186:
324:
806:
In their original paper, CKLS argued that the elasticity of interest rate volatility is 1.5 based on historical data, a result that has been widely cited. Also, they showed that models with
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414:
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216:
81:
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909:
between 1979 and 1982. They have found evidence supporting the square root Cox-Ingersoll-Ross model (CIR SR), a special case of the CKLS model with
1976:
93:
2579:
1572:
1535:
Bliss, Robert R.; Smith, David C. (1998-03-01). "The
Elasticity of Interest Rate Volatility: Chan, Karolyi, Longstaff, and Sanders Revisited".
2109:
2089:
2493:
2843:
833:
228:
2410:
48:. It is named after K. C. Chan, G. Andrew Karolyi, Francis A. Longstaff, and Anthony B. Sanders, with their paper published in 1992.
2094:
499:
are special cases of the CKLS process which can be obtained by setting the CKLS model parameters to specific values. In all cases,
2420:
2104:
417:
1490:"An Evaluation of Contingent Claims Using the CKLS Interest Rate Model: An Analysis of Australia, Japan, and the United Kingdom"
2462:
2359:
2649:
2639:
2485:
2177:
2162:
905:
values. Moreover, by redefining the regime period, Bliss and Smith have shown that there is evidence for regime shift in the
706:
2549:
2513:
2817:
2554:
2466:
1664:
1565:
2853:
2619:
1026:"Are These Shocks for Real? Sensitivity Analysis of the Significance of the Wavelet Response to Some CKLS Processes"
436:
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2470:
2454:
2369:
2197:
2167:
1589:
628:
84:
775:
One question studied in the literature is how to set the model parameters, in particular the elasticity parameter
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1934:
1851:
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2878:
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2137:
2132:
1939:
1836:
45:
1366:
Dinenis, E.; Allegretto, W.; Sorwar, G.; N, Quaderno; Barone-adesi, Giovanni; Dinenis, Elias; Sorwar, Ghulam,
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values (like 0.5) in the CKLS model can capture volatility dependence more accurately compared to higher
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2898:
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2544:
2529:
2172:
2142:
1709:
1607:
342:
1432:
1390:
Koedijk, Kees G.; Nissen, François G. J. A.; Schotman, Peter C.; Wolff, Christian C. P. (1997-04-01).
1328:
1273:
1234:
1161:
1070:"Central limit theorem and moderate deviation principle for CKLS model with small random perturbation"
1069:
2858:
2848:
2624:
2425:
2339:
2324:
2255:
1831:
1714:
1612:
1204:"Gaussian Estimation of Single-Factor Continuous Time Models of The Term Structure of Interest Rates"
658:
393:
1272:
Sikora, Grzegorz; Michalak, Anna; Bielak, Łukasz; Miśta, Paweł; Wyłomańska, Agnieszka (2019-06-01).
2458:
2344:
1846:
1821:
1766:
1376:
809:
800:
416:. Moreover, the MGF can be written as the MGF of the CIR model plus a term that is a solution to a
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1024:
Kokabisaghi, Somayeh; Pauwels, Eric J.; Van
Meulder, Katrien; Dorsman, André B. (2018-09-02).
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Chan, K. C.; Karolyi, G. Andrew; Longstaff, Francis A.; Sanders, Anthony B. (July 1992).
1289:
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Later empirical studies by Bliss and Smith have shown the reverse: sometimes lower
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430:
and deviation principle for the CKLS model while studying its asymptotic behavior.
2769:
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2349:
2027:
1971:
1954:
1699:
1274:"Stochastic modeling of currency exchange rates with novel validation techniques"
951:
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479:
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1816:
1297:
433:
CKLS has been referred to as a time-homogeneous model as usually the parameters
2774:
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2012:
1929:
1898:
1893:
1873:
1863:
1806:
1781:
1761:
1726:
1694:
1677:
1113:
978:"An Empirical Comparison of Alternative Models of the Short-Term Interest Rate"
954:, and this regime change has often been studied in the context of CKLS models.
729:
219:
41:
37:
1801:
1505:
1458:
1408:
1391:
1250:
1085:
728:
The CKLS process is often used to model interest rate dynamics and pricing of
2837:
2676:
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2049:
2007:
1949:
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1627:
1513:
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1878:
1811:
1786:
1622:
757:
596:
181:{\displaystyle dX_{t}=(\alpha +\beta X_{t})dt+\sigma X_{t}^{\gamma }dW_{t}}
1489:
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2017:
1981:
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17:
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2157:
2110:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1550:
1233:
Tangman, D. Y.; Thakoor, N.; Dookhitram, K.; Bhuruth, M. (2011-12-01).
1160:
Dell'Aquila, Rosario; Ronchetti, Elvezio; Trojani, Fabio (2003-05-01).
1002:
2283:
1431:
Mishura, Yuliya; Ralchenko, Kostiantyn; Dehtiar, Olena (2022-05-01).
1329:"Pricing UK and US securities within the CKLS model Further results"
1023:
1449:
1392:"The Dynamics of Short-Term Interest Rate Volatility Reconsidered"
523:
Family of CKLS process under different parametric specifications.
319:{\displaystyle dX_{t}=-k(X_{t}-a)dt+\sigma X_{t}^{\gamma }dW_{t}}
44:. The CKLS process can also be viewed as a generalization of the
33:
1433:"Parameter estimation in CKLS model by continuous observations"
1232:
1159:
1235:"Fast approximations of bond option prices under CKLS models"
1118:"Time-Dependent Diffusion Models for Term Structure Dynamics"
769:
222:. The CKLS process has the following equivalent definition:
1368:
Valuation of
Derivatives Based on CKLS Interest Rate Models
1008:
975:
2090:
Autoregressive conditional heteroskedasticity (ARCH) model
1389:
1162:"Robust GMM analysis of models for the short rate process"
1271:
1618:
Independent and identically distributed random variables
1430:
2095:
Autoregressive integrated moving average (ARIMA) model
1278:
Physica A: Statistical
Mechanics and Its Applications
915:
891:
871:
842:
812:
781:
578:
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536:
505:
439:
402:
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231:
197:
96:
62:
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824:
787:
584:
563:
542:
511:
463:
408:
384:
318:
210:
180:
75:
1111:
803:have been used to measure CKLS model parameters.
423:The CKLS equation has a unique pathwise solution.
2835:
1977:Stochastic chains with memory of variable length
464:{\displaystyle \alpha ,\beta ,\sigma ,\gamma }
36:. In particular it has been used to model the
1566:
1488:Nowman, K. Ben; Sorwar, Ghulam (1999-09-01).
1327:Nowman, K. Ben; Sorwar, Ghulam (1999-03-01).
1487:
1326:
946:The period of 1979-1982 marked a change in
2105:Autoregressive–moving-average (ARMA) model
1573:
1559:
1534:
1333:International Review of Financial Analysis
1030:International Journal of Financial Studies
756:. It has also been used in the pricing of
1448:
1407:
1375:
1068:Cai, Yujie; Wang, Shaochen (2015-03-01).
1041:
993:
723:
1580:
474:The CKLS has also been referred to as a
1067:
418:Nonlinear partial differential equation
2836:
2411:Doob's martingale convergence theorems
1201:
22:Chan–Karolyi–Longstaff–Sanders process
2163:Constant elasticity of variance (CEV)
2153:Chan–Karolyi–Longstaff–Sanders (CKLS)
1554:
1437:Statistics & Probability Letters
1197:
1195:
1155:
1153:
1151:
1107:
1105:
1103:
1074:Statistics & Probability Letters
1063:
1061:
1019:
1017:
971:
969:
967:
396:at a critical moment independent of
385:{\displaystyle X_{t}^{2(1-\gamma )}}
801:nonparametric estimation techniques
426:Cai and Wang (2015) have derived a
13:
2650:Skorokhod's representation theorem
2431:Law of large numbers (weak/strong)
1220:10.1111/j.1540-6261.1997.tb01127.x
995:10.1111/j.1540-6261.1992.tb04011.x
14:
2915:
2844:Stochastic differential equations
2620:Martingale representation theorem
1192:
1148:
1112:Fan, Jianqing; Jiang, Jiancheng;
1100:
1058:
1014:
964:
836:more accurately than models with
764:and has been combined with other
471:are taken to be time-independent.
2665:Stochastic differential equation
2555:Doob's optional stopping theorem
2550:Doob–Meyer decomposition theorem
1202:Nowman, K. B. (September 1997).
486:
85:stochastic differential equation
2535:Convergence of random variables
2421:Fisher–Tippett–Gnedenko theorem
1528:
1481:
1424:
1383:
1359:
1320:
2133:Binomial options pricing model
1494:Asia-Pacific Financial Markets
1265:
1226:
377:
365:
273:
254:
135:
113:
1:
2600:Kolmogorov continuity theorem
2436:Law of the iterated logarithm
1345:10.1016/S1057-5219(99)00019-8
1178:10.1016/S0927-5398(02)00050-6
957:
825:{\displaystyle \gamma \geq 1}
329:
51:
2605:Kolmogorov extension theorem
2284:Generalized queueing network
1792:Interacting particle systems
1166:Journal of Empirical Finance
855:{\displaystyle \gamma <1}
83:is defined by the following
7:
1737:Continuous-time random walk
1298:10.1016/j.physa.2019.04.098
936:{\displaystyle \gamma =1/2}
519:is assumed to be positive.
10:
2920:
2745:Extreme value theory (EVT)
2545:Doob decomposition theorem
1837:Ornstein–Uhlenbeck process
1608:Chinese restaurant process
663:Black–Scholes–Merton model
629:CIR or square root process
343:moment-generating function
46:Ornstein–Uhlenbeck process
2813:
2717:
2625:Optional stopping theorem
2522:
2484:
2426:Large deviation principle
2393:
2307:
2264:
2231:
2178:Heath–Jarrow–Morton (HJM)
2123:
2115:Moving-average (MA) model
2100:Autoregressive (AR) model
2080:
1990:
1925:Hidden Markov model (HMM)
1907:
1859:Schramm–Loewner evolution
1663:
1588:
1459:10.1016/j.spl.2022.109391
1251:10.1016/j.frl.2011.03.002
1086:10.1016/j.spl.2014.11.017
834:short-term interest rates
659:Geometric Brownian motion
2854:Variants of random walks
2540:Doléans-Dade exponential
2370:Progressively measurable
2168:Cox–Ingersoll–Ross (CIR)
1239:Finance Research Letters
1116:; Zhou, Zhenwei (2003).
335:CKLS is an example of a
2864:Financial risk modeling
2760:Mathematical statistics
2750:Large deviations theory
2580:Infinitesimal generator
2441:Maximal ergodic theorem
2360:Piecewise-deterministic
1962:Random dynamical system
1827:Markov additive process
1506:10.1023/A:1010013604561
1409:10.1023/A:1009714314989
898:{\displaystyle \gamma }
878:{\displaystyle \gamma }
788:{\displaystyle \gamma }
738:currency exchange rates
585:{\displaystyle \gamma }
543:{\displaystyle \alpha }
512:{\displaystyle \sigma }
409:{\displaystyle \gamma }
2595:Karhunen–Loève theorem
2530:Cameron–Martin formula
2494:Burkholder–Davis–Gundy
1889:Variance gamma process
1208:The Journal of Finance
982:The Journal of Finance
937:
899:
879:
856:
826:
789:
724:Financial applications
586:
565:
564:{\displaystyle \beta }
544:
513:
465:
410:
386:
337:mean-reverting process
327:
320:
212:
189:
182:
77:
2894:Fixed income analysis
2879:Derivatives (finance)
2725:Actuarial mathematics
2687:Uniform integrability
2682:Stratonovich integral
2610:Lévy–Prokhorov metric
2514:Marcinkiewicz–Zygmund
2401:Central limit theorem
2003:Gaussian random field
1832:McKean–Vlasov process
1752:Dyson Brownian motion
1613:Galton–Watson process
938:
900:
880:
857:
827:
790:
587:
566:
545:
514:
466:
428:central limit theorem
411:
387:
321:
224:
213:
211:{\displaystyle W_{t}}
183:
89:
78:
76:{\displaystyle X_{t}}
32:with applications to
2800:Time series analysis
2755:Mathematical finance
2640:Reflection principle
1967:Regenerative process
1767:Fleming–Viot process
1582:Stochastic processes
913:
889:
869:
840:
810:
779:
678:Brennan and Schwartz
576:
555:
534:
503:
437:
400:
349:
229:
195:
94:
60:
2795:Stochastic analysis
2635:Quadratic variation
2630:Prokhorov's theorem
2565:Feynman–Kac formula
2035:Markov random field
1683:Birth–death process
1290:2019PhyA..523.1202S
1043:10.3390/ijfs6030076
524:
381:
302:
164:
2765:Probability theory
2645:Skorokhod integral
2615:Malliavin calculus
2198:Korn-Kreer-Lenssen
2082:Time series models
2045:Pitman–Yor process
933:
895:
875:
852:
822:
785:
582:
561:
540:
522:
509:
461:
406:
382:
352:
316:
288:
208:
178:
150:
73:
30:stochastic process
2904:Short-rate models
2899:Stochastic models
2884:Equity securities
2874:Options (finance)
2831:
2830:
2785:Signal processing
2504:Doob's upcrossing
2499:Doob's martingale
2463:Engelbert–Schmidt
2406:Donsker's theorem
2340:Feller-continuous
2208:Rendleman–Bartter
1998:Dirichlet process
1915:Branching process
1884:Telegraph process
1777:Geometric process
1757:Empirical process
1747:Diffusion process
1603:Branching process
1598:Bernoulli process
1396:Review of Finance
1122:Statistica Sinica
797:Robust statistics
754:contingent claims
721:
720:
497:short-rate models
56:The CKLS process
2911:
2859:Financial models
2849:Markov processes
2805:Machine learning
2692:Usual hypotheses
2575:Girsanov theorem
2560:Dynkin's formula
2325:Continuous paths
2233:Actuarial models
2173:Garman–Kohlhagen
2143:Black–Karasinski
2138:Black–Derman–Toy
2125:Financial models
1991:Fields and other
1920:Gaussian process
1869:Sigma-martingale
1673:Additive process
1575:
1568:
1561:
1552:
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1545:
1544:
1532:
1526:
1525:
1485:
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1411:
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1357:
1356:
1324:
1318:
1317:
1269:
1263:
1262:
1230:
1224:
1223:
1214:(4): 1695–1706.
1199:
1190:
1189:
1157:
1146:
1145:
1109:
1098:
1097:
1065:
1056:
1055:
1045:
1021:
1012:
1009:Chan et al. 1992
1006:
1000:
999:
997:
988:(3): 1209–1227.
973:
942:
940:
939:
934:
929:
904:
902:
901:
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884:
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881:
876:
861:
859:
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831:
829:
828:
823:
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792:
791:
786:
768:methods such as
591:
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583:
570:
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541:
525:
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510:
476:one-factor model
470:
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109:
108:
82:
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72:
71:
24:(abbreviated as
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2914:
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2912:
2910:
2909:
2908:
2834:
2833:
2832:
2827:
2809:
2770:Queueing theory
2713:
2655:Skorokhod space
2518:
2509:Kunita–Watanabe
2480:
2446:Sanov's theorem
2416:Ergodic theorem
2389:
2385:Time-reversible
2303:
2266:Queueing models
2260:
2256:Sparre–Anderson
2246:Cramér–Lundberg
2227:
2213:SABR volatility
2119:
2076:
2028:Boolean network
1986:
1972:Renewal process
1903:
1852:Non-homogeneous
1842:Poisson process
1732:Contact process
1695:Brownian motion
1665:Continuous time
1659:
1653:Maximal entropy
1584:
1579:
1549:
1548:
1533:
1529:
1486:
1482:
1429:
1425:
1388:
1384:
1364:
1360:
1325:
1321:
1270:
1266:
1231:
1227:
1200:
1193:
1158:
1149:
1114:Zhang, Chunming
1110:
1101:
1066:
1059:
1022:
1015:
1007:
1003:
974:
965:
960:
952:Federal Reserve
948:monetary policy
925:
914:
911:
910:
907:Federal Reserve
890:
887:
886:
870:
867:
866:
841:
838:
837:
811:
808:
807:
780:
777:
776:
772:-class models.
726:
577:
574:
573:
556:
553:
552:
535:
532:
531:
504:
501:
500:
489:
480:Factor analysis
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401:
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361:
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54:
12:
11:
5:
2917:
2907:
2906:
2901:
2896:
2891:
2889:Interest rates
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2846:
2829:
2828:
2826:
2825:
2820:
2818:List of topics
2814:
2811:
2810:
2808:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2775:Renewal theory
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2740:Ergodic theory
2737:
2732:
2730:Control theory
2727:
2721:
2719:
2715:
2714:
2712:
2711:
2710:
2709:
2704:
2694:
2689:
2684:
2679:
2674:
2673:
2672:
2662:
2660:Snell envelope
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
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2577:
2572:
2567:
2562:
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2552:
2547:
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2520:
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2517:
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2511:
2506:
2501:
2496:
2490:
2488:
2482:
2481:
2479:
2478:
2459:Borel–Cantelli
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2397:
2395:
2394:Limit theorems
2391:
2390:
2388:
2387:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2337:
2332:
2327:
2322:
2317:
2311:
2309:
2305:
2304:
2302:
2301:
2296:
2291:
2286:
2281:
2276:
2270:
2268:
2262:
2261:
2259:
2258:
2253:
2248:
2243:
2237:
2235:
2229:
2228:
2226:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2129:
2127:
2121:
2120:
2118:
2117:
2112:
2107:
2102:
2097:
2092:
2086:
2084:
2078:
2077:
2075:
2074:
2069:
2064:
2063:
2062:
2057:
2047:
2042:
2037:
2032:
2031:
2030:
2025:
2015:
2013:Hopfield model
2010:
2005:
2000:
1994:
1992:
1988:
1987:
1985:
1984:
1979:
1974:
1969:
1964:
1959:
1958:
1957:
1952:
1947:
1942:
1932:
1930:Markov process
1927:
1922:
1917:
1911:
1909:
1905:
1904:
1902:
1901:
1899:Wiener sausage
1896:
1894:Wiener process
1891:
1886:
1881:
1876:
1874:Stable process
1871:
1866:
1864:Semimartingale
1861:
1856:
1855:
1854:
1849:
1839:
1834:
1829:
1824:
1819:
1814:
1809:
1807:Jump diffusion
1804:
1799:
1794:
1789:
1784:
1782:Hawkes process
1779:
1774:
1769:
1764:
1762:Feller process
1759:
1754:
1749:
1744:
1739:
1734:
1729:
1727:Cauchy process
1724:
1723:
1722:
1717:
1712:
1707:
1702:
1692:
1691:
1690:
1680:
1678:Bessel process
1675:
1669:
1667:
1661:
1660:
1658:
1657:
1656:
1655:
1650:
1645:
1640:
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1594:
1592:
1586:
1585:
1578:
1577:
1570:
1563:
1555:
1547:
1546:
1527:
1500:(3): 205–219.
1480:
1423:
1402:(1): 105–130.
1382:
1377:10.1.1.24.6963
1358:
1339:(3): 235–245.
1319:
1264:
1245:(4): 206–212.
1225:
1191:
1172:(3): 373–397.
1147:
1128:(4): 965–992.
1099:
1057:
1013:
1001:
962:
961:
959:
956:
932:
928:
924:
921:
918:
894:
874:
851:
848:
845:
821:
818:
815:
784:
725:
722:
719:
718:
715:
712:
709:
703:
702:
699:
696:
693:
689:
688:
685:
682:
679:
675:
674:
671:
668:
665:
655:
654:
651:
648:
645:
641:
640:
637:
634:
631:
625:
624:
621:
618:
615:
609:
608:
605:
602:
599:
593:
592:
581:
571:
560:
550:
539:
529:
508:
488:
485:
484:
483:
472:
460:
457:
454:
451:
448:
445:
442:
431:
424:
421:
405:
379:
376:
373:
370:
367:
364:
359:
355:
339:
331:
328:
313:
309:
305:
300:
295:
291:
287:
284:
281:
278:
275:
272:
269:
264:
260:
256:
253:
250:
247:
242:
238:
234:
220:Wiener process
205:
201:
175:
171:
167:
162:
157:
153:
149:
146:
143:
140:
137:
132:
128:
124:
121:
118:
115:
112:
107:
103:
99:
70:
66:
53:
50:
42:interest rates
38:term structure
9:
6:
4:
3:
2:
2916:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2841:
2839:
2824:
2821:
2819:
2816:
2815:
2812:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2722:
2720:
2716:
2708:
2705:
2703:
2700:
2699:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2677:Stopping time
2675:
2671:
2668:
2667:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2527:
2525:
2521:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2495:
2492:
2491:
2489:
2487:
2483:
2476:
2472:
2468:
2467:Hewitt–Savage
2464:
2460:
2456:
2452:
2451:Zero–one laws
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2398:
2396:
2392:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2312:
2310:
2306:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2271:
2269:
2267:
2263:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2238:
2236:
2234:
2230:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2148:Black–Scholes
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2130:
2128:
2126:
2122:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2087:
2085:
2083:
2079:
2073:
2070:
2068:
2065:
2061:
2058:
2056:
2053:
2052:
2051:
2050:Point process
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2029:
2026:
2024:
2021:
2020:
2019:
2016:
2014:
2011:
2009:
2008:Gibbs measure
2006:
2004:
2001:
1999:
1996:
1995:
1993:
1989:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1937:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1912:
1910:
1906:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1853:
1850:
1848:
1845:
1844:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1797:Itô diffusion
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1772:Gamma process
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1697:
1696:
1693:
1689:
1686:
1685:
1684:
1681:
1679:
1676:
1674:
1671:
1670:
1668:
1666:
1662:
1654:
1651:
1649:
1646:
1644:
1643:Self-avoiding
1641:
1639:
1636:
1635:
1634:
1631:
1629:
1628:Moran process
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1595:
1593:
1591:
1590:Discrete time
1587:
1583:
1576:
1571:
1569:
1564:
1562:
1557:
1556:
1553:
1542:
1538:
1531:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1484:
1476:
1472:
1468:
1464:
1460:
1456:
1451:
1446:
1442:
1438:
1434:
1427:
1419:
1415:
1410:
1405:
1401:
1397:
1393:
1386:
1378:
1373:
1369:
1362:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1323:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1284:: 1202–1215.
1283:
1279:
1275:
1268:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1229:
1221:
1217:
1213:
1209:
1205:
1198:
1196:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1156:
1154:
1152:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1108:
1106:
1104:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1064:
1062:
1053:
1049:
1044:
1039:
1035:
1031:
1027:
1020:
1018:
1010:
1005:
996:
991:
987:
983:
979:
972:
970:
968:
963:
955:
953:
949:
944:
930:
926:
922:
919:
916:
908:
892:
872:
863:
849:
846:
843:
835:
819:
816:
813:
804:
802:
798:
782:
773:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
716:
713:
710:
708:
705:
704:
700:
697:
694:
691:
690:
686:
683:
680:
677:
676:
672:
669:
666:
664:
660:
657:
656:
652:
649:
646:
643:
642:
638:
635:
632:
630:
627:
626:
622:
619:
616:
614:
611:
610:
606:
603:
600:
598:
595:
594:
579:
572:
558:
551:
537:
530:
528:Model/Process
527:
526:
520:
506:
498:
494:
493:interest rate
487:Special cases
481:
477:
473:
458:
455:
452:
449:
446:
443:
440:
432:
429:
425:
422:
419:
403:
395:
374:
371:
368:
362:
357:
353:
344:
340:
338:
334:
333:
326:
311:
307:
303:
298:
293:
289:
285:
282:
279:
276:
270:
267:
262:
258:
251:
248:
245:
240:
236:
232:
223:
221:
203:
199:
188:
173:
169:
165:
160:
155:
151:
147:
144:
141:
138:
130:
126:
122:
119:
116:
110:
105:
101:
97:
88:
86:
68:
64:
49:
47:
43:
39:
35:
31:
27:
23:
19:
2735:Econometrics
2697:Wiener space
2585:Itô integral
2486:Inequalities
2375:Self-similar
2345:Gauss–Markov
2335:Exchangeable
2315:Càdlàg paths
2251:Risk process
2203:LIBOR market
2152:
2072:Random graph
2067:Random field
1879:Superprocess
1817:Lévy process
1812:Jump process
1787:Hunt process
1623:Markov chain
1530:
1497:
1493:
1483:
1440:
1436:
1426:
1399:
1395:
1385:
1367:
1361:
1336:
1332:
1322:
1281:
1277:
1267:
1242:
1238:
1228:
1211:
1207:
1169:
1165:
1125:
1121:
1077:
1073:
1033:
1029:
1004:
985:
981:
945:
864:
805:
774:
758:fixed income
744:, and other
734:bond options
727:
490:
225:
218:denotes the
190:
90:
55:
26:CKLS process
25:
21:
15:
2869:Credit risk
2780:Ruin theory
2718:Disciplines
2590:Itô's lemma
2365:Predictable
2040:Percolation
2023:Potts model
2018:Ising model
1982:White noise
1940:Differences
1802:Itô process
1742:Cox process
1638:Loop-erased
1633:Random walk
766:time series
762:credit risk
750:derivatives
495:models and
394:singularity
18:mathematics
2838:Categories
2790:Statistics
2570:Filtration
2471:Kolmogorov
2455:Blumenthal
2380:Stationary
2320:Continuous
2308:Properties
2193:Hull–White
1935:Martingale
1822:Local time
1710:Fractional
1688:pure birth
1450:2105.13724
1443:: 109391.
958:References
832:can model
742:securities
478:(also see
330:Properties
52:Definition
2702:Classical
1715:Geometric
1705:Excursion
1522:150454155
1514:1573-6946
1475:235248362
1467:0167-7152
1418:1572-3097
1372:CiteSeerX
1353:1057-5219
1314:149884892
1306:0378-4371
1259:1544-6123
1186:0927-5398
1134:1017-0405
1094:0167-7152
1052:2227-7072
1036:(3): 76.
917:γ
893:γ
873:γ
844:γ
817:≥
814:γ
783:γ
580:γ
559:β
538:α
507:σ
459:γ
453:σ
447:β
441:α
404:γ
375:γ
372:−
345:(MGF) of
299:γ
286:σ
268:−
249:−
161:γ
148:σ
123:β
117:α
2823:Category
2707:Abstract
2241:Bühlmann
1847:Compound
1142:24307157
1080:: 6–11.
2330:Ergodic
2218:Vašíček
2060:Poisson
1720:Meander
1286:Bibcode
950:of the
746:options
613:Vasicek
34:finance
28:) is a
2670:Tanaka
2355:Mixing
2350:Markov
2223:Wilkie
2188:Ho–Lee
2183:Heston
1955:Super-
1700:Bridge
1648:Biased
1539:
1520:
1512:
1473:
1465:
1416:
1374:
1351:
1312:
1304:
1257:
1184:
1140:
1132:
1092:
1050:
752:, and
692:CIR VR
644:Dothan
597:Merton
392:has a
191:where
20:, the
2523:Tools
2299:M/M/c
2294:M/M/1
2289:M/G/1
2279:Fluid
1945:Local
1541:99894
1518:S2CID
1471:S2CID
1445:arXiv
1310:S2CID
1138:JSTOR
770:GARCH
730:bonds
491:Many
2475:Lévy
2274:Bulk
2158:Chen
1950:Sub-
1908:Both
1537:SSRN
1510:ISSN
1463:ISSN
1414:ISSN
1349:ISSN
1302:ISSN
1255:ISSN
1182:ISSN
1130:ISSN
1090:ISSN
1048:ISSN
847:<
799:and
760:and
717:Any
701:3/2
639:1/2
341:The
2055:Cox
1502:doi
1455:doi
1441:184
1404:doi
1341:doi
1294:doi
1282:523
1247:doi
1216:doi
1174:doi
1082:doi
1038:doi
990:doi
714:Any
707:CEV
684:Any
681:Any
670:Any
661:or
636:Any
633:Any
620:Any
617:Any
601:Any
40:of
16:In
2840::
2473:,
2469:,
2465:,
2461:,
2457:,
1516:.
1508:.
1496:.
1492:.
1469:.
1461:.
1453:.
1439:.
1435:.
1412:.
1398:.
1394:.
1370:,
1347:.
1335:.
1331:.
1308:.
1300:.
1292:.
1280:.
1276:.
1253:.
1241:.
1237:.
1212:52
1210:.
1206:.
1194:^
1180:.
1170:10
1168:.
1164:.
1150:^
1136:.
1126:13
1124:.
1120:.
1102:^
1088:.
1078:98
1076:.
1072:.
1060:^
1046:.
1032:.
1028:.
1016:^
986:47
984:.
980:.
966:^
943:.
862:.
795:.
748:,
740:,
736:,
732:,
687:1
673:1
653:1
623:0
607:0
482:).
87::
2477:)
2453:(
1574:e
1567:t
1560:v
1543:.
1524:.
1504::
1498:6
1477:.
1457::
1447::
1420:.
1406::
1400:1
1355:.
1343::
1337:8
1316:.
1296::
1288::
1261:.
1249::
1243:8
1222:.
1218::
1188:.
1176::
1144:.
1096:.
1084::
1054:.
1040::
1034:6
1011:.
998:.
992::
931:2
927:/
923:1
920:=
850:1
820:1
711:0
698:0
695:0
667:0
650:0
647:0
604:0
456:,
450:,
444:,
420:.
378:)
369:1
366:(
363:2
358:t
354:X
312:t
308:W
304:d
294:t
290:X
283:+
280:t
277:d
274:)
271:a
263:t
259:X
255:(
252:k
246:=
241:t
237:X
233:d
204:t
200:W
174:t
170:W
166:d
156:t
152:X
145:+
142:t
139:d
136:)
131:t
127:X
120:+
114:(
111:=
106:t
102:X
98:d
69:t
65:X
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