3579:
from turning a short-rate model used for pricing into a forecasting tool, lies in an appropriate partitioning of the dataset into subgroups according to a given distribution ). In there it was shown how the said partitioning enables capturing statistically significant time changes in volatility of interest rates. following the said approach, Orlando et al. (2021) ) compares the Hull–White model with the CIR model in terms of forecasting and prediction of interest rate directionality.
2226:
1172:
1704:
3578:
Even though single factor models such as
Vasicek, CIR and Hull–White model has been devised for pricing, recent research has shown their potential with regard to forecasting. In Orlando et al. (2018, 2019,) was provided a new methodology to forecast future interest rates called CIR#. The ideas, apart
3531:
of today's short rate). Thus knowing how to price caps is also sufficient for pricing swaptions. In the even that the underlying is a compounded backward-looking rate rather than a (forward-looking) LIBOR term rate, Turfus (2020) shows how this formula can be straightforwardly modified to take into
503:
is a deterministic function, typically the identity function (extension of the one-factor version, analytically tractable, and with potentially negative rates), the natural logarithm (extension of Black–Karasinski, not analytically tractable, and with positive interest rates), or combinations
981:
1936:
3507:
1497:
46:. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a
992:
1508:
3105:
217:
There is a degree of ambiguity among practitioners about exactly which parameters in the model are time-dependent or what name to apply to the model in each case. The most commonly accepted naming convention is the following:
812:
2221:{\displaystyle A(S,T)={\frac {P(0,T)}{P(0,S)}}\exp \left(\,-B(S,T){\frac {\partial \log(P(0,S))}{\partial S}}-{\frac {\sigma ^{2}(\exp(-\alpha T)-\exp(-\alpha S))^{2}(\exp(2\alpha S)-1)}{4\alpha ^{3}}}\right).}
1930:
3371:
1312:
476:
2992:
3174:
212:
3298:
605:
1221:
1835:
2675:
2405:
1167:{\displaystyle r(t)\sim {\mathcal {N}}\left(e^{-\alpha t}r(0)+{\frac {\theta }{\alpha }}\left(1-e^{-\alpha t}\right),{\frac {\sigma ^{2}}{2\alpha }}\left(1-e^{-2\alpha t}\right)\right),}
4585:
3516:-bond measure, whereas we did not specify a measure at all for the original Hull–White process. This does not matter — the volatility is all that matters and is measure-independent.
2575:
2795:
4506:
711:
5120:
2437:
524:
501:
1699:{\displaystyle r(t)\sim {\mathcal {N}}\left(e^{-\alpha t}r(0)+\int _{0}^{t}e^{\alpha (s-t)}\theta (s)ds,{\frac {\sigma ^{2}}{2\alpha }}\left(1-e^{-2\alpha t}\right)\right).}
663:
3559:, or other derivatives in a multi-currency context such as Quanto Constant Maturity Swaps, as explained for example in Brigo and Mercurio (2001). The efficient and exact
3328:
1272:
2487:
1304:
727:
describing the current term structure of interest rates. Typically α is left as a user input (for example it may be estimated from historical data). σ is determined via
4944:
800:
780:
760:
633:
289:
269:
239:
3363:
2833:
2717:
2264:
1245:
5547:
4540:
3652:
5077:
5057:
2291:
5461:
3563:
of the Hull–White model with time dependent parameters can be easily performed, see
Ostrovski (2013) and (2016). An open-source implementation of the exact
3003:
4494:
3547:
However, valuing vanilla instruments such as caps and swaptions is useful primarily for calibration. The real use of the model is to value somewhat more
5378:
3523:
are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model.
5388:
5062:
4500:
3711:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (1 January 2019). "A new approach to forecast market interest rates through the CIR model".
5072:
976:{\displaystyle r(t)=e^{-\alpha t}r(0)+{\frac {\theta }{\alpha }}\left(1-e^{-\alpha t}\right)+\sigma e^{-\alpha t}\int _{0}^{t}e^{\alpha u}\,dW(u),}
5430:
5145:
5327:
5617:
5607:
5130:
4107:
4018:
3695:
5517:
5481:
4077:
5434:
5785:
5522:
4430:
3502:{\displaystyle {\sqrt {S}}\sigma _{P}={\frac {\sigma }{\alpha }}(1-\exp(-\alpha (T-S))){\sqrt {\frac {1-\exp(-2\alpha S)}{2\alpha }}}.}
4632:
4533:
4425:
1846:
1492:{\displaystyle r(t)=e^{-\alpha t}r(0)+\int _{0}^{t}e^{\alpha (s-t)}\theta (s)ds+\sigma e^{-\alpha t}\int _{0}^{t}e^{\alpha u}\,dW(u),}
5587:
5165:
5135:
3976:
3593:
317:
5438:
5422:
5632:
5337:
4557:
5537:
5502:
5471:
5466:
5105:
4902:
4819:
5476:
4804:
717:
5816:
5100:
4907:
4420:
4296:
3960:
4826:
5562:
5442:
2845:
5790:
5567:
5403:
5302:
5287:
4699:
4615:
4526:
5577:
5213:
3754:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (19 August 2019). "Interest rates calibration with a CIR model".
3539:
Swaptions can also be priced directly as described in
Henrard (2003). Direct implementations are usually more efficient.
3116:
5572:
92:
5175:
3197:
532:
4759:
4704:
4620:
4236:
3926:
John Hull and Alan White, "One factor interest rate models and the valuation of interest rate derivative securities,"
1180:
732:
5507:
5497:
5140:
5110:
3598:
1736:
504:(proportional to the natural logarithm on small rates and proportional to the identity function on large rates); and
3678:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (2018). "A New
Approach to CIR Short-Term Rates Modelling".
5821:
5512:
4677:
4575:
4276:
714:
5223:
4799:
4580:
3919:
John Hull and Alan White, "The pricing of options on interest rate caps and floors using the Hull–White model" in
2583:
2308:
5826:
5592:
5393:
5307:
5292:
4682:
4100:
3520:
5426:
5312:
4734:
4814:
4789:
4435:
4062:
3948:
62:
51:
5532:
5115:
4650:
2836:
5811:
5727:
5717:
5408:
5190:
4929:
4794:
4605:
3556:
2500:
2267:
47:
5012:
4027:
Henrard, Marc (2003). "Explicit Bond Option and
Swaption Formula in Heath–Jarrow–Morton One Factor Model,"
5669:
5597:
4856:
4369:
4271:
3564:
3560:
2728:
66:
4051:
Ostrovski, Vladimir (2016). Efficient and Exact
Simulation of the Gaussian Affine Interest Rate Models.,
676:
5692:
5674:
5654:
5649:
5368:
5200:
5180:
5027:
4970:
4809:
4719:
4415:
4379:
4216:
4175:
3856:"Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work"
31:
3620:"A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation"
3524:
2413:
5767:
5722:
5712:
5453:
5398:
5373:
5342:
5322:
5082:
5067:
4934:
4093:
507:
484:
4038:
Henrard, Marc (2009). Efficient swaptions price in Hull–White one factor model, arXiv, 0901.1776v1.
5762:
5602:
5527:
5332:
5092:
5002:
4892:
4185:
3968:
1224:
642:
311::657–658) contains an additional disturbance term whose mean reverts to zero, and is of the form:
5732:
5697:
5612:
5582:
5352:
5347:
5170:
5007:
4672:
4610:
4549:
4261:
3836:
3810:
3779:
3736:
3548:
3528:
3365:. A fairly substantial amount of algebra shows that it is related to the original parameters via
3306:
1250:
39:
5413:
2450:
1280:
5752:
5557:
5208:
4965:
4882:
4851:
4744:
4724:
4714:
4570:
4565:
4468:
4394:
4246:
4168:
4046:
4014:
3990:
3982:
3972:
3877:
3828:
3771:
3728:
3691:
3552:
803:
55:
5418:
5155:
785:
765:
745:
618:
274:
254:
224:
5772:
5659:
5542:
4912:
4887:
4836:
4764:
4687:
4640:
4458:
4329:
4301:
4241:
4226:
3952:
3912:
John Hull and Alan White, "Numerical procedures for implementing term structure models II,"
3867:
3820:
3763:
3720:
3683:
3627:
3333:
2803:
2687:
2234:
83:
3905:
John Hull and Alan White, "Numerical procedures for implementing term structure models I,"
5737:
5637:
5622:
5383:
5317:
4995:
4939:
4922:
4667:
4473:
4463:
4266:
4256:
4251:
4180:
3533:
2440:
1230:
5552:
4784:
3953:
5742:
5707:
5627:
5233:
4980:
4897:
4866:
4861:
4841:
4831:
4774:
4769:
4749:
4729:
4694:
4662:
4645:
4374:
4359:
4324:
4311:
4286:
4190:
4158:
4127:
4006:
3798:
5805:
5644:
5185:
5022:
5017:
4975:
4917:
4739:
4655:
4595:
4478:
4453:
4364:
4349:
4339:
4291:
4231:
4221:
4002:
3964:
3840:
3783:
3740:
3588:
1723:
294:
43:
4045:
Ostrovski, Vladimir (2013). Efficient and Exact
Simulation of the Hull–White Model,
3799:"Forecasting interest rates through Vasicek and CIR models: A partitioning approach"
3100:{\displaystyle d_{1}={\frac {\log(F/K)+\sigma _{P}^{2}S/2}{\sigma _{P}{\sqrt {S}}}}}
5702:
5664:
5218:
5150:
5039:
5034:
4846:
4779:
4754:
4590:
4389:
4334:
4281:
4211:
4137:
3527:
applies to Hull–White (as today's value of a swaption in the Hull–White model is a
3330:
is the standard deviation (relative volatility) of the log-normal distribution for
5282:
3682:. Contributions to Management Science. Springer International Publishing: 35–43.
2719:
analytically when working in the Hull–White model. For example, in the case of a
713:
and positive if the current value is small. That is, the stochastic process is a
5747:
5266:
5261:
5256:
5246:
5049:
4990:
4985:
4949:
4709:
4600:
4445:
4384:
4344:
4319:
4195:
4163:
4153:
4116:
4011:
Interest Rate Models — Theory and
Practice with Smile, Inflation and Credit
3687:
728:
724:
17:
5757:
5297:
5241:
5125:
2720:
3881:
3832:
3775:
3767:
3732:
3724:
5251:
4132:
3994:
2279:
4082:, Fixed Income Quant Group, DTCC (detailed numeric example and derivation)
639:-dependence. Neglecting the stochastic term for a moment, notice that for
4399:
4354:
3933:
John Hull and Alan White, "Pricing interest-rate derivative securities",
3631:
736:
5078:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4518:
3872:
3855:
3824:
3797:
Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (July 2020).
4071:
4056:
3619:
4070:
Turfus, Colin (2020). Caplet
Pricing with Backward-Looking Rates.,
3815:
4039:
4032:
3986:
3898:
John Hull and Alan White, "Using Hull–White interest rate trees,"
2835:
is lognormally distributed, the general calculation used for the
4064:
Implementation of Hull–White's No-Arbitrage Term
Structure Model
1925:{\displaystyle B(S,T)={\frac {1-\exp(-\alpha (T-S))}{\alpha }},}
4522:
4089:
4067:, Diploma Thesis, Center for Central European Financial Markets
4053:
International Journal of Financial Engineering, Vol. 3, No. 02.
3951:(2006). "Interest Rate Derivatives: Models of the Short Rate".
4085:
471:{\displaystyle d\,f(r(t))=\leftdt+\sigma _{1}(t)\,dW_{1}(t),}
2647:
2443:. Moreover, standard arbitrage arguments show that the time
2377:
1529:
1186:
1013:
5058:
Autoregressive conditional heteroskedasticity (ARCH) model
69:
in 1990. The model is still popular in the market today.
4586:
Independent and identically distributed random variables
4029:
International Journal of Theoretical and Applied Finance
5063:
Autoregressive integrated moving average (ARIMA) model
4507:
Securities Industry and Financial Markets Association
3374:
3336:
3309:
3200:
3119:
3006:
2987:{\displaystyle {E}_{S}=KN(-d_{2})-F(t,S,T)N(-d_{1}),}
2848:
2806:
2731:
2690:
2586:
2503:
2453:
2416:
2311:
2237:
1939:
1849:
1739:
1511:
1315:
1283:
1253:
1233:
1183:
995:
815:
788:
768:
748:
679:
645:
621:
535:
511:
510:
488:
487:
320:
277:
257:
227:
95:
3567:
following Fries (2016) can be found in finmath lib.
5685:
5490:
5452:
5361:
5275:
5232:
5199:
5091:
5048:
4958:
4875:
4631:
4556:
4487:
4444:
4408:
4310:
4204:
4146:
3169:{\displaystyle d_{2}=d_{1}-\sigma _{P}{\sqrt {S}}.}
526:has an initial value of 0 and follows the process:
3501:
3357:
3322:
3292:
3168:
3099:
2986:
2827:
2789:
2711:
2669:
2569:
2481:
2431:
2399:
2258:
2220:
1924:
1829:
1698:
1491:
1298:
1266:
1239:
1215:
1166:
975:
794:
774:
754:
705:
657:
627:
599:
518:
495:
470:
283:
263:
233:
207:{\displaystyle dr(t)=\left\,dt+\sigma (t)\,dW(t).}
206:
3854:Orlando, Giuseppe; Bufalo, Michele (2021-05-26).
3293:{\displaystyle P(0,S)KN(-d_{2})-P(0,T)N(-d_{1}).}
600:{\displaystyle du=-bu\,dt+\sigma _{2}\,dW_{2}(t)}
4945:Stochastic chains with memory of variable length
3921:Advanced Strategies in Financial Risk Management
1216:{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}
1830:{\displaystyle P(S,T)=A(S,T)\exp(-B(S,T)r(S)),}
3928:Journal of Financial and Quantitative Analysis
2680:Thus it is possible to value many derivatives
4534:
4101:
3930:, Vol 28, No 2, (June 1993) pp. 235–254.
3902:, Vol. 3, No. 3 (Spring 1996), pp. 26–36
2439:is the expectation taken with respect to the
2292:fundamental theorem of arbitrage-free pricing
86:. In general, it has the following dynamics:
8:
2670:{\displaystyle F_{V}(t,T)=\mathbb {E} _{T}.}
2400:{\displaystyle V(t)=P(t,S)\mathbb {E} _{S}.}
2286:bond (which corresponds to switching to the
615:For the rest of this article we assume only
61:The first Hull–White model was described by
3512:Note that this expectation was done in the
5073:Autoregressive–moving-average (ARMA) model
4541:
4527:
4519:
4495:Commercial Mortgage Securities Association
4108:
4094:
4086:
4013:(2nd ed. 2006 ed.). Springer Verlag.
3543:Monte-Carlo simulation, trees and lattices
2231:Note that their terminal distribution for
3871:
3814:
3450:
3395:
3386:
3375:
3373:
3335:
3314:
3308:
3278:
3235:
3199:
3156:
3150:
3137:
3124:
3118:
3087:
3081:
3067:
3058:
3053:
3035:
3020:
3011:
3005:
2972:
2923:
2895:
2855:
2850:
2847:
2805:
2778:
2730:
2689:
2646:
2645:
2621:
2617:
2616:
2591:
2585:
2544:
2508:
2502:
2458:
2452:
2423:
2419:
2418:
2415:
2376:
2375:
2351:
2347:
2346:
2310:
2298:of a derivative which has payoff at time
2236:
2201:
2153:
2095:
2088:
2038:
2016:
1961:
1938:
1871:
1848:
1738:
1668:
1637:
1631:
1589:
1579:
1574:
1543:
1528:
1527:
1510:
1470:
1461:
1451:
1446:
1430:
1381:
1371:
1366:
1335:
1314:
1282:
1258:
1252:
1232:
1204:
1185:
1184:
1182:
1136:
1105:
1099:
1079:
1054:
1027:
1012:
1011:
994:
954:
945:
935:
930:
914:
887:
862:
835:
814:
787:
767:
747:
692:
678:
644:
620:
582:
574:
568:
554:
534:
509:
486:
450:
442:
427:
387:
324:
319:
276:
256:
226:
185:
163:
94:
4501:International Capital Market Association
3955:Options, Futures, and Other Derivatives
3937:, Vol 3, No. 4 (1990) pp. 573–592.
3610:
1710:Bond pricing using the Hull–White model
5379:Doob's martingale convergence theorems
2570:{\displaystyle F_{V}(t,T)=V(t)/P(t,T)}
5131:Constant elasticity of variance (CEV)
5121:Chan–Karolyi–Longstaff–Sanders (CKLS)
7:
3680:New Methods in Fixed Income Modeling
2790:{\displaystyle V(S)=(K-P(S,T))^{+}.}
2290:-forward measure), we have from the
308:
4431:Commercial mortgage-backed security
706:{\displaystyle \theta (t)/\alpha )}
673:is currently "large" (greater than
5618:Skorokhod's representation theorem
5399:Law of large numbers (weak/strong)
4426:Collateralized mortgage obligation
2684:dependent solely on a single bond
2076:
2041:
25:
5588:Martingale representation theorem
739:readily tradeable in the market.
723:θ is calculated from the initial
307:The two-factor Hull–White model (
5633:Stochastic differential equation
5523:Doob's optional stopping theorem
5518:Doob–Meyer decomposition theorem
3713:Studies in Economics and Finance
2432:{\displaystyle \mathbb {E} _{S}}
611:Analysis of the one-factor model
5503:Convergence of random variables
5389:Fisher–Tippett–Gnedenko theorem
3935:The Review of Financial Studies
519:{\displaystyle \displaystyle u}
496:{\displaystyle \displaystyle f}
5101:Binomial options pricing model
4421:Collateralized debt obligation
4297:Reverse convertible securities
3916:, Winter 1994, pp. 37–48.
3481:
3466:
3447:
3444:
3441:
3429:
3420:
3405:
3352:
3340:
3284:
3268:
3262:
3250:
3241:
3225:
3216:
3204:
3043:
3029:
2978:
2962:
2956:
2938:
2929:
2913:
2901:
2892:
2888:
2876:
2864:
2861:
2822:
2810:
2775:
2771:
2759:
2747:
2741:
2735:
2706:
2694:
2661:
2658:
2652:
2639:
2633:
2627:
2609:
2597:
2564:
2552:
2541:
2535:
2526:
2514:
2476:
2464:
2391:
2388:
2382:
2369:
2363:
2357:
2342:
2330:
2321:
2315:
2253:
2241:
2189:
2180:
2168:
2159:
2150:
2146:
2134:
2122:
2110:
2101:
2071:
2068:
2056:
2050:
2035:
2023:
1999:
1987:
1979:
1967:
1955:
1943:
1910:
1907:
1895:
1886:
1865:
1853:
1821:
1818:
1812:
1806:
1794:
1785:
1776:
1764:
1755:
1743:
1619:
1613:
1605:
1593:
1564:
1558:
1521:
1515:
1483:
1477:
1411:
1405:
1397:
1385:
1356:
1350:
1325:
1319:
1293:
1287:
1210:
1191:
1048:
1042:
1005:
999:
967:
961:
856:
850:
825:
819:
700:
689:
683:
594:
588:
462:
456:
439:
433:
406:
403:
397:
391:
384:
378:
363:
357:
343:
340:
334:
328:
198:
192:
182:
176:
155:
149:
143:
137:
128:
122:
108:
102:
27:Model of future interest rates
1:
5568:Kolmogorov continuity theorem
5404:Law of the iterated logarithm
3179:Thus today's value (with the
5573:Kolmogorov extension theorem
5252:Generalized queueing network
4760:Interacting particle systems
3923:, Chapter 4, pp. 59–67.
658:{\displaystyle \alpha >0}
58:can be valued in the model.
4705:Continuous-time random walk
4237:Contingent convertible bond
3909:, Fall 1994, pp. 7–16.
3756:The Journal of Risk Finance
3688:10.1007/978-3-319-95285-7_2
3323:{\displaystyle \sigma _{P}}
1726:has distribution (note the
1714:It turns out that the time-
1267:{\displaystyle \sigma ^{2}}
5843:
5713:Extreme value theory (EVT)
5513:Doob decomposition theorem
4805:Ornstein–Uhlenbeck process
4576:Chinese restaurant process
4277:Inverse floating rate note
2482:{\displaystyle F_{V}(t,T)}
1299:{\displaystyle \theta (t)}
806:can be used to prove that
718:Ornstein–Uhlenbeck process
291:are both time-dependent —
5781:
5593:Optional stopping theorem
5394:Large deviation principle
5146:Heath–Jarrow–Morton (HJM)
5083:Moving-average (MA) model
5068:Autoregressive (AR) model
4893:Hidden Markov model (HMM)
4827:Schramm–Loewner evolution
4123:
3618:Fries, Christian (2016).
3521:interest rate caps/floors
3187:) multiplied back in and
52:interest rate derivatives
5508:Doléans-Dade exponential
5338:Progressively measurable
5136:Cox–Ingersoll–Ross (CIR)
4436:Mortgage-backed security
4205:Types of bonds by payout
4147:Types of bonds by issuer
3768:10.1108/JRF-05-2019-0080
3725:10.1108/SEF-03-2019-0116
3594:Cox–Ingersoll–Ross model
2268:distributed log-normally
5728:Mathematical statistics
5718:Large deviations theory
5548:Infinitesimal generator
5409:Maximal ergodic theorem
5328:Piecewise-deterministic
4930:Random dynamical system
4795:Markov additive process
3961:Upper Saddle River, N.J
3532:account the additional
1502:which has distribution
986:which has distribution
795:{\displaystyle \sigma }
775:{\displaystyle \theta }
755:{\displaystyle \alpha }
628:{\displaystyle \theta }
284:{\displaystyle \alpha }
264:{\displaystyle \theta }
234:{\displaystyle \theta }
5563:Karhunen–Loève theorem
5498:Cameron–Martin formula
5462:Burkholder–Davis–Gundy
4857:Variance gamma process
4370:Option-adjusted spread
4272:Inflation-indexed bond
3914:Journal of Derivatives
3907:Journal of Derivatives
3900:Journal of Derivatives
3860:Journal of Forecasting
3803:Journal of Forecasting
3599:Black–Karasinski model
3565:Monte-Carlo simulation
3561:Monte-Carlo simulation
3503:
3359:
3358:{\displaystyle P(S,T)}
3324:
3294:
3170:
3101:
2988:
2829:
2828:{\displaystyle P(S,T)}
2791:
2713:
2712:{\displaystyle P(S,T)}
2671:
2571:
2483:
2433:
2401:
2260:
2259:{\displaystyle P(S,T)}
2222:
1926:
1831:
1700:
1493:
1300:
1268:
1241:
1217:
1168:
977:
796:
776:
756:
707:
659:
629:
601:
520:
497:
472:
285:
265:
235:
208:
5817:Fixed income analysis
5693:Actuarial mathematics
5655:Uniform integrability
5650:Stratonovich integral
5578:Lévy–Prokhorov metric
5482:Marcinkiewicz–Zygmund
5369:Central limit theorem
4971:Gaussian random field
4800:McKean–Vlasov process
4720:Dyson Brownian motion
4581:Galton–Watson process
4416:Asset-backed security
4380:Weighted-average life
4217:Auction rate security
3653:"HullWhiteModel.java"
3504:
3360:
3325:
3295:
3171:
3102:
2989:
2830:
2792:
2714:
2672:
2572:
2489:for a payoff at time
2484:
2434:
2402:
2261:
2223:
1927:
1832:
1701:
1494:
1301:
1269:
1242:
1218:
1169:
978:
797:
777:
757:
708:
660:
630:
602:
521:
498:
473:
286:
266:
236:
209:
32:financial mathematics
5768:Time series analysis
5723:Mathematical finance
5608:Reflection principle
4935:Regenerative process
4735:Fleming–Viot process
4550:Stochastic processes
4409:Securitized products
4061:Puschkarski, Eugen.
3632:10.2139/ssrn.2737091
3372:
3334:
3307:
3198:
3117:
3004:
2846:
2804:
2729:
2688:
2584:
2501:
2451:
2414:
2309:
2294:, the value at time
2235:
1937:
1847:
1737:
1509:
1313:
1281:
1251:
1240:{\displaystyle \mu }
1231:
1181:
993:
813:
786:
766:
746:
677:
643:
619:
533:
508:
485:
318:
275:
255:
247:the Hull–White model
245:(time) dependence —
225:
93:
5763:Stochastic analysis
5603:Quadratic variation
5598:Prokhorov's theorem
5533:Feynman–Kac formula
5003:Markov random field
4651:Birth–death process
4186:Infrastructure bond
3063:
2837:Black–Scholes model
1584:
1456:
1376:
1306:is time-dependent,
1225:normal distribution
940:
5733:Probability theory
5613:Skorokhod integral
5583:Malliavin calculus
5166:Korn-Kreer-Lenssen
5050:Time series models
5013:Pitman–Yor process
4262:Floating rate note
3893:Primary references
3553:bermudan swaptions
3549:exotic derivatives
3529:monotonic function
3525:Jamshidian's trick
3499:
3355:
3320:
3290:
3166:
3097:
3049:
2984:
2825:
2787:
2709:
2667:
2567:
2479:
2429:
2397:
2274:Derivative pricing
2256:
2218:
1922:
1827:
1696:
1570:
1489:
1442:
1362:
1296:
1264:
1237:
1213:
1164:
973:
926:
792:
772:
752:
703:
655:
625:
597:
516:
515:
493:
492:
468:
281:
261:
231:
204:
56:bermudan swaptions
5822:Short-rate models
5799:
5798:
5753:Signal processing
5472:Doob's upcrossing
5467:Doob's martingale
5431:Engelbert–Schmidt
5374:Donsker's theorem
5308:Feller-continuous
5176:Rendleman–Bartter
4966:Dirichlet process
4883:Branching process
4852:Telegraph process
4745:Geometric process
4725:Empirical process
4715:Diffusion process
4571:Branching process
4566:Bernoulli process
4516:
4515:
4469:Exchangeable bond
4395:Yield to maturity
4247:Exchangeable bond
4169:Subordinated debt
4020:978-3-540-22149-4
3697:978-3-319-95284-0
3494:
3493:
3403:
3380:
3161:
3095:
3092:
2208:
2083:
2003:
1917:
1730:structure here!)
1651:
1119:
1062:
870:
16:(Redirected from
5834:
5827:Financial models
5773:Machine learning
5660:Usual hypotheses
5543:Girsanov theorem
5528:Dynkin's formula
5293:Continuous paths
5201:Actuarial models
5141:Garman–Kohlhagen
5111:Black–Karasinski
5106:Black–Derman–Toy
5093:Financial models
4959:Fields and other
4888:Gaussian process
4837:Sigma-martingale
4641:Additive process
4543:
4536:
4529:
4520:
4459:Convertible bond
4302:Zero-coupon bond
4242:Convertible bond
4227:Commercial paper
4110:
4103:
4096:
4087:
4079:Hull–White Model
4024:
3998:
3959:(6th ed.).
3958:
3942:Other references
3886:
3885:
3875:
3873:10.1002/for.2783
3866:(8): 1566–1580.
3851:
3845:
3844:
3825:10.1002/for.2642
3818:
3794:
3788:
3787:
3751:
3745:
3744:
3708:
3702:
3701:
3675:
3669:
3668:
3666:
3664:
3649:
3643:
3642:
3640:
3638:
3615:
3508:
3506:
3505:
3500:
3495:
3492:
3484:
3452:
3451:
3404:
3396:
3391:
3390:
3381:
3376:
3364:
3362:
3361:
3356:
3329:
3327:
3326:
3321:
3319:
3318:
3299:
3297:
3296:
3291:
3283:
3282:
3240:
3239:
3175:
3173:
3172:
3167:
3162:
3157:
3155:
3154:
3142:
3141:
3129:
3128:
3106:
3104:
3103:
3098:
3096:
3094:
3093:
3088:
3086:
3085:
3075:
3071:
3062:
3057:
3039:
3021:
3016:
3015:
2993:
2991:
2990:
2985:
2977:
2976:
2928:
2927:
2900:
2899:
2860:
2859:
2854:
2834:
2832:
2831:
2826:
2796:
2794:
2793:
2788:
2783:
2782:
2718:
2716:
2715:
2710:
2676:
2674:
2673:
2668:
2651:
2650:
2626:
2625:
2620:
2596:
2595:
2576:
2574:
2573:
2568:
2548:
2513:
2512:
2488:
2486:
2485:
2480:
2463:
2462:
2438:
2436:
2435:
2430:
2428:
2427:
2422:
2406:
2404:
2403:
2398:
2381:
2380:
2356:
2355:
2350:
2278:By selecting as
2265:
2263:
2262:
2257:
2227:
2225:
2224:
2219:
2214:
2210:
2209:
2207:
2206:
2205:
2192:
2158:
2157:
2100:
2099:
2089:
2084:
2082:
2074:
2039:
2004:
2002:
1982:
1962:
1931:
1929:
1928:
1923:
1918:
1913:
1872:
1836:
1834:
1833:
1828:
1705:
1703:
1702:
1697:
1692:
1688:
1687:
1683:
1682:
1681:
1652:
1650:
1642:
1641:
1632:
1609:
1608:
1583:
1578:
1554:
1553:
1533:
1532:
1498:
1496:
1495:
1490:
1469:
1468:
1455:
1450:
1441:
1440:
1401:
1400:
1375:
1370:
1346:
1345:
1305:
1303:
1302:
1297:
1273:
1271:
1270:
1265:
1263:
1262:
1246:
1244:
1243:
1238:
1222:
1220:
1219:
1214:
1209:
1208:
1190:
1189:
1173:
1171:
1170:
1165:
1160:
1156:
1155:
1151:
1150:
1149:
1120:
1118:
1110:
1109:
1100:
1095:
1091:
1090:
1089:
1063:
1055:
1038:
1037:
1017:
1016:
982:
980:
979:
974:
953:
952:
939:
934:
925:
924:
903:
899:
898:
897:
871:
863:
846:
845:
801:
799:
798:
793:
781:
779:
778:
773:
761:
759:
758:
753:
712:
710:
709:
704:
696:
664:
662:
661:
656:
634:
632:
631:
626:
606:
604:
603:
598:
587:
586:
573:
572:
525:
523:
522:
517:
502:
500:
499:
494:
477:
475:
474:
469:
455:
454:
432:
431:
413:
409:
303:Two-factor model
290:
288:
287:
282:
270:
268:
267:
262:
240:
238:
237:
232:
213:
211:
210:
205:
162:
158:
84:short-rate model
78:One-factor model
36:Hull–White model
21:
18:Hull-White model
5842:
5841:
5837:
5836:
5835:
5833:
5832:
5831:
5802:
5801:
5800:
5795:
5777:
5738:Queueing theory
5681:
5623:Skorokhod space
5486:
5477:Kunita–Watanabe
5448:
5414:Sanov's theorem
5384:Ergodic theorem
5357:
5353:Time-reversible
5271:
5234:Queueing models
5228:
5224:Sparre–Anderson
5214:Cramér–Lundberg
5195:
5181:SABR volatility
5087:
5044:
4996:Boolean network
4954:
4940:Renewal process
4871:
4820:Non-homogeneous
4810:Poisson process
4700:Contact process
4663:Brownian motion
4633:Continuous time
4627:
4621:Maximal entropy
4552:
4547:
4517:
4512:
4483:
4474:Extendible bond
4464:Embedded option
4440:
4404:
4306:
4267:High-yield debt
4257:Fixed rate bond
4252:Extendible bond
4200:
4181:Government bond
4176:Distressed debt
4142:
4119:
4114:
4031:, 6(1), 57–72.
4021:
4001:
3979:
3947:
3890:
3889:
3853:
3852:
3848:
3796:
3795:
3791:
3753:
3752:
3748:
3710:
3709:
3705:
3698:
3677:
3676:
3672:
3662:
3660:
3651:
3650:
3646:
3636:
3634:
3617:
3616:
3612:
3607:
3585:
3576:
3570:
3545:
3485:
3453:
3382:
3370:
3369:
3332:
3331:
3310:
3305:
3304:
3274:
3231:
3196:
3195:
3146:
3133:
3120:
3115:
3114:
3077:
3076:
3022:
3007:
3002:
3001:
2968:
2919:
2891:
2849:
2844:
2843:
2802:
2801:
2774:
2727:
2726:
2686:
2685:
2615:
2587:
2582:
2581:
2504:
2499:
2498:
2454:
2449:
2448:
2441:forward measure
2417:
2412:
2411:
2345:
2307:
2306:
2276:
2233:
2232:
2197:
2193:
2149:
2091:
2090:
2075:
2040:
2015:
2011:
1983:
1963:
1935:
1934:
1873:
1845:
1844:
1735:
1734:
1712:
1664:
1657:
1653:
1643:
1633:
1585:
1539:
1538:
1534:
1507:
1506:
1457:
1426:
1377:
1331:
1311:
1310:
1279:
1278:
1254:
1249:
1248:
1229:
1228:
1200:
1179:
1178:
1132:
1125:
1121:
1111:
1101:
1075:
1068:
1064:
1023:
1022:
1018:
991:
990:
941:
910:
883:
876:
872:
831:
811:
810:
784:
783:
764:
763:
744:
743:
675:
674:
669:is negative if
641:
640:
617:
616:
613:
578:
564:
531:
530:
506:
505:
483:
482:
446:
423:
353:
349:
316:
315:
305:
273:
272:
253:
252:
223:
222:
118:
114:
91:
90:
82:The model is a
80:
75:
48:tree or lattice
28:
23:
22:
15:
12:
11:
5:
5840:
5838:
5830:
5829:
5824:
5819:
5814:
5812:Interest rates
5804:
5803:
5797:
5796:
5794:
5793:
5788:
5786:List of topics
5782:
5779:
5778:
5776:
5775:
5770:
5765:
5760:
5755:
5750:
5745:
5743:Renewal theory
5740:
5735:
5730:
5725:
5720:
5715:
5710:
5708:Ergodic theory
5705:
5700:
5698:Control theory
5695:
5689:
5687:
5683:
5682:
5680:
5679:
5678:
5677:
5672:
5662:
5657:
5652:
5647:
5642:
5641:
5640:
5630:
5628:Snell envelope
5625:
5620:
5615:
5610:
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5565:
5560:
5555:
5550:
5545:
5540:
5535:
5530:
5525:
5520:
5515:
5510:
5505:
5500:
5494:
5492:
5488:
5487:
5485:
5484:
5479:
5474:
5469:
5464:
5458:
5456:
5450:
5449:
5447:
5446:
5427:Borel–Cantelli
5416:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5371:
5365:
5363:
5362:Limit theorems
5359:
5358:
5356:
5355:
5350:
5345:
5340:
5335:
5330:
5325:
5320:
5315:
5310:
5305:
5300:
5295:
5290:
5285:
5279:
5277:
5273:
5272:
5270:
5269:
5264:
5259:
5254:
5249:
5244:
5238:
5236:
5230:
5229:
5227:
5226:
5221:
5216:
5211:
5205:
5203:
5197:
5196:
5194:
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5123:
5118:
5113:
5108:
5103:
5097:
5095:
5089:
5088:
5086:
5085:
5080:
5075:
5070:
5065:
5060:
5054:
5052:
5046:
5045:
5043:
5042:
5037:
5032:
5031:
5030:
5025:
5015:
5010:
5005:
5000:
4999:
4998:
4993:
4983:
4981:Hopfield model
4978:
4973:
4968:
4962:
4960:
4956:
4955:
4953:
4952:
4947:
4942:
4937:
4932:
4927:
4926:
4925:
4920:
4915:
4910:
4900:
4898:Markov process
4895:
4890:
4885:
4879:
4877:
4873:
4872:
4870:
4869:
4867:Wiener sausage
4864:
4862:Wiener process
4859:
4854:
4849:
4844:
4842:Stable process
4839:
4834:
4832:Semimartingale
4829:
4824:
4823:
4822:
4817:
4807:
4802:
4797:
4792:
4787:
4782:
4777:
4775:Jump diffusion
4772:
4767:
4762:
4757:
4752:
4750:Hawkes process
4747:
4742:
4737:
4732:
4730:Feller process
4727:
4722:
4717:
4712:
4707:
4702:
4697:
4695:Cauchy process
4692:
4691:
4690:
4685:
4680:
4675:
4670:
4660:
4659:
4658:
4648:
4646:Bessel process
4643:
4637:
4635:
4629:
4628:
4626:
4625:
4624:
4623:
4618:
4613:
4608:
4598:
4593:
4588:
4583:
4578:
4573:
4568:
4562:
4560:
4554:
4553:
4548:
4546:
4545:
4538:
4531:
4523:
4514:
4513:
4511:
4510:
4504:
4498:
4491:
4489:
4485:
4484:
4482:
4481:
4476:
4471:
4466:
4461:
4456:
4450:
4448:
4442:
4441:
4439:
4438:
4433:
4428:
4423:
4418:
4412:
4410:
4406:
4405:
4403:
4402:
4397:
4392:
4387:
4382:
4377:
4375:Risk-free bond
4372:
4367:
4362:
4360:Mortgage yield
4357:
4352:
4347:
4342:
4337:
4332:
4327:
4322:
4316:
4314:
4312:Bond valuation
4308:
4307:
4305:
4304:
4299:
4294:
4289:
4287:Perpetual bond
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4208:
4206:
4202:
4201:
4199:
4198:
4193:
4191:Municipal bond
4188:
4183:
4178:
4173:
4172:
4171:
4166:
4159:Corporate bond
4156:
4150:
4148:
4144:
4143:
4141:
4140:
4135:
4130:
4124:
4121:
4120:
4115:
4113:
4112:
4105:
4098:
4090:
4084:
4083:
4074:
4072:Preprint SSRN.
4068:
4059:
4057:Preprint SSRN.
4049:
4047:Preprint SSRN.
4043:
4040:Preprint arXiv
4036:
4025:
4019:
4007:Fabio Mercurio
3999:
3977:
3944:
3943:
3939:
3938:
3931:
3924:
3917:
3910:
3903:
3895:
3894:
3888:
3887:
3846:
3809:(4): 569–579.
3789:
3762:(4): 370–387.
3746:
3719:(2): 267–292.
3703:
3696:
3670:
3644:
3609:
3608:
3606:
3603:
3602:
3601:
3596:
3591:
3584:
3581:
3575:
3572:
3544:
3541:
3510:
3509:
3498:
3491:
3488:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3402:
3399:
3394:
3389:
3385:
3379:
3354:
3351:
3348:
3345:
3342:
3339:
3317:
3313:
3301:
3300:
3289:
3286:
3281:
3277:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3238:
3234:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3191:set to 0) is:
3177:
3176:
3165:
3160:
3153:
3149:
3145:
3140:
3136:
3132:
3127:
3123:
3108:
3107:
3091:
3084:
3080:
3074:
3070:
3066:
3061:
3056:
3052:
3048:
3045:
3042:
3038:
3034:
3031:
3028:
3025:
3019:
3014:
3010:
2995:
2994:
2983:
2980:
2975:
2971:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2922:
2918:
2915:
2912:
2909:
2906:
2903:
2898:
2894:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2858:
2853:
2824:
2821:
2818:
2815:
2812:
2809:
2798:
2797:
2786:
2781:
2777:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2708:
2705:
2702:
2699:
2696:
2693:
2678:
2677:
2666:
2663:
2660:
2657:
2654:
2649:
2644:
2641:
2638:
2635:
2632:
2629:
2624:
2619:
2614:
2611:
2608:
2605:
2602:
2599:
2594:
2590:
2566:
2563:
2560:
2557:
2554:
2551:
2547:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2511:
2507:
2478:
2475:
2472:
2469:
2466:
2461:
2457:
2447:forward price
2426:
2421:
2408:
2407:
2396:
2393:
2390:
2387:
2384:
2379:
2374:
2371:
2368:
2365:
2362:
2359:
2354:
2349:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2275:
2272:
2255:
2252:
2249:
2246:
2243:
2240:
2229:
2228:
2217:
2213:
2204:
2200:
2196:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2156:
2152:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2098:
2094:
2087:
2081:
2078:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2014:
2010:
2007:
2001:
1998:
1995:
1992:
1989:
1986:
1981:
1978:
1975:
1972:
1969:
1966:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1932:
1921:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1838:
1837:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1711:
1708:
1707:
1706:
1695:
1691:
1686:
1680:
1677:
1674:
1671:
1667:
1663:
1660:
1656:
1649:
1646:
1640:
1636:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1607:
1604:
1601:
1598:
1595:
1592:
1588:
1582:
1577:
1573:
1569:
1566:
1563:
1560:
1557:
1552:
1549:
1546:
1542:
1537:
1531:
1526:
1523:
1520:
1517:
1514:
1500:
1499:
1488:
1485:
1482:
1479:
1476:
1473:
1467:
1464:
1460:
1454:
1449:
1445:
1439:
1436:
1433:
1429:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1399:
1396:
1393:
1390:
1387:
1384:
1380:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1349:
1344:
1341:
1338:
1334:
1330:
1327:
1324:
1321:
1318:
1295:
1292:
1289:
1286:
1261:
1257:
1236:
1212:
1207:
1203:
1199:
1196:
1193:
1188:
1175:
1174:
1163:
1159:
1154:
1148:
1145:
1142:
1139:
1135:
1131:
1128:
1124:
1117:
1114:
1108:
1104:
1098:
1094:
1088:
1085:
1082:
1078:
1074:
1071:
1067:
1061:
1058:
1053:
1050:
1047:
1044:
1041:
1036:
1033:
1030:
1026:
1021:
1015:
1010:
1007:
1004:
1001:
998:
984:
983:
972:
969:
966:
963:
960:
957:
951:
948:
944:
938:
933:
929:
923:
920:
917:
913:
909:
906:
902:
896:
893:
890:
886:
882:
879:
875:
869:
866:
861:
858:
855:
852:
849:
844:
841:
838:
834:
830:
827:
824:
821:
818:
802:are constant,
791:
771:
751:
715:mean-reverting
702:
699:
695:
691:
688:
685:
682:
665:the change in
654:
651:
648:
624:
612:
609:
608:
607:
596:
593:
590:
585:
581:
577:
571:
567:
563:
560:
557:
553:
550:
547:
544:
541:
538:
514:
491:
479:
478:
467:
464:
461:
458:
453:
449:
445:
441:
438:
435:
430:
426:
422:
419:
416:
412:
408:
405:
402:
399:
396:
393:
390:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
352:
348:
345:
342:
339:
336:
333:
330:
327:
323:
304:
301:
300:
299:
280:
260:
250:
230:
215:
214:
203:
200:
197:
194:
191:
188:
184:
181:
178:
175:
172:
169:
166:
161:
157:
154:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
121:
117:
113:
110:
107:
104:
101:
98:
79:
76:
74:
71:
44:interest rates
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5839:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5809:
5807:
5792:
5789:
5787:
5784:
5783:
5780:
5774:
5771:
5769:
5766:
5764:
5761:
5759:
5756:
5754:
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5690:
5688:
5684:
5676:
5673:
5671:
5668:
5667:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5646:
5645:Stopping time
5643:
5639:
5636:
5635:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5496:
5495:
5493:
5489:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5459:
5457:
5455:
5451:
5444:
5440:
5436:
5435:Hewitt–Savage
5432:
5428:
5424:
5420:
5419:Zero–one laws
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5367:
5366:
5364:
5360:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5289:
5286:
5284:
5281:
5280:
5278:
5274:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5239:
5237:
5235:
5231:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5206:
5204:
5202:
5198:
5192:
5189:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5116:Black–Scholes
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5098:
5096:
5094:
5090:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5055:
5053:
5051:
5047:
5041:
5038:
5036:
5033:
5029:
5026:
5024:
5021:
5020:
5019:
5018:Point process
5016:
5014:
5011:
5009:
5006:
5004:
5001:
4997:
4994:
4992:
4989:
4988:
4987:
4984:
4982:
4979:
4977:
4976:Gibbs measure
4974:
4972:
4969:
4967:
4964:
4963:
4961:
4957:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4905:
4904:
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4880:
4878:
4874:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4821:
4818:
4816:
4813:
4812:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4765:Itô diffusion
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4740:Gamma process
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4665:
4664:
4661:
4657:
4654:
4653:
4652:
4649:
4647:
4644:
4642:
4639:
4638:
4636:
4634:
4630:
4622:
4619:
4617:
4614:
4612:
4611:Self-avoiding
4609:
4607:
4604:
4603:
4602:
4599:
4597:
4596:Moran process
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4563:
4561:
4559:
4558:Discrete time
4555:
4551:
4544:
4539:
4537:
4532:
4530:
4525:
4524:
4521:
4508:
4505:
4502:
4499:
4496:
4493:
4492:
4490:
4486:
4480:
4479:Puttable bond
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4454:Callable bond
4452:
4451:
4449:
4447:
4443:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4413:
4411:
4407:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4365:Nominal yield
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4340:Current yield
4338:
4336:
4335:Credit spread
4333:
4331:
4328:
4326:
4323:
4321:
4318:
4317:
4315:
4313:
4309:
4303:
4300:
4298:
4295:
4293:
4292:Puttable bond
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4222:Callable bond
4220:
4218:
4215:
4213:
4210:
4209:
4207:
4203:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4170:
4167:
4165:
4162:
4161:
4160:
4157:
4155:
4152:
4151:
4149:
4145:
4139:
4136:
4134:
4131:
4129:
4126:
4125:
4122:
4118:
4111:
4106:
4104:
4099:
4097:
4092:
4091:
4088:
4081:
4080:
4076:Letian Wang,
4075:
4073:
4069:
4066:
4065:
4060:
4058:
4054:
4050:
4048:
4044:
4041:
4037:
4034:
4033:Preprint SSRN
4030:
4026:
4022:
4016:
4012:
4008:
4004:
4003:Damiano Brigo
4000:
3996:
3992:
3988:
3984:
3980:
3978:0-13-149908-4
3974:
3970:
3966:
3965:Prentice Hall
3962:
3957:
3956:
3950:
3949:Hull, John C.
3946:
3945:
3941:
3940:
3936:
3932:
3929:
3925:
3922:
3918:
3915:
3911:
3908:
3904:
3901:
3897:
3896:
3892:
3891:
3883:
3879:
3874:
3869:
3865:
3861:
3857:
3850:
3847:
3842:
3838:
3834:
3830:
3826:
3822:
3817:
3812:
3808:
3804:
3800:
3793:
3790:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3750:
3747:
3742:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3707:
3704:
3699:
3693:
3689:
3685:
3681:
3674:
3671:
3659:. finmath.net
3658:
3654:
3648:
3645:
3633:
3629:
3625:
3621:
3614:
3611:
3604:
3600:
3597:
3595:
3592:
3590:
3589:Vasicek model
3587:
3586:
3582:
3580:
3573:
3571:
3568:
3566:
3562:
3558:
3554:
3550:
3542:
3540:
3537:
3535:
3530:
3526:
3522:
3517:
3515:
3496:
3489:
3486:
3478:
3475:
3472:
3469:
3463:
3460:
3457:
3454:
3438:
3435:
3432:
3426:
3423:
3417:
3414:
3411:
3408:
3400:
3397:
3392:
3387:
3383:
3377:
3368:
3367:
3366:
3349:
3346:
3343:
3337:
3315:
3311:
3287:
3279:
3275:
3271:
3265:
3259:
3256:
3253:
3247:
3244:
3236:
3232:
3228:
3222:
3219:
3213:
3210:
3207:
3201:
3194:
3193:
3192:
3190:
3186:
3182:
3163:
3158:
3151:
3147:
3143:
3138:
3134:
3130:
3125:
3121:
3113:
3112:
3111:
3089:
3082:
3078:
3072:
3068:
3064:
3059:
3054:
3050:
3046:
3040:
3036:
3032:
3026:
3023:
3017:
3012:
3008:
3000:
2999:
2998:
2981:
2973:
2969:
2965:
2959:
2953:
2950:
2947:
2944:
2941:
2935:
2932:
2924:
2920:
2916:
2910:
2907:
2904:
2896:
2885:
2882:
2879:
2873:
2870:
2867:
2856:
2851:
2842:
2841:
2840:
2838:
2819:
2816:
2813:
2807:
2784:
2779:
2768:
2765:
2762:
2756:
2753:
2750:
2744:
2738:
2732:
2725:
2724:
2723:
2722:
2703:
2700:
2697:
2691:
2683:
2664:
2655:
2642:
2636:
2630:
2622:
2612:
2606:
2603:
2600:
2592:
2588:
2580:
2579:
2578:
2561:
2558:
2555:
2549:
2545:
2538:
2532:
2529:
2523:
2520:
2517:
2509:
2505:
2497:must satisfy
2496:
2492:
2473:
2470:
2467:
2459:
2455:
2446:
2442:
2424:
2394:
2385:
2372:
2366:
2360:
2352:
2339:
2336:
2333:
2327:
2324:
2318:
2312:
2305:
2304:
2303:
2301:
2297:
2293:
2289:
2285:
2281:
2273:
2271:
2269:
2250:
2247:
2244:
2238:
2215:
2211:
2202:
2198:
2194:
2186:
2183:
2177:
2174:
2171:
2165:
2162:
2154:
2143:
2140:
2137:
2131:
2128:
2125:
2119:
2116:
2113:
2107:
2104:
2096:
2092:
2085:
2079:
2065:
2062:
2059:
2053:
2047:
2044:
2032:
2029:
2026:
2020:
2017:
2012:
2008:
2005:
1996:
1993:
1990:
1984:
1976:
1973:
1970:
1964:
1958:
1952:
1949:
1946:
1940:
1933:
1919:
1914:
1904:
1901:
1898:
1892:
1889:
1883:
1880:
1877:
1874:
1868:
1862:
1859:
1856:
1850:
1843:
1842:
1841:
1824:
1815:
1809:
1803:
1800:
1797:
1791:
1788:
1782:
1779:
1773:
1770:
1767:
1761:
1758:
1752:
1749:
1746:
1740:
1733:
1732:
1731:
1729:
1725:
1724:discount bond
1721:
1718:value of the
1717:
1709:
1693:
1689:
1684:
1678:
1675:
1672:
1669:
1665:
1661:
1658:
1654:
1647:
1644:
1638:
1634:
1628:
1625:
1622:
1616:
1610:
1602:
1599:
1596:
1590:
1586:
1580:
1575:
1571:
1567:
1561:
1555:
1550:
1547:
1544:
1540:
1535:
1524:
1518:
1512:
1505:
1504:
1503:
1486:
1480:
1474:
1471:
1465:
1462:
1458:
1452:
1447:
1443:
1437:
1434:
1431:
1427:
1423:
1420:
1417:
1414:
1408:
1402:
1394:
1391:
1388:
1382:
1378:
1372:
1367:
1363:
1359:
1353:
1347:
1342:
1339:
1336:
1332:
1328:
1322:
1316:
1309:
1308:
1307:
1290:
1284:
1275:
1259:
1255:
1247:and variance
1234:
1226:
1205:
1201:
1197:
1194:
1161:
1157:
1152:
1146:
1143:
1140:
1137:
1133:
1129:
1126:
1122:
1115:
1112:
1106:
1102:
1096:
1092:
1086:
1083:
1080:
1076:
1072:
1069:
1065:
1059:
1056:
1051:
1045:
1039:
1034:
1031:
1028:
1024:
1019:
1008:
1002:
996:
989:
988:
987:
970:
964:
958:
955:
949:
946:
942:
936:
931:
927:
921:
918:
915:
911:
907:
904:
900:
894:
891:
888:
884:
880:
877:
873:
867:
864:
859:
853:
847:
842:
839:
836:
832:
828:
822:
816:
809:
808:
807:
805:
789:
769:
749:
740:
738:
734:
730:
726:
721:
719:
716:
697:
693:
686:
680:
672:
668:
652:
649:
646:
638:
622:
610:
591:
583:
579:
575:
569:
565:
561:
558:
555:
551:
548:
545:
542:
539:
536:
529:
528:
527:
512:
489:
465:
459:
451:
447:
443:
436:
428:
424:
420:
417:
414:
410:
400:
394:
388:
381:
375:
372:
369:
366:
360:
354:
350:
346:
337:
331:
325:
321:
314:
313:
312:
310:
302:
297:
296:
295:Vasicek model
293:the extended
278:
258:
251:
248:
244:
228:
221:
220:
219:
201:
195:
189:
186:
179:
173:
170:
167:
164:
159:
152:
146:
140:
134:
131:
125:
119:
115:
111:
105:
99:
96:
89:
88:
87:
85:
77:
72:
70:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
19:
5703:Econometrics
5665:Wiener space
5553:Itô integral
5454:Inequalities
5343:Self-similar
5313:Gauss–Markov
5303:Exchangeable
5283:Càdlàg paths
5219:Risk process
5171:LIBOR market
5160:
5040:Random graph
5035:Random field
4847:Superprocess
4785:Lévy process
4780:Jump process
4755:Hunt process
4591:Markov chain
4488:Institutions
4446:Bond options
4390:Yield spread
4282:Lottery bond
4212:Accrual bond
4138:Fixed income
4078:
4063:
4052:
4028:
4010:
3954:
3934:
3927:
3920:
3913:
3906:
3899:
3863:
3859:
3849:
3806:
3802:
3792:
3759:
3755:
3749:
3716:
3712:
3706:
3679:
3673:
3661:. Retrieved
3656:
3647:
3635:. Retrieved
3623:
3613:
3577:
3569:
3546:
3538:
3518:
3513:
3511:
3302:
3188:
3184:
3180:
3178:
3109:
2996:
2799:
2681:
2679:
2494:
2490:
2444:
2409:
2299:
2295:
2287:
2283:
2277:
2230:
1839:
1727:
1719:
1715:
1713:
1501:
1276:
1176:
985:
741:
731:to a set of
722:
670:
666:
636:
614:
480:
306:
292:
246:
242:
216:
81:
63:John C. Hull
60:
35:
29:
5748:Ruin theory
5686:Disciplines
5558:Itô's lemma
5333:Predictable
5008:Percolation
4991:Potts model
4986:Ising model
4950:White noise
4908:Differences
4770:Itô process
4710:Cox process
4606:Loop-erased
4601:Random walk
4385:Yield curve
4345:Dirty price
4320:Clean price
4196:Global bond
4164:Senior debt
4154:Agency bond
4117:Bond market
3967:. pp.
3663:October 15,
3657:finmath lib
3637:October 15,
3574:Forecasting
2839:shows that
1728:affine term
804:Itô's lemma
729:calibration
725:yield curve
5806:Categories
5758:Statistics
5538:Filtration
5439:Kolmogorov
5423:Blumenthal
5348:Stationary
5288:Continuous
5276:Properties
5161:Hull–White
4903:Martingale
4790:Local time
4678:Fractional
4656:pure birth
3987:2005047692
3816:1901.02246
3605:References
1722:-maturity
1227:with mean
67:Alan White
42:of future
5670:Classical
4683:Geometric
4673:Excursion
4325:Convexity
4133:Debenture
3882:0277-6693
3841:126507446
3833:0277-6693
3784:204435499
3776:1526-5943
3741:204424299
3733:1086-7376
3534:convexity
3490:α
3476:α
3470:−
3464:
3458:−
3436:−
3427:α
3424:−
3418:
3412:−
3401:α
3398:σ
3384:σ
3312:σ
3272:−
3245:−
3229:−
3148:σ
3144:−
3079:σ
3051:σ
3027:
2966:−
2933:−
2917:−
2871:−
2754:−
2643:∣
2493:given by
2373:∣
2282:the time-
2280:numeraire
2199:α
2184:−
2175:α
2166:
2141:α
2138:−
2132:
2126:−
2117:α
2114:−
2108:
2093:σ
2086:−
2077:∂
2048:
2042:∂
2018:−
2009:
1915:α
1902:−
1893:α
1890:−
1884:
1878:−
1789:−
1783:
1676:α
1670:−
1662:−
1648:α
1635:σ
1611:θ
1600:−
1591:α
1572:∫
1548:α
1545:−
1525:∼
1463:α
1444:∫
1435:α
1432:−
1424:σ
1403:θ
1392:−
1383:α
1364:∫
1340:α
1337:−
1285:θ
1256:σ
1235:μ
1202:σ
1195:μ
1144:α
1138:−
1130:−
1116:α
1103:σ
1084:α
1081:−
1073:−
1060:α
1057:θ
1032:α
1029:−
1009:∼
947:α
928:∫
919:α
916:−
908:σ
892:α
889:−
881:−
868:α
865:θ
840:α
837:−
790:σ
770:θ
750:α
737:swaptions
698:α
681:θ
647:α
623:θ
566:σ
546:−
425:σ
376:α
373:−
355:θ
309:Hull 2006
279:α
259:θ
229:θ
174:σ
135:α
132:−
120:θ
73:The model
5791:Category
5675:Abstract
5209:Bühlmann
4815:Compound
4400:Z-spread
4355:I-spread
4350:Duration
4009:(2001).
3995:60321487
3583:See also
3551:such as
3519:Because
2800:Because
2721:bond put
54:such as
5298:Ergodic
5186:Vašíček
5028:Poisson
4688:Meander
4509:(SIFMA)
3557:lattice
2577:, thus
1223:is the
733:caplets
50:and so
5638:Tanaka
5323:Mixing
5318:Markov
5191:Wilkie
5156:Ho–Lee
5151:Heston
4923:Super-
4668:Bridge
4616:Biased
4503:(ICMA)
4497:(CMSA)
4330:Coupon
4232:Consol
4017:
3993:
3985:
3975:
3971:–658.
3880:
3839:
3831:
3782:
3774:
3739:
3731:
3694:
2997:where
2410:Here,
1840:where
1177:where
782:, and
481:where
34:, the
5491:Tools
5267:M/M/c
5262:M/M/1
5257:M/G/1
5247:Fluid
4913:Local
3837:S2CID
3811:arXiv
3780:S2CID
3737:S2CID
3555:on a
3303:Here
1277:When
742:When
40:model
38:is a
5443:Lévy
5242:Bulk
5126:Chen
4918:Sub-
4876:Both
4128:Bond
4015:ISBN
3991:OCLC
3983:LCCN
3973:ISBN
3878:ISSN
3829:ISSN
3772:ISSN
3729:ISSN
3692:ISBN
3665:2023
3639:2023
3624:SSRN
3110:and
2495:V(T)
735:and
650:>
635:has
271:and
241:has
65:and
5023:Cox
3969:657
3868:doi
3821:doi
3764:doi
3721:doi
3684:doi
3628:doi
3461:exp
3415:exp
3183:(0,
3024:log
2266:is
2163:exp
2129:exp
2105:exp
2045:log
2006:exp
1881:exp
1780:exp
30:In
5808::
5441:,
5437:,
5433:,
5429:,
5425:,
4005:,
3989:.
3981:.
3963::
3876:.
3864:40
3862:.
3858:.
3835:.
3827:.
3819:.
3807:39
3805:.
3801:.
3778:.
3770:.
3760:20
3758:.
3735:.
3727:.
3717:37
3715:.
3690:.
3655:.
3626:.
3622:.
3536:.
2302:.
2270:.
1274:.
762:,
720:.
5445:)
5421:(
4542:e
4535:t
4528:v
4109:e
4102:t
4095:v
4055:,
4042:.
4035:.
4023:.
3997:.
3884:.
3870::
3843:.
3823::
3813::
3786:.
3766::
3743:.
3723::
3700:.
3686::
3667:.
3641:.
3630::
3514:S
3497:.
3487:2
3482:)
3479:S
3473:2
3467:(
3455:1
3448:)
3445:)
3442:)
3439:S
3433:T
3430:(
3421:(
3409:1
3406:(
3393:=
3388:P
3378:S
3353:)
3350:T
3347:,
3344:S
3341:(
3338:P
3316:P
3288:.
3285:)
3280:1
3276:d
3269:(
3266:N
3263:)
3260:T
3257:,
3254:0
3251:(
3248:P
3242:)
3237:2
3233:d
3226:(
3223:N
3220:K
3217:)
3214:S
3211:,
3208:0
3205:(
3202:P
3189:t
3185:S
3181:P
3164:.
3159:S
3152:P
3139:1
3135:d
3131:=
3126:2
3122:d
3090:S
3083:P
3073:2
3069:/
3065:S
3060:2
3055:P
3047:+
3044:)
3041:K
3037:/
3033:F
3030:(
3018:=
3013:1
3009:d
2982:,
2979:)
2974:1
2970:d
2963:(
2960:N
2957:)
2954:T
2951:,
2948:S
2945:,
2942:t
2939:(
2936:F
2930:)
2925:2
2921:d
2914:(
2911:N
2908:K
2905:=
2902:]
2897:+
2893:)
2889:)
2886:T
2883:,
2880:S
2877:(
2874:P
2868:K
2865:(
2862:[
2857:S
2852:E
2823:)
2820:T
2817:,
2814:S
2811:(
2808:P
2785:.
2780:+
2776:)
2772:)
2769:T
2766:,
2763:S
2760:(
2757:P
2751:K
2748:(
2745:=
2742:)
2739:S
2736:(
2733:V
2707:)
2704:T
2701:,
2698:S
2695:(
2692:P
2682:V
2665:.
2662:]
2659:)
2656:t
2653:(
2648:F
2640:)
2637:T
2634:(
2631:V
2628:[
2623:T
2618:E
2613:=
2610:)
2607:T
2604:,
2601:t
2598:(
2593:V
2589:F
2565:)
2562:T
2559:,
2556:t
2553:(
2550:P
2546:/
2542:)
2539:t
2536:(
2533:V
2530:=
2527:)
2524:T
2521:,
2518:t
2515:(
2510:V
2506:F
2491:T
2477:)
2474:T
2471:,
2468:t
2465:(
2460:V
2456:F
2445:T
2425:S
2420:E
2395:.
2392:]
2389:)
2386:t
2383:(
2378:F
2370:)
2367:S
2364:(
2361:V
2358:[
2353:S
2348:E
2343:)
2340:S
2337:,
2334:t
2331:(
2328:P
2325:=
2322:)
2319:t
2316:(
2313:V
2300:S
2296:t
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2284:S
2254:)
2251:T
2248:,
2245:S
2242:(
2239:P
2216:.
2212:)
2203:3
2195:4
2190:)
2187:1
2181:)
2178:S
2172:2
2169:(
2160:(
2155:2
2151:)
2147:)
2144:S
2135:(
2123:)
2120:T
2111:(
2102:(
2097:2
2080:S
2072:)
2069:)
2066:S
2063:,
2060:0
2057:(
2054:P
2051:(
2036:)
2033:T
2030:,
2027:S
2024:(
2021:B
2013:(
2000:)
1997:S
1994:,
1991:0
1988:(
1985:P
1980:)
1977:T
1974:,
1971:0
1968:(
1965:P
1959:=
1956:)
1953:T
1950:,
1947:S
1944:(
1941:A
1920:,
1911:)
1908:)
1905:S
1899:T
1896:(
1887:(
1875:1
1869:=
1866:)
1863:T
1860:,
1857:S
1854:(
1851:B
1825:,
1822:)
1819:)
1816:S
1813:(
1810:r
1807:)
1804:T
1801:,
1798:S
1795:(
1792:B
1786:(
1777:)
1774:T
1771:,
1768:S
1765:(
1762:A
1759:=
1756:)
1753:T
1750:,
1747:S
1744:(
1741:P
1720:T
1716:S
1694:.
1690:)
1685:)
1679:t
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1659:1
1655:(
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1626:s
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1617:s
1614:(
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1594:(
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1536:(
1530:N
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1519:t
1516:(
1513:r
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1472:d
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1329:=
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1323:t
1320:(
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1014:N
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1000:(
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895:t
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878:1
874:(
860:+
857:)
854:0
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848:r
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829:=
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694:/
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595:)
592:t
589:(
584:2
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576:d
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562:+
559:t
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552:u
549:b
543:=
540:u
537:d
513:u
490:f
466:,
463:)
460:t
457:(
452:1
448:W
444:d
440:)
437:t
434:(
429:1
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415:d
411:]
407:)
404:)
401:t
398:(
395:r
392:(
389:f
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367:+
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347:=
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298:.
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202:.
199:)
196:t
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190:W
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171:+
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160:]
156:)
153:t
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147:r
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129:)
126:t
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116:[
112:=
109:)
106:t
103:(
100:r
97:d
20:)
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