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40:. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness. It belongs to the class of no-arbitrage models, i.e. it can fit today's
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The main state variable of the model is the short rate, which is assumed to follow the stochastic differential equation (under the
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has shown how analytic prices can be conveniently deduced in many such circumstances, as well as for interest rate options.
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Perturbation
Expansion for Arrow–Debreu Pricing with Hull-White Interest Rates and Black–Karasinski Credit Intensity
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280:, where the Black–Karasinski short rate expresses the (stochastic) intensity of default events driven by a
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276:(usually trees) are used in the calibration stage as well as for pricing. It can also be used in modeling
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Black, F.; Karasinski, P. (July–August 1991). "Bond and Option pricing when Short rates are
Lognormal".
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In the original article by
Fischer Black and Piotr Karasinski the model was implemented using a
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284:; the guaranteed positive rates are an important feature of the model here. Recent work on
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prices, and in its most general form, today's prices for a set of caps, floors or
European
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implementation is more common in practice, typically a log-normal application of the
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Interest Rate Models – Theory and
Practice with Smile, Inflation and Credit
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Generalized autoregressive conditional heteroskedasticity (GARCH) model
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Exact Arrow-Debreu
Pricing for the Black–Karasinski Short Rate Model
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45:
359:
Analytic Option Prices for the Black–Karasinski Short Rate Model
319:
927:
Autoregressive conditional heteroskedasticity (ARCH) model
357:
Blanka
Horvath, Antoine Jacquier and Colin Turfus (2017).
211:
of the money-market account is infinite for any maturity.
455:
Independent and identically distributed random variables
365:
Analytic
Swaption Pricing in the Black–Karasinski Model
932:
Autoregressive integrated moving average (ARIMA) model
77:
298:
179:
1672:
814:Stochastic chains with memory of variable length
180:{\displaystyle d\ln(r)=\,dt+\sigma _{t}\,dW_{t}}
403:
234:The model is used mainly for the pricing of
383:The Black-Karasinski Model: Thirty Years On
942:Autoregressive–moving-average (ARMA) model
410:
396:
381:Colin Turfus and Piotr Karasinski (2021).
324:(2nd ed. 2006 ed.). Springer Verlag.
286:Perturbation Methods in Credit Derivatives
163:
143:
417:
1673:
1248:Doob's martingale convergence theorems
346:Simon Benninga and Zvi Wiener (1998).
320:Damiano Brigo, Fabio Mercurio (2001).
1000:Constant elasticity of variance (CEV)
990:Chan–Karolyi–Longstaff–Sanders (CKLS)
391:
352:Mathematica in Education and Research
207:for the short rate and therefore the
13:
1487:Skorokhod's representation theorem
1268:Law of large numbers (weak/strong)
14:
1697:
1457:Martingale representation theorem
340:
1502:Stochastic differential equation
1392:Doob's optional stopping theorem
1387:Doob–Meyer decomposition theorem
1372:Convergence of random variables
1258:Fisher–Tippett–Gnedenko theorem
229:
970:Binomial options pricing model
348:Binomial Term Structure Models
140:
137:
131:
99:
93:
87:
48:. The model was introduced by
1:
1437:Kolmogorov continuity theorem
1273:Law of the iterated logarithm
291:
218:with variable spacing, but a
1442:Kolmogorov extension theorem
1121:Generalized queueing network
629:Interacting particle systems
7:
574:Continuous-time random walk
10:
1702:
1582:Extreme value theory (EVT)
1382:Doob decomposition theorem
674:Ornstein–Uhlenbeck process
445:Chinese restaurant process
301:Financial Analysts Journal
1650:
1554:
1462:Optional stopping theorem
1359:
1321:
1263:Large deviation principle
1230:
1144:
1101:
1068:
1015:Heath–Jarrow–Morton (HJM)
960:
952:Moving-average (MA) model
937:Autoregressive (AR) model
917:
827:
762:Hidden Markov model (HMM)
744:
696:Schramm–Loewner evolution
500:
425:
239:interest rate derivatives
1377:Doléans-Dade exponential
1207:Progressively measurable
1005:Cox–Ingersoll–Ross (CIR)
59:
1597:Mathematical statistics
1587:Large deviations theory
1417:Infinitesimal generator
1278:Maximal ergodic theorem
1197:Piecewise-deterministic
799:Random dynamical system
664:Markov additive process
205:log-normal distribution
1432:Karhunen–Loève theorem
1367:Cameron–Martin formula
1331:Burkholder–Davis–Gundy
726:Variance gamma process
203:. The model implies a
181:
22:Black–Karasinski model
1562:Actuarial mathematics
1524:Uniform integrability
1519:Stratonovich integral
1447:Lévy–Prokhorov metric
1351:Marcinkiewicz–Zygmund
1238:Central limit theorem
840:Gaussian random field
669:McKean–Vlasov process
589:Dyson Brownian motion
450:Galton–Watson process
375:Colin Turfus (2019).
369:Colin Turfus (2018).
363:Colin Turfus (2018).
313:10.2469/faj.v47.n4.52
182:
18:financial mathematics
1637:Time series analysis
1592:Mathematical finance
1477:Reflection principle
804:Regenerative process
604:Fleming–Viot process
419:Stochastic processes
258:implied volatilities
75:
66:risk-neutral measure
1632:Stochastic analysis
1472:Quadratic variation
1467:Prokhorov's theorem
1402:Feynman–Kac formula
872:Markov random field
520:Birth–death process
354:, Vol. 7 No. 3 1998
278:credit default risk
1602:Probability theory
1482:Skorokhod integral
1452:Malliavin calculus
1035:Korn-Kreer-Lenssen
919:Time series models
882:Pitman–Yor process
224:Hull–White lattice
177:
26:mathematical model
1681:Short-rate models
1668:
1667:
1622:Signal processing
1341:Doob's upcrossing
1336:Doob's martingale
1300:Engelbert–Schmidt
1243:Donsker's theorem
1177:Feller-continuous
1045:Rendleman–Bartter
835:Dirichlet process
752:Branching process
721:Telegraph process
614:Geometric process
594:Empirical process
584:Diffusion process
440:Branching process
435:Bernoulli process
331:978-3-540-22149-4
274:Numerical methods
1693:
1686:Financial models
1642:Machine learning
1529:Usual hypotheses
1412:Girsanov theorem
1397:Dynkin's formula
1162:Continuous paths
1070:Actuarial models
1010:Garman–Kohlhagen
980:Black–Karasinski
975:Black–Derman–Toy
962:Financial models
828:Fields and other
757:Gaussian process
706:Sigma-martingale
510:Additive process
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54:Piotr Karasinski
42:zero-coupon bond
38:short-rate model
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1607:Queueing theory
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1492:Skorokhod space
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1346:Kunita–Watanabe
1317:
1283:Sanov's theorem
1253:Ergodic theorem
1226:
1222:Time-reversible
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1103:Queueing models
1097:
1093:Sparre–Anderson
1083:Cramér–Lundberg
1064:
1050:SABR volatility
956:
913:
865:Boolean network
823:
809:Renewal process
740:
689:Non-homogeneous
679:Poisson process
569:Contact process
532:Brownian motion
502:Continuous time
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490:Maximal entropy
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201:Brownian motion
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1655:List of topics
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1612:Renewal theory
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1577:Ergodic theory
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1567:Control theory
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850:Hopfield model
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767:Markov process
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736:Wiener sausage
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731:Wiener process
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711:Stable process
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701:Semimartingale
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644:Jump diffusion
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619:Hawkes process
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599:Feller process
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564:Cauchy process
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515:Bessel process
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341:External links
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220:trinomial tree
209:expected value
199:is a standard
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34:interest rates
30:term structure
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1514:Stopping time
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1304:Hewitt–Savage
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1288:Zero–one laws
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985:Black–Scholes
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887:Point process
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845:Gibbs measure
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634:Itô diffusion
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609:Gamma process
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480:Self-avoiding
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465:Moran process
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427:Discrete time
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217:
216:binomial tree
212:
210:
206:
202:
197:
193:
172:
168:
164:
158:
154:
150:
147:
144:
134:
128:
125:
120:
116:
112:
107:
103:
96:
90:
84:
81:
78:
71:
70:
69:
67:
57:
55:
51:
50:Fischer Black
47:
43:
39:
35:
31:
27:
23:
19:
1572:Econometrics
1534:Wiener space
1422:Itô integral
1323:Inequalities
1212:Self-similar
1182:Gauss–Markov
1172:Exchangeable
1152:Càdlàg paths
1088:Risk process
1040:LIBOR market
979:
909:Random graph
904:Random field
716:Superprocess
654:Lévy process
649:Jump process
624:Hunt process
460:Markov chain
351:
321:
307:(4): 52–59.
304:
300:
250:bond options
233:
230:Applications
213:
195:
191:
189:
63:
21:
15:
1617:Ruin theory
1555:Disciplines
1427:Itô's lemma
1202:Predictable
877:Percolation
860:Potts model
855:Ising model
819:White noise
777:Differences
639:Itô process
579:Cox process
475:Loop-erased
470:Random walk
282:Cox process
272:swaptions.
1675:Categories
1627:Statistics
1407:Filtration
1308:Kolmogorov
1292:Blumenthal
1217:Stationary
1157:Continuous
1145:Properties
1030:Hull–White
772:Martingale
659:Local time
547:Fractional
525:pure birth
292:References
1539:Classical
552:Geometric
542:Excursion
254:swaptions
155:σ
129:
117:ϕ
113:−
104:θ
85:
56:in 1991.
46:swaptions
1660:Category
1544:Abstract
1078:Bühlmann
684:Compound
270:European
247:Bermudan
243:American
241:such as
1167:Ergodic
1055:Vašíček
897:Poisson
557:Meander
28:of the
1507:Tanaka
1192:Mixing
1187:Markov
1060:Wilkie
1025:Ho–Lee
1020:Heston
792:Super-
537:Bridge
485:Biased
328:
266:floors
236:exotic
190:where
36:; see
20:, the
1360:Tools
1136:M/M/c
1131:M/M/1
1126:M/G/1
1116:Fluid
782:Local
60:Model
24:is a
1312:Lévy
1111:Bulk
995:Chen
787:Sub-
745:Both
326:ISBN
262:caps
252:and
245:and
52:and
892:Cox
309:doi
268:or
260:of
68:):
32:of
16:In
1677::
1310:,
1306:,
1302:,
1298:,
1294:,
350:,
305:47
303:.
264:,
226:.
192:dW
126:ln
82:ln
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1290:(
411:e
404:t
397:v
334:.
315:.
311::
196:t
173:t
169:W
165:d
159:t
151:+
148:t
145:d
141:]
138:)
135:r
132:(
121:t
108:t
100:[
97:=
94:)
91:r
88:(
79:d
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