6132:Financial models
6073:Machine learning
5960:Usual hypotheses
5843:Girsanov theorem
5828:Dynkin's formula
5593:Continuous paths
5501:Actuarial models
5441:Garman–Kohlhagen
5411:Black–Karasinski
5406:Black–Derman–Toy
5393:Financial models
5259:Fields and other
5188:Gaussian process
5137:Sigma-martingale
4941:Additive process
4843:
4836:
4829:
4820:
4759:Convertible bond
4602:Zero-coupon bond
4542:Convertible bond
4527:Commercial paper
4410:
4403:
4396:
4387:
4382:
4363:10.1002/for.2642
4356:
4330:
4297:
4278:
4257:
4251:
4243:
4212:
4196:
4177:
4176:
4174:
4164:
4140:
4134:
4133:
4123:
4113:
4089:
4083:
4082:
4072:
4063:(4): 1216–1238.
4048:
4042:
4041:
4031:
4029:10.1002/for.2783
4022:(8): 1566–1580.
4007:
4001:
4000:
3981:10.1002/for.2642
3974:
3950:
3944:
3943:
3907:
3901:
3900:
3864:
3858:
3857:
3831:
3825:
3824:
3784:
3778:
3777:
3745:
3739:
3736:
3730:
3729:
3727:
3726:
3712:
3687:Hull–White model
3644:
3642:
3641:
3636:
3634:
3632:
3631:
3616:
3615:
3606:
3588:
3586:
3585:
3580:
3578:
3576:
3566:
3565:
3513:
3503:
3502:
3471:
3438:
3436:
3435:
3430:
3428:
3427:
3426:
3425:
3416:
3401:
3397:
3395:
3385:
3384:
3332:
3331:
3330:
3326:
3279:
3238:
3236:
3235:
3230:
3227:
3226:
3225:
3224:
3096:
3094:
3093:
3088:
3086:
3085:
3065:
3063:
3062:
3057:
3052:
3051:
3042:
3034:
3019:
3018:
3013:
3008:
2999:
2997:
2996:
2991:
2986:
2983:
2969:
2964:
2962:
2961:
2956:
2951:
2948:
2947:
2946:
2934:
2933:
2914:
2893:
2891:
2890:
2885:
2880:
2879:
2870:
2862:
2861:
2852:
2833:
2832:
2808:
2807:
2795:
2794:
2747:
2745:
2744:
2739:
2734:
2733:
2724:
2723:
2699:
2697:
2689:
2688:
2687:
2674:
2666:
2665:
2644:
2643:
2625:
2620:
2619:
2610:
2608:
2607:
2592:
2591:
2579:
2578:
2551:
2549:
2548:
2543:
2531:
2529:
2528:
2523:
2518:
2517:
2487:
2486:
2471:
2470:
2455:
2454:
2442:
2441:
2414:
2412:
2411:
2406:
2404:
2403:
2375:
2373:
2372:
2367:
2365:
2364:
2345:
2343:
2342:
2337:
2335:
2333:
2332:
2323:
2291:
2289:
2288:
2283:
2263:
2262:
2243:
2241:
2240:
2235:
2233:
2232:
2217:
2216:
2198:
2197:
2175:
2173:
2172:
2167:
2165:
2164:
2149:
2147:
2136:
2125:
2123:
2118:
2107:
2095:
2093:
2092:
2087:
2085:
2084:
2075:
2051:
2049:
2048:
2043:
2041:
2040:
2031:
2001:
1999:
1998:
1993:
1991:
1987:
1986:
1984:
1976:
1975:
1974:
1961:
1953:
1952:
1938:
1937:
1928:
1926:
1918:
1913:
1912:
1875:
1873:
1872:
1867:
1856:
1855:
1836:
1834:
1833:
1828:
1814:
1812:
1811:
1810:
1797:
1796:
1787:
1785:
1784:
1775:
1773:
1765:
1733:
1731:
1720:
1715:
1713:
1705:
1697:
1681:
1679:
1678:
1673:
1671:
1670:
1644:
1638:
1636:
1635:
1630:
1628:
1627:
1618:
1591:
1589:
1588:
1583:
1581:
1580:
1571:
1543:
1541:
1540:
1535:
1530:
1529:
1528:
1527:
1506:
1495:
1486:
1484:
1470:
1469:
1460:
1434:
1433:
1406:will approach a
1405:
1403:
1402:
1397:
1395:
1394:
1368:
1366:
1365:
1360:
1348:
1346:
1345:
1340:
1335:
1327:
1319:
1318:
1302:
1300:
1299:
1294:
1292:
1291:
1260:
1258:
1257:
1252:
1250:
1249:
1234:
1233:
1208:
1206:
1205:
1200:
1192:
1190:
1189:
1180:
1169:
1148:
1146:
1145:
1140:
1132:
1124:
1116:
1115:
1106:
1105:
1101:
1092:
1088:
1080:
1073:
1072:
1026:
1025:
1013:
1012:
978:
976:
975:
970:
968:
967:
952:
951:
929:
927:
926:
921:
919:
917:
916:
907:
896:
879:
877:
876:
871:
869:
867:
866:
865:
853:
852:
827:
819:
800:
798:
797:
792:
787:
785:
774:
769:
768:
718:
716:
715:
710:
707:
706:
671:
669:
668:
663:
661:
659:
658:
649:
637:
635:
634:
629:
627:
626:
607:
605:
604:
599:
596:
595:
564:
562:
561:
556:
544:
542:
541:
536:
524:
522:
521:
516:
514:
512:
511:
502:
483:
481:
480:
475:
463:
461:
460:
455:
439:
437:
436:
431:
426:
425:
397:
395:
394:
389:
376:
374:
373:
368:
356:
354:
353:
348:
337:. The parameter
332:
330:
329:
324:
311:
309:
308:
303:
291:
289:
288:
283:
267:
265:
264:
259:
257:
256:
237:
235:
234:
229:
224:
223:
210:
208:
207:
198:
180:
179:
155:
154:
124:
122:
121:
116:
114:
113:
52:short-rate model
21:
6147:
6146:
6142:
6141:
6140:
6138:
6137:
6136:
6102:
6101:
6100:
6095:
6077:
6038:Queueing theory
5981:
5923:Skorokhod space
5786:
5777:Kunita–Watanabe
5748:
5714:Sanov's theorem
5684:Ergodic theorem
5657:
5653:Time-reversible
5571:
5534:Queueing models
5528:
5524:Sparre–Anderson
5514:Cramér–Lundberg
5495:
5481:SABR volatility
5387:
5344:
5296:Boolean network
5254:
5240:Renewal process
5171:
5120:Non-homogeneous
5110:Poisson process
5000:Contact process
4963:Brownian motion
4933:Continuous time
4927:
4921:Maximal entropy
4852:
4847:
4817:
4812:
4783:
4774:Extendible bond
4764:Embedded option
4740:
4704:
4606:
4567:High-yield debt
4557:Fixed rate bond
4552:Extendible bond
4500:
4481:Government bond
4476:Distressed debt
4442:
4419:
4414:
4338:
4300:
4294:
4281:
4260:
4244:
4232:10.2307/1911242
4215:
4209:
4188:
4185:
4180:
4142:
4141:
4137:
4091:
4090:
4086:
4050:
4049:
4045:
4009:
4008:
4004:
3952:
3951:
3947:
3909:
3908:
3904:
3866:
3865:
3861:
3854:
3833:
3832:
3828:
3786:
3785:
3781:
3747:
3746:
3742:
3737:
3733:
3724:
3722:
3714:
3713:
3709:
3705:
3683:
3651:
3623:
3607:
3594:
3593:
3542:
3514:
3479:
3472:
3444:
3443:
3417:
3361:
3333:
3287:
3280:
3274:
3273:
3247:
3246:
3216:
3190:
3146:
3145:
3135:
3114:
3077:
3072:
3071:
3043:
3006:
2984:
2970:
2949:
2938:
2916:
2915:
2908:
2907:
2871:
2853:
2824:
2799:
2777:
2772:
2771:
2754:
2725:
2709:
2690:
2679:
2675:
2648:
2629:
2611:
2599:
2583:
2570:
2556:
2555:
2534:
2533:
2503:
2472:
2462:
2446:
2433:
2419:
2418:
2395:
2378:
2377:
2356:
2351:
2350:
2324:
2314:
2313:
2303:
2298:
2254:
2249:
2248:
2247:Over the range
2218:
2202:
2189:
2184:
2183:
2156:
2108:
2101:
2100:
2076:
2054:
2053:
2032:
2007:
2006:
1966:
1962:
1944:
1943:
1939:
1929:
1904:
1881:
1880:
1847:
1842:
1841:
1802:
1788:
1776:
1698:
1691:
1690:
1662:
1657:
1656:
1649:
1619:
1594:
1593:
1572:
1550:
1549:
1519:
1508:
1471:
1461:
1425:
1414:
1413:
1386:
1381:
1380:
1351:
1350:
1310:
1305:
1304:
1277:
1263:
1262:
1235:
1225:
1211:
1210:
1181:
1170:
1157:
1156:
1107:
1075:
1074:
1055:
1017:
998:
987:
986:
953:
943:
932:
931:
908:
897:
890:
889:
857:
838:
828:
820:
807:
806:
778:
754:
749:
748:
737:
698:
681:
680:
650:
640:
639:
618:
613:
612:
587:
570:
569:
547:
546:
527:
526:
503:
493:
492:
466:
465:
446:
445:
417:
400:
399:
379:
378:
359:
358:
339:
338:
314:
313:
294:
293:
274:
273:
248:
243:
242:
215:
199:
171:
146:
138:
137:
105:
100:
99:
88:
72:Stephen A. Ross
28:
23:
22:
15:
12:
11:
5:
6145:
6143:
6135:
6134:
6129:
6124:
6119:
6114:
6112:Interest rates
6104:
6103:
6097:
6096:
6094:
6093:
6088:
6086:List of topics
6082:
6079:
6078:
6076:
6075:
6070:
6065:
6060:
6055:
6050:
6045:
6043:Renewal theory
6040:
6035:
6030:
6025:
6020:
6015:
6010:
6008:Ergodic theory
6005:
6000:
5998:Control theory
5995:
5989:
5987:
5983:
5982:
5980:
5979:
5978:
5977:
5972:
5962:
5957:
5952:
5947:
5942:
5941:
5940:
5930:
5928:Snell envelope
5925:
5920:
5915:
5910:
5905:
5900:
5895:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5855:
5850:
5845:
5840:
5835:
5830:
5825:
5820:
5815:
5810:
5805:
5800:
5794:
5792:
5788:
5787:
5785:
5784:
5779:
5774:
5769:
5764:
5758:
5756:
5750:
5749:
5747:
5746:
5727:Borel–Cantelli
5716:
5711:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5665:
5663:
5662:Limit theorems
5659:
5658:
5656:
5655:
5650:
5645:
5640:
5635:
5630:
5625:
5620:
5615:
5610:
5605:
5600:
5595:
5590:
5585:
5579:
5577:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5538:
5536:
5530:
5529:
5527:
5526:
5521:
5516:
5511:
5505:
5503:
5497:
5496:
5494:
5493:
5488:
5483:
5478:
5473:
5468:
5463:
5458:
5453:
5448:
5443:
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5403:
5397:
5395:
5389:
5388:
5386:
5385:
5380:
5375:
5370:
5365:
5360:
5354:
5352:
5346:
5345:
5343:
5342:
5337:
5332:
5331:
5330:
5325:
5315:
5310:
5305:
5300:
5299:
5298:
5293:
5283:
5281:Hopfield model
5278:
5273:
5268:
5262:
5260:
5256:
5255:
5253:
5252:
5247:
5242:
5237:
5232:
5227:
5226:
5225:
5220:
5215:
5210:
5200:
5198:Markov process
5195:
5190:
5185:
5179:
5177:
5173:
5172:
5170:
5169:
5167:Wiener sausage
5164:
5162:Wiener process
5159:
5154:
5149:
5144:
5142:Stable process
5139:
5134:
5132:Semimartingale
5129:
5124:
5123:
5122:
5117:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5075:Jump diffusion
5072:
5067:
5062:
5057:
5052:
5050:Hawkes process
5047:
5042:
5037:
5032:
5030:Feller process
5027:
5022:
5017:
5012:
5007:
5002:
4997:
4995:Cauchy process
4992:
4991:
4990:
4985:
4980:
4975:
4970:
4960:
4959:
4958:
4948:
4946:Bessel process
4943:
4937:
4935:
4929:
4928:
4926:
4925:
4924:
4923:
4918:
4913:
4908:
4898:
4893:
4888:
4883:
4878:
4873:
4868:
4862:
4860:
4854:
4853:
4848:
4846:
4845:
4838:
4831:
4823:
4814:
4813:
4811:
4810:
4804:
4798:
4791:
4789:
4785:
4784:
4782:
4781:
4776:
4771:
4766:
4761:
4756:
4750:
4748:
4742:
4741:
4739:
4738:
4733:
4728:
4723:
4718:
4712:
4710:
4706:
4705:
4703:
4702:
4697:
4692:
4687:
4682:
4677:
4675:Risk-free bond
4672:
4667:
4662:
4660:Mortgage yield
4657:
4652:
4647:
4642:
4637:
4632:
4627:
4622:
4616:
4614:
4612:Bond valuation
4608:
4607:
4605:
4604:
4599:
4594:
4589:
4587:Perpetual bond
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4534:
4529:
4524:
4519:
4514:
4508:
4506:
4502:
4501:
4499:
4498:
4493:
4491:Municipal bond
4488:
4483:
4478:
4473:
4472:
4471:
4466:
4459:Corporate bond
4456:
4450:
4448:
4444:
4443:
4441:
4440:
4435:
4430:
4424:
4421:
4420:
4415:
4413:
4412:
4405:
4398:
4390:
4384:
4383:
4347:(4): 569–579.
4336:
4331:
4313:(3): 369–388.
4298:
4292:
4279:
4258:
4226:(2): 385–407.
4213:
4207:
4184:
4181:
4179:
4178:
4135:
4104:(4): 593–618.
4084:
4043:
4002:
3965:(4): 569–579.
3945:
3918:(4): 370–387.
3902:
3875:(2): 267–292.
3859:
3852:
3826:
3799:(3): 369–387.
3779:
3740:
3731:
3706:
3704:
3701:
3700:
3699:
3694:
3689:
3682:
3679:
3650:
3647:
3646:
3645:
3630:
3626:
3622:
3619:
3614:
3610:
3604:
3601:
3590:
3589:
3575:
3572:
3569:
3564:
3561:
3558:
3555:
3552:
3549:
3545:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3512:
3509:
3506:
3501:
3498:
3495:
3492:
3489:
3486:
3482:
3478:
3475:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3440:
3439:
3424:
3420:
3415:
3411:
3408:
3405:
3400:
3394:
3391:
3388:
3383:
3380:
3377:
3374:
3371:
3368:
3364:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3329:
3325:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3290:
3286:
3283:
3277:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3240:
3239:
3223:
3219:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3193:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3134:
3131:
3130:
3129:
3126:
3124:Discretization
3113:
3110:
3109:
3108:
3103:
3099:
3098:
3084:
3080:
3068:
3067:
3066:
3055:
3050:
3046:
3040:
3037:
3031:
3028:
3025:
3022:
3017:
3011:
3005:
3002:
2995:
2989:
2982:
2979:
2976:
2973:
2967:
2960:
2954:
2945:
2941:
2937:
2932:
2929:
2926:
2923:
2919:
2902:
2901:
2897:
2896:
2895:
2894:
2883:
2878:
2874:
2868:
2865:
2860:
2856:
2849:
2846:
2843:
2840:
2836:
2831:
2827:
2823:
2820:
2817:
2814:
2811:
2806:
2802:
2798:
2793:
2790:
2787:
2784:
2780:
2766:
2765:
2761:
2760:
2753:
2750:
2749:
2748:
2737:
2732:
2728:
2722:
2719:
2716:
2712:
2708:
2705:
2702:
2696:
2693:
2686:
2682:
2678:
2672:
2669:
2664:
2661:
2658:
2655:
2651:
2647:
2642:
2639:
2636:
2632:
2628:
2623:
2618:
2614:
2606:
2602:
2598:
2595:
2590:
2586:
2582:
2577:
2573:
2569:
2566:
2563:
2553:
2541:
2521:
2516:
2513:
2510:
2506:
2502:
2499:
2496:
2493:
2490:
2485:
2482:
2479:
2475:
2469:
2465:
2461:
2458:
2453:
2449:
2445:
2440:
2436:
2432:
2429:
2426:
2416:
2402:
2398:
2394:
2391:
2388:
2385:
2363:
2359:
2347:
2331:
2327:
2321:
2310:
2307:Mean reversion
2302:
2299:
2295:
2294:
2281:
2278:
2275:
2272:
2269:
2266:
2261:
2257:
2245:
2244:
2231:
2228:
2225:
2221:
2215:
2212:
2209:
2205:
2201:
2196:
2192:
2177:
2176:
2163:
2159:
2155:
2152:
2146:
2143:
2139:
2134:
2131:
2128:
2122:
2117:
2114:
2111:
2083:
2079:
2074:
2070:
2067:
2064:
2061:
2039:
2035:
2030:
2026:
2023:
2020:
2017:
2014:
2003:
2002:
1990:
1983:
1980:
1973:
1969:
1965:
1959:
1956:
1951:
1947:
1942:
1936:
1932:
1925:
1921:
1916:
1911:
1907:
1903:
1900:
1897:
1894:
1891:
1888:
1865:
1862:
1859:
1854:
1850:
1838:
1837:
1826:
1823:
1820:
1817:
1809:
1805:
1801:
1795:
1791:
1783:
1779:
1772:
1768:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1730:
1727:
1723:
1718:
1712:
1709:
1704:
1701:
1669:
1665:
1651:
1650:
1647:
1642:
1641:
1640:
1626:
1622:
1617:
1613:
1610:
1607:
1604:
1601:
1579:
1575:
1570:
1566:
1563:
1560:
1557:
1546:
1545:
1544:
1533:
1526:
1522:
1518:
1515:
1511:
1505:
1502:
1499:
1494:
1490:
1483:
1480:
1477:
1474:
1468:
1464:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1432:
1428:
1424:
1421:
1393:
1389:
1376:
1375:
1371:
1370:
1358:
1338:
1333:
1330:
1325:
1322:
1317:
1313:
1290:
1287:
1284:
1280:
1276:
1273:
1270:
1248:
1245:
1242:
1238:
1232:
1228:
1224:
1221:
1218:
1198:
1195:
1188:
1184:
1179:
1176:
1173:
1167:
1164:
1152:
1151:
1150:
1149:
1138:
1135:
1130:
1127:
1122:
1119:
1114:
1110:
1104:
1100:
1096:
1091:
1086:
1083:
1078:
1071:
1068:
1065:
1062:
1058:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1024:
1020:
1016:
1011:
1008:
1005:
1001:
997:
994:
981:
980:
966:
963:
960:
956:
950:
946:
942:
939:
915:
911:
906:
903:
900:
864:
860:
856:
851:
848:
845:
841:
837:
834:
831:
826:
823:
817:
814:
803:
802:
801:
790:
784:
781:
777:
772:
767:
764:
761:
757:
742:
741:
736:
733:
705:
701:
697:
694:
691:
688:
657:
653:
647:
625:
621:
609:
608:
594:
590:
586:
583:
580:
577:
554:
534:
510:
506:
500:
473:
453:
442:mean reversion
429:
424:
420:
416:
413:
410:
407:
386:
366:
346:
321:
301:
281:
270:Wiener process
255:
251:
239:
238:
227:
222:
218:
214:
206:
202:
196:
193:
190:
187:
183:
178:
174:
170:
167:
164:
161:
158:
153:
149:
145:
112:
108:
87:
84:
48:interest rates
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6144:
6133:
6130:
6128:
6125:
6123:
6120:
6118:
6115:
6113:
6110:
6109:
6107:
6092:
6089:
6087:
6084:
6083:
6080:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6026:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5996:
5994:
5991:
5990:
5988:
5984:
5976:
5973:
5971:
5968:
5967:
5966:
5963:
5961:
5958:
5956:
5953:
5951:
5948:
5946:
5945:Stopping time
5943:
5939:
5936:
5935:
5934:
5931:
5929:
5926:
5924:
5921:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5899:
5896:
5894:
5891:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5799:
5796:
5795:
5793:
5789:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5759:
5757:
5755:
5751:
5744:
5740:
5736:
5735:Hewitt–Savage
5732:
5728:
5724:
5720:
5719:Zero–one laws
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5666:
5664:
5660:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5539:
5537:
5535:
5531:
5525:
5522:
5520:
5517:
5515:
5512:
5510:
5507:
5506:
5504:
5502:
5498:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5427:
5424:
5422:
5419:
5417:
5416:Black–Scholes
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5398:
5396:
5394:
5390:
5384:
5381:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5355:
5353:
5351:
5347:
5341:
5338:
5336:
5333:
5329:
5326:
5324:
5321:
5320:
5319:
5318:Point process
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5297:
5294:
5292:
5289:
5288:
5287:
5284:
5282:
5279:
5277:
5276:Gibbs measure
5274:
5272:
5269:
5267:
5264:
5263:
5261:
5257:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5224:
5221:
5219:
5216:
5214:
5211:
5209:
5206:
5205:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5180:
5178:
5174:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5121:
5118:
5116:
5113:
5112:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5065:Itô diffusion
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5040:Gamma process
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4989:
4986:
4984:
4981:
4979:
4976:
4974:
4971:
4969:
4966:
4965:
4964:
4961:
4957:
4954:
4953:
4952:
4949:
4947:
4944:
4942:
4939:
4938:
4936:
4934:
4930:
4922:
4919:
4917:
4914:
4912:
4911:Self-avoiding
4909:
4907:
4904:
4903:
4902:
4899:
4897:
4896:Moran process
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4863:
4861:
4859:
4858:Discrete time
4855:
4851:
4844:
4839:
4837:
4832:
4830:
4825:
4824:
4821:
4808:
4805:
4802:
4799:
4796:
4793:
4792:
4790:
4786:
4780:
4779:Puttable bond
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4754:Callable bond
4752:
4751:
4749:
4747:
4743:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4713:
4711:
4707:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4665:Nominal yield
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4640:Current yield
4638:
4636:
4635:Credit spread
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4617:
4615:
4613:
4609:
4603:
4600:
4598:
4595:
4593:
4592:Puttable bond
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4522:Callable bond
4520:
4518:
4515:
4513:
4510:
4509:
4507:
4503:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4470:
4467:
4465:
4462:
4461:
4460:
4457:
4455:
4452:
4451:
4449:
4445:
4439:
4436:
4434:
4431:
4429:
4426:
4425:
4422:
4418:
4411:
4406:
4404:
4399:
4397:
4392:
4391:
4388:
4380:
4376:
4372:
4368:
4364:
4360:
4355:
4350:
4346:
4342:
4337:
4335:
4332:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4299:
4295:
4289:
4285:
4280:
4276:
4272:
4269:(6): 89–109.
4268:
4264:
4259:
4255:
4249:
4241:
4237:
4233:
4229:
4225:
4221:
4220:
4214:
4210:
4208:0-13-009056-5
4204:
4200:
4199:Prentice Hall
4195:
4194:
4187:
4186:
4182:
4173:
4168:
4163:
4158:
4154:
4150:
4146:
4139:
4136:
4131:
4127:
4122:
4117:
4112:
4107:
4103:
4099:
4095:
4088:
4085:
4080:
4076:
4071:
4066:
4062:
4058:
4054:
4047:
4044:
4039:
4035:
4030:
4025:
4021:
4017:
4013:
4006:
4003:
3998:
3994:
3990:
3986:
3982:
3978:
3973:
3968:
3964:
3960:
3956:
3949:
3946:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3913:
3906:
3903:
3898:
3894:
3890:
3886:
3882:
3878:
3874:
3870:
3863:
3860:
3855:
3849:
3845:
3841:
3837:
3830:
3827:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3783:
3780:
3775:
3771:
3767:
3763:
3760:(1): 89–109.
3759:
3755:
3751:
3744:
3741:
3735:
3732:
3721:
3717:
3711:
3708:
3702:
3698:
3695:
3693:
3692:Vasicek model
3690:
3688:
3685:
3684:
3680:
3678:
3676:
3672:
3668:
3663:
3660:
3656:
3648:
3628:
3624:
3620:
3617:
3612:
3608:
3602:
3599:
3592:
3591:
3570:
3567:
3559:
3556:
3553:
3547:
3543:
3533:
3530:
3527:
3521:
3518:
3515:
3507:
3504:
3496:
3493:
3490:
3484:
3480:
3473:
3467:
3461:
3458:
3455:
3449:
3442:
3441:
3422:
3418:
3413:
3409:
3406:
3403:
3398:
3389:
3386:
3378:
3375:
3372:
3366:
3362:
3352:
3349:
3346:
3340:
3337:
3334:
3327:
3323:
3316:
3313:
3310:
3301:
3298:
3295:
3288:
3284:
3281:
3275:
3270:
3264:
3261:
3258:
3252:
3245:
3244:
3243:
3221:
3217:
3210:
3207:
3204:
3198:
3195:
3191:
3184:
3181:
3178:
3172:
3169:
3163:
3160:
3157:
3151:
3144:
3143:
3142:
3140:
3132:
3127:
3125:
3122:
3121:
3120:
3118:
3111:
3107:
3104:
3101:
3100:
3082:
3078:
3069:
3053:
3048:
3044:
3038:
3029:
3026:
3023:
3015:
3009:
3003:
3000:
2993:
2987:
2980:
2974:
2971:
2965:
2958:
2952:
2943:
2939:
2935:
2930:
2924:
2921:
2917:
2906:
2905:
2904:
2903:
2899:
2898:
2881:
2876:
2872:
2866:
2858:
2854:
2847:
2844:
2841:
2829:
2825:
2821:
2818:
2812:
2809:
2804:
2800:
2796:
2791:
2785:
2782:
2778:
2770:
2769:
2768:
2767:
2763:
2762:
2759:
2756:
2755:
2751:
2735:
2730:
2720:
2717:
2714:
2710:
2706:
2703:
2694:
2691:
2684:
2680:
2676:
2670:
2662:
2659:
2656:
2653:
2649:
2645:
2640:
2637:
2634:
2630:
2621:
2616:
2612:
2604:
2600:
2596:
2588:
2584:
2580:
2575:
2571:
2564:
2561:
2554:
2539:
2514:
2511:
2508:
2504:
2500:
2497:
2491:
2488:
2483:
2480:
2477:
2473:
2467:
2463:
2459:
2451:
2447:
2443:
2438:
2434:
2427:
2417:
2400:
2396:
2392:
2389:
2386:
2383:
2361:
2357:
2348:
2329:
2325:
2319:
2311:
2308:
2305:
2304:
2300:
2293:
2273:
2270:
2264:
2255:
2229:
2226:
2223:
2219:
2213:
2210:
2207:
2203:
2199:
2190:
2182:
2181:
2180:
2157:
2153:
2150:
2144:
2141:
2137:
2132:
2129:
2126:
2120:
2115:
2112:
2109:
2099:
2098:
2097:
2081:
2077:
2072:
2068:
2065:
2062:
2059:
2037:
2033:
2028:
2024:
2021:
2018:
2015:
2012:
1988:
1981:
1978:
1967:
1963:
1957:
1954:
1945:
1940:
1934:
1930:
1923:
1919:
1914:
1905:
1898:
1895:
1892:
1886:
1879:
1878:
1877:
1863:
1857:
1852:
1821:
1818:
1807:
1803:
1793:
1781:
1777:
1770:
1766:
1761:
1755:
1749:
1746:
1743:
1737:
1728:
1716:
1710:
1702:
1689:
1688:
1687:
1685:
1663:
1653:
1652:
1646:
1645:
1624:
1620:
1615:
1611:
1608:
1605:
1602:
1599:
1577:
1573:
1568:
1564:
1561:
1558:
1555:
1547:
1531:
1520:
1516:
1513:
1509:
1503:
1500:
1497:
1488:
1478:
1466:
1462:
1456:
1450:
1447:
1444:
1441:
1438:
1435:
1426:
1419:
1412:
1411:
1409:
1387:
1378:
1377:
1373:
1372:
1356:
1331:
1328:
1323:
1315:
1311:
1288:
1285:
1282:
1278:
1274:
1271:
1268:
1246:
1243:
1240:
1236:
1230:
1226:
1222:
1219:
1216:
1196:
1193:
1186:
1182:
1177:
1174:
1171:
1165:
1162:
1154:
1153:
1136:
1128:
1125:
1120:
1112:
1108:
1102:
1098:
1094:
1089:
1084:
1081:
1076:
1069:
1066:
1063:
1060:
1056:
1051:
1048:
1042:
1039:
1036:
1033:
1030:
1027:
1022:
1018:
1014:
1009:
1006:
1003:
999:
992:
985:
984:
983:
982:
964:
961:
958:
954:
948:
944:
940:
937:
913:
909:
904:
901:
898:
887:
883:
862:
858:
849:
846:
843:
839:
835:
832:
824:
821:
815:
812:
804:
788:
782:
779:
775:
770:
765:
762:
759:
755:
747:
746:
744:
743:
739:
738:
734:
732:
730:
726:
722:
703:
699:
695:
692:
689:
686:
677:
675:
655:
651:
645:
623:
619:
592:
588:
584:
581:
578:
575:
568:
567:
566:
552:
532:
508:
504:
498:
490:
485:
471:
451:
443:
422:
418:
414:
411:
405:
384:
364:
344:
336:
319:
299:
279:
271:
253:
249:
225:
220:
216:
212:
204:
200:
194:
191:
188:
185:
176:
172:
168:
165:
159:
156:
151:
147:
143:
136:
135:
134:
132:
128:
110:
106:
92:
85:
83:
81:
77:
76:Vasicek model
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
6003:Econometrics
5965:Wiener space
5853:Itô integral
5754:Inequalities
5643:Self-similar
5613:Gauss–Markov
5603:Exchangeable
5583:Càdlàg paths
5519:Risk process
5471:LIBOR market
5435:
5340:Random graph
5335:Random field
5147:Superprocess
5085:Lévy process
5080:Jump process
5055:Hunt process
4891:Markov chain
4788:Institutions
4746:Bond options
4690:Yield spread
4582:Lottery bond
4512:Accrual bond
4438:Fixed income
4344:
4340:
4310:
4306:
4283:
4266:
4262:
4248:cite journal
4223:
4219:Econometrica
4217:
4192:
4172:11585/916048
4152:
4148:
4138:
4101:
4098:SeMa Journal
4097:
4087:
4060:
4056:
4046:
4019:
4015:
4005:
3962:
3958:
3948:
3915:
3911:
3905:
3872:
3868:
3862:
3835:
3829:
3796:
3792:
3782:
3757:
3753:
3743:
3734:
3723:. Retrieved
3719:
3710:
3669:is given by
3664:
3652:
3241:
3136:
3133:Bond pricing
3115:
2246:
2178:
2004:
1839:
1654:
881:
735:Distribution
729:Heston model
679:In the case
678:
610:
486:
240:
97:
78:, itself an
43:
37:
6048:Ruin theory
5986:Disciplines
5858:Itô's lemma
5633:Predictable
5308:Percolation
5291:Potts model
5286:Ising model
5250:White noise
5208:Differences
5070:Itô process
5010:Cox process
4906:Loop-erased
4901:Random walk
4685:Yield curve
4645:Dirty price
4620:Clean price
4496:Global bond
4464:Senior debt
4454:Agency bond
4417:Bond market
2752:Calibration
674:equilibrium
94:CIR process
64:John C. Cox
56:market risk
18:CIR process
6106:Categories
6058:Statistics
5838:Filtration
5739:Kolmogorov
5723:Blumenthal
5648:Stationary
5588:Continuous
5576:Properties
5461:Hull–White
5203:Martingale
5090:Local time
4978:Fractional
4956:pure birth
4354:1901.02246
4111:2106.03716
3972:1901.02246
3725:2023-10-14
3703:References
3697:Chen model
3675:Chen model
3649:Extensions
3137:Under the
3112:Simulation
2301:Properties
335:parameters
5970:Classical
4983:Geometric
4973:Excursion
4625:Convexity
4433:Debenture
4379:126507446
4371:1099-131X
4155:(2): 38.
4130:235358123
4079:2029-4921
4038:0277-6693
3997:126507446
3989:0277-6693
3940:204435499
3932:1526-5943
3897:204424299
3889:1086-7376
3813:0949-2984
3774:0960-1627
3625:σ
3568:−
3557:−
3505:−
3494:−
3419:σ
3387:−
3376:−
3314:−
3196:−
3079:ε
3070:provided
3045:ε
3036:Δ
3030:σ
3021:Δ
3001:−
2978:Δ
2936:−
2928:Δ
2873:ε
2864:Δ
2848:σ
2839:Δ
2822:−
2797:−
2789:Δ
2715:−
2707:−
2681:σ
2654:−
2646:−
2635:−
2613:σ
2581:∣
2565:
2509:−
2501:−
2478:−
2444:∣
2428:
2397:σ
2393:≥
2320:σ
2277:∞
2265:∈
2260:∞
2227:β
2224:−
2211:−
2208:α
2200:∝
2195:∞
2162:∞
2154:
2130:β
2127:−
2113:−
2110:α
2078:σ
2060:β
2034:σ
2013:α
2005:Defining
1972:∞
1950:∞
1931:σ
1910:∞
1896:−
1861:→
1849:∂
1800:∂
1790:∂
1778:σ
1747:−
1726:∂
1722:∂
1708:∂
1700:∂
1668:∞
1621:σ
1600:α
1574:σ
1556:β
1525:∞
1517:β
1514:−
1501:−
1498:α
1493:∞
1479:α
1473:Γ
1467:α
1463:β
1451:σ
1431:∞
1392:∞
1241:−
1194:−
1183:σ
1067:−
1061:−
1043:σ
959:−
910:σ
859:σ
844:−
836:−
700:σ
646:σ
589:σ
585:≥
499:σ
415:−
385:σ
320:σ
195:σ
169:−
86:The model
6091:Category
5975:Abstract
5509:Bühlmann
5115:Compound
4700:Z-spread
4655:I-spread
4650:Duration
4327:35316609
3821:35316609
3681:See also
3671:Lin Chen
723:. It is
491:factor,
333:are the
5598:Ergodic
5486:Vašíček
5328:Poisson
4988:Meander
4809:(SIFMA)
4240:1911242
725:ergodic
125:with a
5938:Tanaka
5623:Mixing
5618:Markov
5491:Wilkie
5456:Ho–Lee
5451:Heston
5223:Super-
4968:Bridge
4916:Biased
4803:(ICMA)
4797:(CMSA)
4630:Coupon
4532:Consol
4377:
4369:
4325:
4290:
4238:
4205:
4128:
4077:
4036:
3995:
3987:
3938:
3930:
3895:
3887:
3850:
3819:
3811:
3772:
3242:where
1548:where
1303:, and
1155:where
880:, and
805:where
377:, and
312:, and
241:where
127:Feller
42:, the
5791:Tools
5567:M/M/c
5562:M/M/1
5557:M/G/1
5547:Fluid
5213:Local
4375:S2CID
4349:arXiv
4323:S2CID
4236:JSTOR
4126:S2CID
4106:arXiv
3993:S2CID
3967:arXiv
3936:S2CID
3893:S2CID
3817:S2CID
3128:Exact
888:with
884:is a
268:is a
5743:Lévy
5542:Bulk
5426:Chen
5218:Sub-
5176:Both
4428:Bond
4367:ISSN
4288:ISBN
4254:link
4203:ISBN
4075:ISSN
4034:ISSN
3985:ISSN
3928:ISSN
3885:ISSN
3848:ISBN
3809:ISSN
3770:ISSN
2052:and
1592:and
545:and
487:The
70:and
5323:Cox
4359:doi
4315:doi
4271:doi
4228:doi
4167:hdl
4157:doi
4116:doi
4065:doi
4024:doi
3977:doi
3920:doi
3877:doi
3840:doi
3801:doi
3762:doi
2562:Var
2151:log
676:).
133:is
38:In
6108::
5741:,
5737:,
5733:,
5729:,
5725:,
4373:.
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3609:a
3603:=
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3308:(
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3027:+
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2925:+
2922:t
2918:r
2882:,
2877:t
2867:t
2859:t
2855:r
2845:+
2842:t
2835:)
2830:t
2826:r
2819:b
2816:(
2813:a
2810:=
2805:t
2801:r
2792:t
2786:+
2783:t
2779:r
2736:.
2731:2
2727:)
2721:t
2718:a
2711:e
2704:1
2701:(
2695:a
2692:2
2685:2
2677:b
2671:+
2668:)
2663:t
2660:a
2657:2
2650:e
2641:t
2638:a
2631:e
2627:(
2622:a
2617:2
2605:0
2601:r
2597:=
2594:]
2589:0
2585:r
2576:t
2572:r
2568:[
2552:,
2540:b
2520:)
2515:t
2512:a
2505:e
2498:1
2495:(
2492:b
2489:+
2484:t
2481:a
2474:e
2468:0
2464:r
2460:=
2457:]
2452:0
2448:r
2439:t
2435:r
2431:[
2425:E
2401:2
2390:b
2387:a
2384:2
2362:0
2358:r
2330:t
2326:r
2309:,
2280:]
2274:,
2271:0
2268:(
2256:p
2230:r
2220:e
2214:1
2204:r
2191:p
2158:p
2145:r
2142:d
2138:d
2133:=
2121:r
2116:1
2082:2
2073:/
2069:a
2066:2
2063:=
2038:2
2029:/
2025:b
2022:a
2019:2
2016:=
1989:)
1982:r
1979:d
1968:p
1964:d
1958:r
1955:+
1946:p
1941:(
1935:2
1924:2
1920:1
1915:=
1906:p
1902:)
1899:r
1893:b
1890:(
1887:a
1864:0
1858:p
1853:t
1825:)
1822:p
1819:r
1816:(
1808:2
1804:r
1794:2
1782:2
1771:2
1767:1
1762:=
1759:]
1756:p
1753:)
1750:r
1744:b
1741:(
1738:a
1735:[
1729:r
1717:+
1711:t
1703:p
1664:p
1639:.
1625:2
1616:/
1612:b
1609:a
1606:2
1603:=
1578:2
1569:/
1565:a
1562:2
1559:=
1532:,
1521:r
1510:e
1504:1
1489:r
1482:)
1476:(
1457:=
1454:)
1448:,
1445:b
1442:,
1439:a
1436:;
1427:r
1423:(
1420:f
1388:r
1369:.
1357:q
1337:)
1332:v
1329:u
1324:2
1321:(
1316:q
1312:I
1289:T
1286:+
1283:t
1279:r
1275:c
1272:=
1269:v
1247:T
1244:a
1237:e
1231:t
1227:r
1223:c
1220:=
1217:u
1197:1
1187:2
1178:b
1175:a
1172:2
1166:=
1163:q
1137:,
1134:)
1129:v
1126:u
1121:2
1118:(
1113:q
1109:I
1103:2
1099:/
1095:q
1090:)
1085:u
1082:v
1077:(
1070:v
1064:u
1057:e
1052:c
1049:=
1046:)
1040:,
1037:b
1034:,
1031:a
1028:,
1023:t
1019:r
1015:;
1010:T
1007:+
1004:t
1000:r
996:(
993:f
965:T
962:a
955:e
949:t
945:r
941:c
938:2
914:2
905:b
902:a
899:4
882:Y
863:2
855:)
850:T
847:a
840:e
833:1
830:(
825:a
822:2
816:=
813:c
789:,
783:c
780:2
776:Y
771:=
766:T
763:+
760:t
756:r
704:2
696:=
693:b
690:a
687:4
656:t
652:r
624:t
620:r
593:2
582:b
579:a
576:2
553:b
533:a
509:t
505:r
472:a
452:b
428:)
423:t
419:r
412:b
409:(
406:a
365:b
345:a
300:b
280:a
254:t
250:W
226:,
221:t
217:W
213:d
205:t
201:r
192:+
189:t
186:d
182:)
177:t
173:r
166:b
163:(
160:a
157:=
152:t
148:r
144:d
111:t
107:r
20:)
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