3336:(HJM). Unlike the short rate models described above, this class of models is generally non-Markovian. This makes general HJM models computationally intractable for most purposes. The great advantage of HJM models is that they give an analytical description of the entire yield curve, rather than just the short rate. For some purposes (e.g., valuation of mortgage backed securities), this can be a big simplification. The Cox–Ingersoll–Ross and Hull–White models in one or more dimensions can both be straightforwardly expressed in the HJM framework. Other short rate models do not have any simple dual HJM representation.
799:
closest to the market prices. This does not allow for fitting options like caps, floors and swaptions as the parameters have been used to fit linear instruments instead. This problem is overcome by allowing the parameters to vary deterministically with time, or by adding a deterministic shift to the endogenous model. In this way, exogenous models such as Ho-Lee and subsequent models, can be calibrated to market data, meaning that these can exactly return the price of bonds comprising the yield curve, and the remaining parameters can be used for options calibration. The implementation is usually via a (
17:
3311:
2261:. The model may be seen as the lognormal application of Hull–White; its lattice-based implementation is similarly trinomial (binomial requiring varying time-steps). The model has no closed form solutions, and even basic calibration to the initial term structure has to be done with numerical methods to generate the zero coupon bond prices. This model too suffers of the issue of explosion of the expected bank account in finite time.
5419:
2980:
582:, is an output of the model, so it is "inside the model" (endogenous) and is determined by the model parameters. Exogenous short rate models are models where such term structure is an input, as the model involves some time dependent functions or shifts that allow for inputing a given market term structure, so that the term structure comes from outside (exogenous).
3320:
obtain the short rate. This model allows for exact calibration of the term structure, semi-closed form solutions for options, control of the volatility term structure for instantaneous forward rates through the correlation parameter, and especially for negative rates, which has become important as rates turned negative in financial markets.
2533:
floor and swaptions through
Jamshidian's trick. The model allows for maintaining positive rates if the shift is constrained to be positive, or allows for negative rates if the shift is allowed to go negative. It has been applied often in credit risk too, for credit default swap and swaptions, in this original version or with jumps.
2852:
3306:{\displaystyle {\begin{aligned}dr_{t}&=(\theta _{t}-\alpha _{t})\,dt+{\sqrt {r_{t}}}\,\sigma _{t}\,dW_{t},\\d\alpha _{t}&=(\zeta _{t}-\alpha _{t})\,dt+{\sqrt {\alpha _{t}}}\,\sigma _{t}\,dW_{t},\\d\sigma _{t}&=(\beta _{t}-\sigma _{t})\,dt+{\sqrt {\sigma _{t}}}\,\eta _{t}\,dW_{t}.\end{aligned}}}
2532:
is a deterministic shift. The shift can be used to absorb the market term structure and make the model fully consistent with this. This model preserves the analytical tractability of the basic CIR model, allowing for closed form solutions for bonds and all linear products, and options such as caps,
798:
values in such a way that the model coincides with a few observed market prices ("calibration") of zero coupon bonds or linear products such as forward rate agreements or swaps, typically, or a best fit is done to these linear products to find the endogenous short rate models parameters that are
3319:
The two-factor Hull-White or G2++ models are models that have been used due to their tractability. These models are summarized and shown to be equivalent in Brigo and
Mercurio (2006). This model is based on adding two possibly correlated Ornstein-Uhlenbeck (Vasicek) processes plus a shift to
1888:
is the most commonly used and it allows for closed form solutions for bond prices, bond options, caps and floors, and swaptions through
Jamshidian's trick. This model allows for an exact calibration of the initial term structure of interest rates through the time dependent function
1169:
is used for the short rate. This model allows for negative rates, because the probability distribution of the short rate is
Gaussian. Also, this model allows for closed form solutions for the bond price and for bond options and caps/floors, and using
2614:
1268:. This model does not have closed form formulas for options and it is not mean reverting. Moreover, it has the problem of an infinite expected bank account after a short time. The same problem will be present in all lognormal short rate models
2510:
1553:, one can also obtain a formula for swaptions. Both this model and the Vasicek model are called affine models, because the formula for the continuously compounded spot rate for a finite maturity T at time t is an affine function of
1373:
1548:
ensures strictly positive short rates. This model follows a Feller square root process and has non-negative rates, and it allows for closed form solutions for the bond price and for bond options and caps/floors, and using
1800:
2959:
4155:
Giacomo Burro, Pier
Giuseppe Giribone, Simone Ligato, Martina Mulas, and Francesca Querci (2017). Negative interest rates effects on option pricing: Back to basics? International Journal of Financial Engineering 4(2),
1473:
4079:
536:
2985:
1018:
4104:
1689:
allows for the initial term structure of interest rates or bond prices to be an input of the model. This model follows again an arithmetic
Brownian motion with time dependent deterministic drift parameter.
1262:
789:
of interest rates, these models can be thought of as specific cases of
Ornstein–Uhlenbeck processes. The Vasicek, Rendleman–Bartter and CIR models are endogenous models and have only a finite number of
2619:
2348:
2138:
otherwise; the model is lognormal. The model has no closed form formulas for options. Also, as all lognormal models, it suffers from the issue of explosion of the expected bank account in finite time.
811:, although some short rate models have closed form solutions for zero coupon bonds, and even caps or floors, easing the calibration task considerably. We list the following endogenous models first.
2136:
1660:
2365:
in 2001, and formulated also earlier by Scott (1995) used the CIR model but instead of introducing time dependent parameters in the dynamics, it adds an external shift. The model is formulated as
1509:
factor precludes (generally) the possibility of negative interest rates. The interpretation of the parameters, in the second formulation, is the same as in the
Vasicek model. The Feller condition
1103:
401:
541:
Short rate models are often classified as endogenous and exogenous. Endogenous short rate models are short rate models where the term structure of interest rates, or of zero-coupon bond prices
2597:", these multi-factor short-rate models are sometimes preferred over One-factor models, as they produce scenarios which are, in general, better "consistent with actual yield curve movements".
2259:
886:
4137:
Lin Chen (1996). "Stochastic Mean and
Stochastic Volatility — A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives".
2061:
2593:
two factor model and the Chen three factor model (also called "stochastic mean and stochastic volatility model"). Note that for the purposes of risk management, "to create realistic
1507:
771:
1546:
3451:
2847:{\displaystyle {\begin{aligned}dX_{t}&=(a_{t}-bX_{t})\,dt+{\sqrt {X_{t}}}\,c_{t}\,dW_{1t},\\dY_{t}&=(d_{t}-eY_{t})\,dt+{\sqrt {Y_{t}}}\,f_{t}\,dW_{2t},\end{aligned}}}
1826:
580:
428:
4062:
Brigo, D. and El-Bachir, N. (2010). An exact formula for default swaptions pricing in the SSRJD stochastic intensity model. Mathematical
Finance. July 2010, pp. 365-382,
1914:
1687:
3503:
Brigo, D. and Mercurio, F. (2001). A deterministic–shift extension of analytically–tractable and time–homogeneous short–rate models. Finance and Stochastics 5, 369–387.
4074:
4053:
Scott, L. (1995). The valuation of interest rate derivatives in a multi-factor term-structure model with deterministic components. University of Georgia. Working paper.
918:
655:
2537:
The idea of a deterministic shift can be applied also to other models that have desirable properties in their endogenous form. For example, one could apply the shift
1886:
1866:
1846:
1163:
722:
616:
187:
127:
77:
2368:
2555:
2530:
1578:
690:
3885:
4452:
2575:
1143:
1123:
258:
234:
214:
151:
5256:
5168:
1278:
2350:, a lognormal analogue to the Ho–Lee model, and a special case of the Black–Derman–Toy model. This approach is effectively similar to "the original
785:
factor – the short rate – determines the future evolution of all interest rates. Other than Rendleman–Bartter and Ho–Lee, which do not capture the
4317:
3731:
1699:
2863:
3627:
3344:
1378:
4962:
4537:
452:
4967:
1848:
are not time-dependent. The distribution of the short rate is normal, and the model allows for negative rates. The model with constant
938:
3479:
3446:
1184:
5302:
2271:
925:
4992:
2265:
808:
4490:
4416:
4394:
4371:
4338:
4279:
4250:
4227:
4204:
4185:
4038:
3431:
3333:
2585:
Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are the
2066:
1595:
1023:
804:
5124:
3340:
266:
2594:
3898:
4463:
2148:
133:, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time
4860:
821:
1934:
5362:
4530:
4330:
1272:
658:
4427:
3389:
5297:
4942:
3949:
3588:
1928:
1166:
800:
693:
4294:
1105:. The second form is the more common, and makes the parameters interpretation more direct, with the parameter
5454:
27:(black vs red): the short rate is the top value; the development of the bond value shows pull-to-par clearly
1478:
130:
2602:
2557:
to the Vasicek model, but due to linearity of the Ornstein-Uhlenbeck process, this is equivalent to making
727:
442:. Thus, specifying a model for the short rate specifies future bond prices. This means that instantaneous
5337:
4875:
4729:
4523:
3802:
1178:
3691:
1512:
5449:
5193:
5134:
4956:
4482:
4242:
4177:
3955:
3893:
2142:
4501:
5238:
5049:
4288:
1805:
1265:
544:
36:
4003:
Effective duration of callable bonds: the Salomon Brothers term structure-based option pricing model
3850:
3602:
409:
5357:
5352:
3789:
3687:
3371:
786:
5444:
5307:
5007:
4977:
4952:
4835:
4676:
4608:
1917:
1892:
1665:
20:
5104:
5089:
5054:
4997:
3845:
3793:
3597:
2505:{\displaystyle dx_{t}=a(b-x_{t})\,dt+{\sqrt {x_{t}}}\,\sigma \,dW_{t},\ \ r_{t}=x_{t}+\phi (t)}
891:
24:
4024:
629:
5317:
5084:
4982:
4661:
3725:
3683:
1871:
1851:
1831:
1550:
1171:
1148:
699:
593:
164:
104:
54:
4443:
5271:
5228:
5218:
5208:
5203:
4929:
4870:
4805:
4759:
4754:
4628:
4588:
4555:
4386:
3474:
3463:
2540:
2515:
1693:
1556:
668:
623:
194:
3830:
8:
5276:
5064:
4987:
4810:
4357:
4271:
4265:
4105:"Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model"
3745:
3679:
3487:
3669:
Dothan, L.U. (1978). On the term structure of interest rates. Jour. of Fin. Ec., 6:59–69
161:
show that, under some fairly relaxed technical conditions, if we model the evolution of
5327:
5312:
5281:
5266:
5233:
5099:
4890:
4855:
4618:
4583:
4546:
4311:
4303:
4123:
3771:
3754:
3713:
3652:
3644:
3565:
3459:
2560:
1128:
1108:
921:
431:
243:
219:
199:
190:
136:
40:
4347:
5332:
5322:
5261:
5248:
5223:
5109:
4895:
4691:
4486:
4474:
4412:
4390:
4367:
4334:
4290:
4275:
4246:
4223:
4200:
4181:
4096:
4034:
4006:
3656:
3611:
3427:
2586:
435:
4260:
5213:
5152:
5147:
5129:
5059:
4825:
4820:
4792:
4744:
4623:
4563:
4439:
4119:
4100:
4063:
3983:
3971:
3932:
3916:
3886:"A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options"
3855:
3811:
3763:
3705:
3636:
3607:
3555:
3547:
3535:
3419:
3359:
2590:
2351:
1589:
815:
237:
1368:{\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}
5423:
5393:
5388:
5342:
5178:
5173:
5119:
5029:
4937:
4910:
4850:
4845:
4815:
4764:
4749:
4666:
4646:
4083:
3583:
3483:
3455:
3752:(1986). "Term structure movements and pricing interest rate contingent claims".
3625:
Rendleman, R.; Bartter, B. (1980). "The Pricing of Options on Debt Securities".
3339:
The HJM framework with multiple sources of randomness, including as it does the
2971:(1996) which has a stochastic mean and volatility of the short rate, is given by
5398:
5383:
5183:
5094:
5044:
5021:
5002:
4830:
4772:
4739:
4734:
4714:
4638:
4001:
Kopprasch, R.; Boyce, W.; Koenigsberg, M.; Tatevassian, A.; Yampol, M. (1987).
3967:
3920:
3877:
3797:
2362:
1921:
1920:
for Bermudan swaptions and for products without analytical formulas is usually
791:
619:
91:
3859:
3423:
2577:
a time dependent function, and would thus coincide with the Hull-White model.
5438:
5378:
5347:
5188:
5114:
5074:
5069:
4905:
4777:
4724:
4719:
4701:
4598:
4578:
4404:
4296:
An Empirical Comparison of Alternative Models of the Short-Term Interest Rate
4219:
3873:
3518:
3351:
2358:
1795:{\displaystyle dr_{t}=(\theta _{t}-\alpha _{t}r_{t})\,dt+\sigma _{t}\,dW_{t}}
932:
95:
44:
4157:
4076:
Pitfalls in Asset and Liability Management: One Factor Term Structure Models
4010:
2954:{\displaystyle dr_{t}=(\mu X+\theta Y)\,dt+\sigma _{t}{\sqrt {Y}}\,dW_{3t}.}
5198:
4972:
4900:
4880:
4840:
4709:
4681:
4671:
4613:
4197:
Interest Rate Models – Theory and Practice with Smile, Inflation and Credit
3831:"Efficient Calibration of Trinomial Trees for One-Factor Short Rate Models"
3749:
3696:
443:
3987:
3936:
3815:
16:
5079:
4947:
4918:
4914:
4865:
4656:
4651:
3881:
3355:
154:
3560:
5403:
5039:
5034:
4800:
4686:
3775:
3717:
3678:
3648:
3569:
2968:
782:
88:
1468:{\displaystyle dr_{t}=a(b-r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}
4593:
4515:
4409:
Modelling Fixed Income Securities and Interest Rate Options (2nd ed.)
3504:
795:
662:
439:
158:
98:
4000:
3767:
3709:
3640:
3551:
5163:
4885:
4782:
4603:
531:{\displaystyle f(t,T)=-{\frac {\partial }{\partial T}}\ln(P(t,T)).}
4509:
Royal Bank of Scotland Quantitative Research Centre Working Paper
3586:(1977). "An Equilibrium Characterisation of the Term Structure".
1013:{\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+\sigma \,dW_{t}}
4327:
Interest Rate Dynamics, Derivatives Pricing, and Risk Management
1165:
being the instantaneous volatility. In this short rate model an
153:. Specifying the current short rate does not specify the entire
5418:
4348:
Rajna Gibson, François-Serge Lhabitant and Denis Talay (1999).
4095:
3332:
The other major framework for interest rate modelling is the
1257:{\displaystyle dr_{t}=\theta r_{t}\,dt+\sigma r_{t}\,dW_{t}}
4213:
4194:
2343:{\displaystyle d\ln(r_{t})=\theta _{t}\,dt+\sigma \,dW_{t}}
310:
4350:
Modeling the Term Structure of Interest Rates: An overview
4171:
4028:
3974:(1993). "A Model for Valuing Bonds and Embedded Options".
3409:
3407:
3405:
3403:
3828:
3538:, Robert C. (1973). "Theory of Rational Option Pricing".
3395:
1181:(1980) or Dothan model (1978) explains the short rate as
4380:
4358:
The Past, Present and Future of Term Structure Modelling
3921:"Bond and Option pricing when Short rates are Lognormal"
805:
Lattice model (finance) § Interest rate derivatives
3400:
2131:{\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}
4453:"Implementing Interest Rate Models: a Practical Guide"
3966:
3872:
3744:
1802:. In many presentations one or more of the parameters
1655:{\displaystyle dr_{t}=\theta _{t}\,dt+\sigma \,dW_{t}}
1584:
We now list a number of exogenous short rate models.
3347:, is often preferred for models of higher dimension.
2983:
2866:
2617:
2563:
2543:
2518:
2371:
2357:
The CIR++ model, introduced and studied in detail by
2274:
2151:
2069:
1937:
1895:
1874:
1854:
1834:
1808:
1702:
1696:(1990)—also called the extended Vasicek model—posits
1668:
1598:
1559:
1515:
1481:
1381:
1281:
1187:
1151:
1131:
1111:
1098:{\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}}
1026:
941:
894:
824:
730:
702:
671:
632:
596:
547:
455:
412:
269:
246:
222:
202:
167:
139:
107:
57:
2605:(1992) supposes the short rate dynamics are given by
920:
is a one-dimensional Brownian motion under the spot
781:
Following are the one-factor models, where a single
3914:
3788:
3525:, Vol. 7 No. 3 1998. Simon Benninga and Zvi Wiener.
434:for the process. The interest rates implied by the
396:{\displaystyle P(t,T)=\operatorname {E} ^{Q}\left,}
4236:
4086:, Dr. Donald R. van Deventer, Kamakura Corporation
4031:Fixed Income Securities: Tools for Today's Markets
3692:"A Theory of the Term Structure of Interest Rates"
3305:
2953:
2846:
2569:
2549:
2524:
2504:
2354:model" (1987), also a lognormal variant on Ho-Lee.
2342:
2253:
2130:
2055:
1908:
1880:
1860:
1840:
1820:
1794:
1681:
1654:
1572:
1540:
1501:
1467:
1367:
1256:
1157:
1137:
1117:
1097:
1012:
912:
880:
765:
716:
684:
649:
610:
574:
530:
422:
395:
252:
228:
208:
181:
145:
121:
71:
4499:
4473:
4139:Financial Markets, Institutions & Instruments
3418:. Springer Finance. Heidelberg: Springer-Verlag.
2580:
1125:being the speed of mean reversion, the parameter
5436:
4064:https://doi.org/10.1111/j.1467-9965.2010.00401.x
3624:
3582:
3540:Bell Journal of Economics and Management Science
3514:
3512:
2254:{\displaystyle d\ln(r)=\,dt+\sigma _{t}\,dW_{t}}
881:{\displaystyle r_{t}=r_{0}+at+\sigma W_{t}^{*}}
776:
585:
3628:Journal of Financial and Quantitative Analysis
2056:{\displaystyle d\ln(r)=dt+\sigma _{t}\,dW_{t}}
924:. In this approach, the short rate follows an
4531:
4425:
4403:
3798:"Pricing interest-rate derivative securities"
3509:
3413:
3327:
2063:for time-dependent short rate volatility and
4316:: CS1 maint: multiple names: authors list (
3730:: CS1 maint: multiple names: authors list (
1145:being the long term mean, and the parameter
4479:Modern Pricing of Interest-Rate Derivatives
4361:
4214:Gerald Buetow & James Sochacki (2001).
4149:
1174:, one can also get a formula for swaptions.
794:and so it is not possible to specify these
4538:
4524:
4216:Term-Structure Models Using Binomial Trees
4199:(2nd ed. 2006 ed.). Springer Verlag.
4172:Martin Baxter & Andrew Rennie (1996).
4029:Tuckman, Bruce & Angel Serrat (2011).
3499:
3497:
3495:
3448:An Overview of Interest-Rate Option Models
3358:are used when interest rates approach the
47:by describing the future evolution of the
4450:
4352:. The Journal of Risk, 1(3): 37–62, 1999.
4158:https://doi.org/10.1142/S2424786317500347
3849:
3601:
3559:
3416:Interest rate models: theory and practice
3282:
3271:
3247:
3177:
3166:
3142:
3072:
3061:
3037:
2931:
2904:
2820:
2809:
2785:
2709:
2698:
2674:
2441:
2437:
2413:
2326:
2313:
2237:
2217:
2114:
2101:
2039:
1778:
1758:
1638:
1625:
1451:
1447:
1423:
1351:
1347:
1323:
1264:. In this model the short rate follows a
1240:
1217:
1081:
1068:
996:
983:
762:
713:
646:
607:
438:form a yield curve, or more precisely, a
353:
178:
118:
68:
4381:Jessica James & Nick Webber (2000).
4324:
4136:
3414:Brigo, Damiano; Mercurio, Fabio (2006).
818:model (1973) explains the short rate as
446:are also specified by the usual formula
15:
5363:Power reverse dual-currency note (PRDC)
5303:Constant proportion portfolio insurance
4444:10.1146/annurev.financial.050808.114513
3492:
43:that describes the future evolution of
5437:
4545:
4428:"The Term Structure of Interest Rates"
4239:Interest Rate Models – An Introduction
4195:Damiano Brigo; Fabio Mercurio (2001).
3534:
1502:{\displaystyle \sigma {\sqrt {r_{t}}}}
809:Monte Carlo methods for option pricing
803:) short rate tree or simulation; see
4519:
4047:
3523:Mathematica in Education and Research
5298:Collateralized debt obligation (CDO)
4432:Annual Review of Financial Economics
3829:Markus Leippold; Zvi Wiener (2004).
766:{\displaystyle r_{t}=\exp {X_{t}}\,}
4218:. The Research Foundation of AIMR (
2858:where the short rate is defined as
13:
4411:. Stanford Economics and Finance.
4267:Encyclopaedia of Actuarial Science
4165:
4124:10.1111/j.1540-6261.1992.tb04657.x
3505:https://doi.org/10.1007/PL00013541
1541:{\displaystyle 2ab>\sigma ^{2}}
486:
482:
415:
374:
292:
14:
5466:
4502:"Term-Structure Models: a Review"
3476:Continuous-Time Short Rate Models
82:
5417:
2145:(1991), which is lognormal, has
1592:(1986) models the short rate as
935:(1977) models the short rate as
4289:K. C. Chan, G. Andrew Karolyi,
4130:
4089:
4068:
4056:
4017:
3994:
3960:
3943:
3908:
3866:
3822:
3782:
3738:
3672:
3663:
1821:{\displaystyle \theta ,\alpha }
575:{\displaystyle T\mapsto P(0,T)}
260:with a payoff of 1 is given by
5125:Year-on-year inflation-indexed
4306:, Vol. XLVII, No. 3 July 1992.
4293:, and Anthony Sanders (1992).
3838:Review of Derivatives Research
3618:
3589:Journal of Financial Economics
3576:
3528:
3519:Binomial Term Structure Models
3468:
3440:
3383:
3244:
3218:
3139:
3113:
3034:
3008:
2901:
2883:
2782:
2753:
2671:
2642:
2581:Multi-factor short-rate models
2499:
2493:
2410:
2391:
2297:
2284:
2266:Kalotay–Williams–Fabozzi model
2214:
2211:
2205:
2173:
2167:
2161:
2085:
2079:
2017:
2014:
2008:
1959:
1953:
1947:
1755:
1719:
1420:
1401:
1320:
1298:
1065:
1046:
980:
958:
569:
557:
551:
522:
519:
507:
501:
471:
459:
423:{\displaystyle {\mathcal {F}}}
285:
273:
87:Under a short rate model, the
1:
5135:Zero-coupon inflation-indexed
3377:
3334:Heath–Jarrow–Morton framework
2268:(1993) has the short rate as
3954:, Professor Ser-Huang Poon,
3612:10.1016/0304-405X(77)90016-2
1918:Lattice-based implementation
777:One-factor short-rate models
586:Particular short-rate models
7:
5338:Foreign exchange derivative
4730:Callable bull/bear contract
4259:Andrew J.G. Cairns (2004).
4237:Andrew J.G. Cairns (2004).
3803:Review of Financial Studies
3394:, Prof. Andrew Lesniewski,
3365:
3341:Brace–Gatarek–Musiela model
1909:{\displaystyle \theta _{t}}
1682:{\displaystyle \theta _{t}}
10:
5471:
4500:Riccardo Rebonato (2003).
4483:Princeton University Press
4243:Princeton University Press
4178:Cambridge University Press
3976:Financial Analysts Journal
3956:Manchester Business School
3925:Financial Analysts Journal
3894:Financial Analysts Journal
3328:Other interest rate models
1167:Ornstein–Uhlenbeck process
926:arithmetic Brownian motion
694:Ornstein–Uhlenbeck process
5412:
5371:
5290:
5247:
5239:Stock market index future
5143:
5020:
4928:
4791:
4700:
4637:
4571:
4562:
4553:
4460:CMPR Research Publication
3860:10.1007/s11147-004-4810-8
3424:10.1007/978-3-540-34604-3
2595:interest rate simulations
1266:geometric Brownian motion
913:{\displaystyle W_{t}^{*}}
216:, then the price at time
37:interest rate derivatives
5358:Mortgage-backed security
5353:Interest rate derivative
5328:Equity-linked note (ELN)
5313:Credit-linked note (CLN)
3372:Fixed-income attribution
2603:Longstaff–Schwartz model
1273:Cox–Ingersoll–Ross model
692:is assumed to follow an
650:{\displaystyle dW_{t}\,}
626:probability measure and
590:Throughout this section
5308:Contract for difference
4609:Risk-free interest rate
4383:Interest Rate Modelling
3970:; Williams, George O.;
3897:: 24–32. Archived from
1881:{\displaystyle \sigma }
1861:{\displaystyle \alpha }
1841:{\displaystyle \sigma }
1179:Rendleman–Bartter model
1158:{\displaystyle \sigma }
717:{\displaystyle r_{t}\,}
611:{\displaystyle W_{t}\,}
182:{\displaystyle r_{t}\,}
131:continuously compounded
122:{\displaystyle r_{t}\,}
72:{\displaystyle r_{t}\,}
5090:Forward Rate Agreement
4426:Robert Jarrow (2009).
4364:Modern Risk Management
4356:Lane Hughston (2003).
4033:. Hoboken, NJ: Wiley.
3307:
2955:
2848:
2571:
2551:
2526:
2506:
2344:
2255:
2143:Black–Karasinski model
2132:
2057:
1929:Black–Derman–Toy model
1910:
1882:
1862:
1842:
1822:
1796:
1683:
1656:
1574:
1542:
1503:
1469:
1375:, it is often written
1369:
1258:
1159:
1139:
1119:
1099:
1020:; it is often written
1014:
914:
882:
767:
718:
686:
651:
618:represents a standard
612:
576:
532:
424:
397:
254:
230:
210:
183:
159:no-arbitrage arguments
147:
123:
73:
28:
5318:Credit default option
4662:Employee stock option
3988:10.2469/faj.v49.n3.35
3937:10.2469/faj.v47.n4.52
3486:, Prof Martin Haugh,
3308:
2956:
2849:
2572:
2552:
2550:{\displaystyle \phi }
2527:
2525:{\displaystyle \phi }
2507:
2345:
2256:
2133:
2058:
1911:
1883:
1863:
1843:
1823:
1797:
1684:
1657:
1575:
1573:{\displaystyle r_{t}}
1543:
1504:
1470:
1370:
1259:
1160:
1140:
1120:
1100:
1015:
915:
883:
768:
724:is assumed to follow
719:
687:
685:{\displaystyle X_{t}}
661:. Where the model is
652:
613:
577:
533:
425:
398:
255:
231:
211:
184:
148:
124:
74:
19:
5455:Mathematical finance
5272:Inflation derivative
5257:Commodity derivative
5229:Single-stock futures
5219:Normal backwardation
5209:Interest rate future
5050:Conditional variance
4556:Derivative (finance)
4362:Peter Field (2003).
4261:Interest-Rate Models
3464:University of Twente
2981:
2864:
2615:
2561:
2541:
2516:
2369:
2272:
2149:
2067:
1935:
1893:
1872:
1852:
1832:
1806:
1700:
1666:
1596:
1557:
1513:
1479:
1379:
1279:
1185:
1149:
1129:
1109:
1024:
939:
892:
822:
728:
700:
669:
630:
594:
545:
453:
410:
267:
244:
220:
200:
195:risk-neutral measure
165:
137:
105:
55:
35:, in the context of
5424:Business portal
5277:Property derivative
4272:John Wiley and Sons
3816:10.1093/rfs/3.4.573
3488:Columbia University
1989:
909:
877:
342:
94:is taken to be the
5282:Weather derivative
5267:Freight derivative
5249:Exotic derivatives
5169:Commodities future
4856:Intermarket spread
4619:Synthetic position
4547:Derivatives market
4451:F.C. Park (2004).
4304:Journal of Finance
4174:Financial Calculus
4112:Journal of Finance
4082:2012-04-03 at the
3968:Kalotay, Andrew J.
3755:Journal of Finance
3482:2012-01-23 at the
3460:Farshid Jamshidian
3454:2012-04-06 at the
3303:
3301:
2951:
2844:
2842:
2567:
2547:
2522:
2502:
2340:
2251:
2128:
2053:
1977:
1906:
1878:
1858:
1838:
1818:
1792:
1679:
1652:
1570:
1551:Jamshidian's trick
1538:
1499:
1465:
1365:
1254:
1172:Jamshidian's trick
1155:
1135:
1115:
1095:
1010:
922:martingale measure
910:
895:
878:
863:
763:
714:
682:
647:
608:
572:
528:
432:natural filtration
420:
393:
328:
250:
226:
206:
191:stochastic process
179:
143:
119:
101:. The short rate,
69:
51:, usually written
41:mathematical model
29:
5450:Short-rate models
5432:
5431:
5333:Equity derivative
5323:Credit derivative
5291:Other derivatives
5262:Energy derivative
5224:Perpetual futures
5105:Overnight indexed
5055:Constant maturity
5016:
5015:
4963:Finite difference
4896:Protective option
4492:978-0-691-08973-7
4475:Riccardo Rebonato
4418:978-0-8047-4438-6
4396:978-0-471-97523-6
4373:978-1-906348-30-4
4340:978-3-540-60814-1
4325:Lin Chen (1996).
4291:Francis Longstaff
4281:978-0-470-84676-6
4252:978-0-691-11894-9
4229:978-0-943205-53-3
4206:978-3-540-22149-4
4187:978-0-521-55289-9
4040:978-0-470-89169-8
3972:Fabozzi, Frank J.
3951:Short Rate Models
3433:978-3-540-22149-4
3391:Short rate models
3269:
3164:
3059:
2929:
2807:
2696:
2570:{\displaystyle b}
2463:
2460:
2435:
2000:
1497:
1445:
1345:
1138:{\displaystyle b}
1118:{\displaystyle a}
493:
436:zero coupon bonds
253:{\displaystyle T}
240:maturing at time
229:{\displaystyle t}
209:{\displaystyle Q}
146:{\displaystyle t}
5462:
5422:
5421:
5194:Forwards pricing
4968:Garman–Kohlhagen
4569:
4568:
4540:
4533:
4526:
4517:
4516:
4512:
4506:
4496:
4470:
4468:
4462:. Archived from
4457:
4447:
4422:
4400:
4377:
4353:
4344:
4321:
4315:
4307:
4301:
4285:
4256:
4233:
4210:
4191:
4160:
4153:
4147:
4146:
4134:
4128:
4127:
4109:
4093:
4087:
4072:
4066:
4060:
4054:
4051:
4045:
4044:
4021:
4015:
4014:
4005:. Salomon Bros.
3998:
3992:
3991:
3964:
3958:
3947:
3941:
3940:
3912:
3906:
3905:
3903:
3890:
3870:
3864:
3863:
3853:
3835:
3826:
3820:
3819:
3786:
3780:
3779:
3762:(5): 1011–1029.
3742:
3736:
3735:
3729:
3721:
3676:
3670:
3667:
3661:
3660:
3622:
3616:
3615:
3605:
3584:Vasicek, Oldrich
3580:
3574:
3573:
3563:
3532:
3526:
3516:
3507:
3501:
3490:
3472:
3466:
3444:
3438:
3437:
3411:
3398:
3387:
3360:zero lower bound
3350:Models based on
3312:
3310:
3309:
3304:
3302:
3295:
3294:
3281:
3280:
3270:
3268:
3267:
3258:
3243:
3242:
3230:
3229:
3210:
3209:
3190:
3189:
3176:
3175:
3165:
3163:
3162:
3153:
3138:
3137:
3125:
3124:
3105:
3104:
3085:
3084:
3071:
3070:
3060:
3058:
3057:
3048:
3033:
3032:
3020:
3019:
3000:
2999:
2960:
2958:
2957:
2952:
2947:
2946:
2930:
2925:
2923:
2922:
2879:
2878:
2853:
2851:
2850:
2845:
2843:
2836:
2835:
2819:
2818:
2808:
2806:
2805:
2796:
2781:
2780:
2765:
2764:
2745:
2744:
2725:
2724:
2708:
2707:
2697:
2695:
2694:
2685:
2670:
2669:
2654:
2653:
2634:
2633:
2576:
2574:
2573:
2568:
2556:
2554:
2553:
2548:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2486:
2485:
2473:
2472:
2461:
2458:
2454:
2453:
2436:
2434:
2433:
2424:
2409:
2408:
2384:
2383:
2352:Salomon Brothers
2349:
2347:
2346:
2341:
2339:
2338:
2312:
2311:
2296:
2295:
2260:
2258:
2257:
2252:
2250:
2249:
2236:
2235:
2198:
2197:
2185:
2184:
2137:
2135:
2134:
2129:
2127:
2126:
2100:
2099:
2062:
2060:
2059:
2054:
2052:
2051:
2038:
2037:
2001:
1999:
1998:
1985:
1976:
1971:
1970:
1915:
1913:
1912:
1907:
1905:
1904:
1887:
1885:
1884:
1879:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1827:
1825:
1824:
1819:
1801:
1799:
1798:
1793:
1791:
1790:
1777:
1776:
1754:
1753:
1744:
1743:
1731:
1730:
1715:
1714:
1694:Hull–White model
1688:
1686:
1685:
1680:
1678:
1677:
1662:. The parameter
1661:
1659:
1658:
1653:
1651:
1650:
1624:
1623:
1611:
1610:
1579:
1577:
1576:
1571:
1569:
1568:
1547:
1545:
1544:
1539:
1537:
1536:
1508:
1506:
1505:
1500:
1498:
1496:
1495:
1486:
1474:
1472:
1471:
1466:
1464:
1463:
1446:
1444:
1443:
1434:
1419:
1418:
1394:
1393:
1374:
1372:
1371:
1366:
1364:
1363:
1346:
1344:
1343:
1334:
1319:
1318:
1294:
1293:
1275:(1985) supposes
1263:
1261:
1260:
1255:
1253:
1252:
1239:
1238:
1216:
1215:
1200:
1199:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1094:
1093:
1064:
1063:
1039:
1038:
1019:
1017:
1016:
1011:
1009:
1008:
979:
978:
954:
953:
919:
917:
916:
911:
908:
903:
887:
885:
884:
879:
876:
871:
847:
846:
834:
833:
772:
770:
769:
764:
761:
760:
759:
740:
739:
723:
721:
720:
715:
712:
711:
691:
689:
688:
683:
681:
680:
656:
654:
653:
648:
645:
644:
617:
615:
614:
609:
606:
605:
581:
579:
578:
573:
537:
535:
534:
529:
494:
492:
481:
429:
427:
426:
421:
419:
418:
402:
400:
399:
394:
389:
385:
384:
383:
378:
377:
370:
366:
365:
364:
360:
352:
351:
341:
336:
300:
299:
259:
257:
256:
251:
238:zero-coupon bond
235:
233:
232:
227:
215:
213:
212:
207:
188:
186:
185:
180:
177:
176:
152:
150:
149:
144:
129:, then, is the (
128:
126:
125:
120:
117:
116:
78:
76:
75:
70:
67:
66:
33:short-rate model
5470:
5469:
5465:
5464:
5463:
5461:
5460:
5459:
5435:
5434:
5433:
5428:
5416:
5408:
5394:Great Recession
5389:Government debt
5367:
5343:Fund derivative
5286:
5243:
5204:Futures pricing
5179:Dividend future
5174:Currency future
5157:
5139:
5012:
4988:Put–call parity
4924:
4911:Vertical spread
4846:Diagonal spread
4816:Calendar spread
4787:
4696:
4633:
4558:
4549:
4544:
4504:
4493:
4466:
4455:
4419:
4397:
4374:
4341:
4309:
4308:
4299:
4282:
4264:
4253:
4230:
4207:
4188:
4168:
4166:Further reading
4163:
4154:
4150:
4135:
4131:
4107:
4097:Longstaff, F.A.
4094:
4090:
4084:Wayback Machine
4073:
4069:
4061:
4057:
4052:
4048:
4041:
4022:
4018:
3999:
3995:
3965:
3961:
3948:
3944:
3913:
3909:
3901:
3888:
3871:
3867:
3851:10.1.1.203.4729
3833:
3827:
3823:
3787:
3783:
3768:10.2307/2328161
3743:
3739:
3723:
3722:
3710:10.2307/1911242
3677:
3673:
3668:
3664:
3641:10.2307/2979016
3623:
3619:
3603:10.1.1.456.1407
3581:
3577:
3552:10.2307/3003143
3533:
3529:
3517:
3510:
3502:
3493:
3484:Wayback Machine
3473:
3469:
3456:Wayback Machine
3445:
3441:
3434:
3412:
3401:
3388:
3384:
3380:
3368:
3330:
3324:
3300:
3299:
3290:
3286:
3276:
3272:
3263:
3259:
3257:
3238:
3234:
3225:
3221:
3211:
3205:
3201:
3195:
3194:
3185:
3181:
3171:
3167:
3158:
3154:
3152:
3133:
3129:
3120:
3116:
3106:
3100:
3096:
3090:
3089:
3080:
3076:
3066:
3062:
3053:
3049:
3047:
3028:
3024:
3015:
3011:
3001:
2995:
2991:
2984:
2982:
2979:
2978:
2939:
2935:
2924:
2918:
2914:
2874:
2870:
2865:
2862:
2861:
2841:
2840:
2828:
2824:
2814:
2810:
2801:
2797:
2795:
2776:
2772:
2760:
2756:
2746:
2740:
2736:
2730:
2729:
2717:
2713:
2703:
2699:
2690:
2686:
2684:
2665:
2661:
2649:
2645:
2635:
2629:
2625:
2618:
2616:
2613:
2612:
2583:
2562:
2559:
2558:
2542:
2539:
2538:
2517:
2514:
2513:
2481:
2477:
2468:
2464:
2449:
2445:
2429:
2425:
2423:
2404:
2400:
2379:
2375:
2370:
2367:
2366:
2334:
2330:
2307:
2303:
2291:
2287:
2273:
2270:
2269:
2245:
2241:
2231:
2227:
2193:
2189:
2180:
2176:
2150:
2147:
2146:
2122:
2118:
2095:
2091:
2068:
2065:
2064:
2047:
2043:
2033:
2029:
1994:
1990:
1981:
1975:
1966:
1962:
1936:
1933:
1932:
1900:
1896:
1894:
1891:
1890:
1873:
1870:
1869:
1853:
1850:
1849:
1833:
1830:
1829:
1807:
1804:
1803:
1786:
1782:
1772:
1768:
1749:
1745:
1739:
1735:
1726:
1722:
1710:
1706:
1701:
1698:
1697:
1673:
1669:
1667:
1664:
1663:
1646:
1642:
1619:
1615:
1606:
1602:
1597:
1594:
1593:
1564:
1560:
1558:
1555:
1554:
1532:
1528:
1514:
1511:
1510:
1491:
1487:
1485:
1480:
1477:
1476:
1459:
1455:
1439:
1435:
1433:
1414:
1410:
1389:
1385:
1380:
1377:
1376:
1359:
1355:
1339:
1335:
1333:
1314:
1310:
1289:
1285:
1280:
1277:
1276:
1248:
1244:
1234:
1230:
1211:
1207:
1195:
1191:
1186:
1183:
1182:
1150:
1147:
1146:
1130:
1127:
1126:
1110:
1107:
1106:
1089:
1085:
1059:
1055:
1034:
1030:
1025:
1022:
1021:
1004:
1000:
974:
970:
949:
945:
940:
937:
936:
904:
899:
893:
890:
889:
872:
867:
842:
838:
829:
825:
823:
820:
819:
792:free parameters
779:
755:
751:
750:
735:
731:
729:
726:
725:
707:
703:
701:
698:
697:
676:
672:
670:
667:
666:
640:
636:
631:
628:
627:
620:Brownian motion
601:
597:
595:
592:
591:
588:
546:
543:
542:
485:
480:
454:
451:
450:
414:
413:
411:
408:
407:
379:
373:
372:
371:
347:
343:
337:
332:
324:
320:
319:
312:
309:
308:
304:
295:
291:
268:
265:
264:
245:
242:
241:
221:
218:
217:
201:
198:
197:
172:
168:
166:
163:
162:
138:
135:
134:
112:
108:
106:
103:
102:
85:
62:
58:
56:
53:
52:
12:
11:
5:
5468:
5458:
5457:
5452:
5447:
5445:Interest rates
5430:
5429:
5427:
5426:
5413:
5410:
5409:
5407:
5406:
5401:
5399:Municipal debt
5396:
5391:
5386:
5384:Corporate debt
5381:
5375:
5373:
5369:
5368:
5366:
5365:
5360:
5355:
5350:
5345:
5340:
5335:
5330:
5325:
5320:
5315:
5310:
5305:
5300:
5294:
5292:
5288:
5287:
5285:
5284:
5279:
5274:
5269:
5264:
5259:
5253:
5251:
5245:
5244:
5242:
5241:
5236:
5231:
5226:
5221:
5216:
5211:
5206:
5201:
5196:
5191:
5186:
5184:Forward market
5181:
5176:
5171:
5166:
5160:
5158:
5156:
5155:
5150:
5144:
5141:
5140:
5138:
5137:
5132:
5127:
5122:
5117:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5065:Credit default
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5026:
5024:
5018:
5017:
5014:
5013:
5011:
5010:
5005:
5000:
4995:
4990:
4985:
4980:
4975:
4970:
4965:
4960:
4950:
4945:
4940:
4934:
4932:
4926:
4925:
4923:
4922:
4908:
4903:
4898:
4893:
4888:
4883:
4878:
4873:
4868:
4863:
4861:Iron butterfly
4858:
4853:
4848:
4843:
4838:
4833:
4831:Covered option
4828:
4823:
4818:
4813:
4808:
4803:
4797:
4795:
4789:
4788:
4786:
4785:
4780:
4775:
4770:
4769:Mountain range
4767:
4762:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4722:
4717:
4712:
4706:
4704:
4698:
4697:
4695:
4694:
4689:
4684:
4679:
4674:
4669:
4664:
4659:
4654:
4649:
4643:
4641:
4635:
4634:
4632:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4575:
4573:
4566:
4560:
4559:
4554:
4551:
4550:
4543:
4542:
4535:
4528:
4520:
4514:
4513:
4497:
4491:
4471:
4469:on 2010-08-16.
4448:
4423:
4417:
4401:
4395:
4378:
4372:
4366:. Risk Books.
4354:
4345:
4339:
4322:
4286:
4280:
4257:
4251:
4234:
4228:
4211:
4205:
4192:
4186:
4167:
4164:
4162:
4161:
4148:
4129:
4118:(4): 1259–82.
4101:Schwartz, E.S.
4088:
4067:
4055:
4046:
4039:
4016:
3993:
3959:
3942:
3917:Karasinski, P.
3907:
3904:on 2008-09-10.
3865:
3844:(3): 213–239.
3821:
3810:(4): 573–592.
3781:
3737:
3704:(2): 385–407.
3684:J.E. Ingersoll
3671:
3662:
3617:
3596:(2): 177–188.
3575:
3546:(1): 141–183.
3527:
3508:
3491:
3467:
3439:
3432:
3399:
3381:
3379:
3376:
3375:
3374:
3367:
3364:
3329:
3326:
3322:
3321:
3316:
3315:
3314:
3313:
3298:
3293:
3289:
3285:
3279:
3275:
3266:
3262:
3256:
3253:
3250:
3246:
3241:
3237:
3233:
3228:
3224:
3220:
3217:
3214:
3212:
3208:
3204:
3200:
3197:
3196:
3193:
3188:
3184:
3180:
3174:
3170:
3161:
3157:
3151:
3148:
3145:
3141:
3136:
3132:
3128:
3123:
3119:
3115:
3112:
3109:
3107:
3103:
3099:
3095:
3092:
3091:
3088:
3083:
3079:
3075:
3069:
3065:
3056:
3052:
3046:
3043:
3040:
3036:
3031:
3027:
3023:
3018:
3014:
3010:
3007:
3004:
3002:
2998:
2994:
2990:
2987:
2986:
2973:
2972:
2964:
2963:
2962:
2961:
2950:
2945:
2942:
2938:
2934:
2928:
2921:
2917:
2913:
2910:
2907:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2877:
2873:
2869:
2856:
2855:
2854:
2839:
2834:
2831:
2827:
2823:
2817:
2813:
2804:
2800:
2794:
2791:
2788:
2784:
2779:
2775:
2771:
2768:
2763:
2759:
2755:
2752:
2749:
2747:
2743:
2739:
2735:
2732:
2731:
2728:
2723:
2720:
2716:
2712:
2706:
2702:
2693:
2689:
2683:
2680:
2677:
2673:
2668:
2664:
2660:
2657:
2652:
2648:
2644:
2641:
2638:
2636:
2632:
2628:
2624:
2621:
2620:
2607:
2606:
2582:
2579:
2566:
2546:
2535:
2534:
2521:
2501:
2498:
2495:
2492:
2489:
2484:
2480:
2476:
2471:
2467:
2457:
2452:
2448:
2444:
2440:
2432:
2428:
2422:
2419:
2416:
2412:
2407:
2403:
2399:
2396:
2393:
2390:
2387:
2382:
2378:
2374:
2355:
2337:
2333:
2329:
2325:
2322:
2319:
2316:
2310:
2306:
2302:
2299:
2294:
2290:
2286:
2283:
2280:
2277:
2262:
2248:
2244:
2240:
2234:
2230:
2226:
2223:
2220:
2216:
2213:
2210:
2207:
2204:
2201:
2196:
2192:
2188:
2183:
2179:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2139:
2125:
2121:
2117:
2113:
2110:
2107:
2104:
2098:
2094:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2050:
2046:
2042:
2036:
2032:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
1997:
1993:
1988:
1984:
1980:
1974:
1969:
1965:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1925:
1903:
1899:
1877:
1857:
1837:
1817:
1814:
1811:
1789:
1785:
1781:
1775:
1771:
1767:
1764:
1761:
1757:
1752:
1748:
1742:
1738:
1734:
1729:
1725:
1721:
1718:
1713:
1709:
1705:
1690:
1676:
1672:
1649:
1645:
1641:
1637:
1634:
1631:
1628:
1622:
1618:
1614:
1609:
1605:
1601:
1582:
1581:
1567:
1563:
1535:
1531:
1527:
1524:
1521:
1518:
1494:
1490:
1484:
1462:
1458:
1454:
1450:
1442:
1438:
1432:
1429:
1426:
1422:
1417:
1413:
1409:
1406:
1403:
1400:
1397:
1392:
1388:
1384:
1362:
1358:
1354:
1350:
1342:
1338:
1332:
1329:
1326:
1322:
1317:
1313:
1309:
1306:
1303:
1300:
1297:
1292:
1288:
1284:
1269:
1251:
1247:
1243:
1237:
1233:
1229:
1226:
1223:
1220:
1214:
1210:
1206:
1203:
1198:
1194:
1190:
1175:
1154:
1134:
1114:
1092:
1088:
1084:
1080:
1077:
1074:
1071:
1067:
1062:
1058:
1054:
1051:
1048:
1045:
1042:
1037:
1033:
1029:
1007:
1003:
999:
995:
992:
989:
986:
982:
977:
973:
969:
966:
963:
960:
957:
952:
948:
944:
929:
907:
902:
898:
875:
870:
866:
862:
859:
856:
853:
850:
845:
841:
837:
832:
828:
787:mean reversion
778:
775:
758:
754:
749:
746:
743:
738:
734:
710:
706:
679:
675:
643:
639:
635:
604:
600:
587:
584:
571:
568:
565:
562:
559:
556:
553:
550:
539:
538:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
491:
488:
484:
479:
476:
473:
470:
467:
464:
461:
458:
417:
404:
403:
392:
388:
382:
376:
369:
363:
359:
356:
350:
346:
340:
335:
331:
327:
323:
318:
315:
311:
307:
303:
298:
294:
290:
287:
284:
281:
278:
275:
272:
249:
225:
205:
175:
171:
142:
115:
111:
92:state variable
84:
83:The short rate
81:
65:
61:
45:interest rates
23:returning the
9:
6:
4:
3:
2:
5467:
5456:
5453:
5451:
5448:
5446:
5443:
5442:
5440:
5425:
5420:
5415:
5414:
5411:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5379:Consumer debt
5377:
5376:
5374:
5372:Market issues
5370:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5348:Fund of funds
5346:
5344:
5341:
5339:
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5295:
5293:
5289:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5254:
5252:
5250:
5246:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5200:
5197:
5195:
5192:
5190:
5189:Forward price
5187:
5185:
5182:
5180:
5177:
5175:
5172:
5170:
5167:
5165:
5162:
5161:
5159:
5154:
5151:
5149:
5146:
5145:
5142:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5100:Interest rate
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5027:
5025:
5023:
5019:
5009:
5006:
5004:
5001:
4999:
4996:
4994:
4993:MC Simulation
4991:
4989:
4986:
4984:
4981:
4979:
4976:
4974:
4971:
4969:
4966:
4964:
4961:
4958:
4954:
4953:Black–Scholes
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4935:
4933:
4931:
4927:
4920:
4916:
4912:
4909:
4907:
4906:Risk reversal
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4836:Credit spread
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4798:
4796:
4794:
4790:
4784:
4781:
4779:
4776:
4774:
4771:
4768:
4766:
4763:
4761:
4760:Interest rate
4758:
4756:
4755:Forward start
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4707:
4705:
4703:
4699:
4693:
4690:
4688:
4685:
4683:
4682:Option styles
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4644:
4642:
4640:
4636:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4599:Open interest
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4579:Delta neutral
4577:
4576:
4574:
4570:
4567:
4565:
4561:
4557:
4552:
4548:
4541:
4536:
4534:
4529:
4527:
4522:
4521:
4518:
4510:
4503:
4498:
4494:
4488:
4484:
4480:
4476:
4472:
4465:
4461:
4454:
4449:
4445:
4441:
4437:
4433:
4429:
4424:
4420:
4414:
4410:
4406:
4405:Robert Jarrow
4402:
4398:
4392:
4388:
4387:Wiley Finance
4384:
4379:
4375:
4369:
4365:
4359:
4355:
4351:
4346:
4342:
4336:
4332:
4328:
4323:
4319:
4313:
4305:
4298:
4297:
4292:
4287:
4283:
4277:
4273:
4269:
4268:
4262:
4258:
4254:
4248:
4244:
4240:
4235:
4231:
4225:
4221:
4220:CFA Institute
4217:
4212:
4208:
4202:
4198:
4193:
4189:
4183:
4179:
4175:
4170:
4169:
4159:
4152:
4144:
4140:
4133:
4125:
4121:
4117:
4113:
4106:
4102:
4098:
4092:
4085:
4081:
4078:
4077:
4071:
4065:
4059:
4050:
4042:
4036:
4032:
4026:
4020:
4012:
4008:
4004:
3997:
3989:
3985:
3981:
3977:
3973:
3969:
3963:
3957:
3953:
3952:
3946:
3938:
3934:
3930:
3926:
3922:
3918:
3911:
3900:
3896:
3895:
3887:
3883:
3879:
3875:
3869:
3861:
3857:
3852:
3847:
3843:
3839:
3832:
3825:
3817:
3813:
3809:
3805:
3804:
3799:
3795:
3791:
3785:
3777:
3773:
3769:
3765:
3761:
3757:
3756:
3751:
3747:
3741:
3733:
3727:
3719:
3715:
3711:
3707:
3703:
3699:
3698:
3693:
3689:
3685:
3681:
3675:
3666:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3629:
3621:
3613:
3609:
3604:
3599:
3595:
3591:
3590:
3585:
3579:
3571:
3567:
3562:
3557:
3553:
3549:
3545:
3541:
3537:
3531:
3524:
3520:
3515:
3513:
3506:
3500:
3498:
3496:
3489:
3485:
3481:
3478:
3477:
3471:
3465:
3461:
3457:
3453:
3450:
3449:
3443:
3435:
3429:
3425:
3421:
3417:
3410:
3408:
3406:
3404:
3397:
3393:
3392:
3386:
3382:
3373:
3370:
3369:
3363:
3361:
3357:
3353:
3352:Fischer Black
3348:
3346:
3345:market models
3342:
3337:
3335:
3325:
3318:
3317:
3296:
3291:
3287:
3283:
3277:
3273:
3264:
3260:
3254:
3251:
3248:
3239:
3235:
3231:
3226:
3222:
3215:
3213:
3206:
3202:
3198:
3191:
3186:
3182:
3178:
3172:
3168:
3159:
3155:
3149:
3146:
3143:
3134:
3130:
3126:
3121:
3117:
3110:
3108:
3101:
3097:
3093:
3086:
3081:
3077:
3073:
3067:
3063:
3054:
3050:
3044:
3041:
3038:
3029:
3025:
3021:
3016:
3012:
3005:
3003:
2996:
2992:
2988:
2977:
2976:
2975:
2974:
2970:
2966:
2965:
2948:
2943:
2940:
2936:
2932:
2926:
2919:
2915:
2911:
2908:
2905:
2898:
2895:
2892:
2889:
2886:
2880:
2875:
2871:
2867:
2860:
2859:
2857:
2837:
2832:
2829:
2825:
2821:
2815:
2811:
2802:
2798:
2792:
2789:
2786:
2777:
2773:
2769:
2766:
2761:
2757:
2750:
2748:
2741:
2737:
2733:
2726:
2721:
2718:
2714:
2710:
2704:
2700:
2691:
2687:
2681:
2678:
2675:
2666:
2662:
2658:
2655:
2650:
2646:
2639:
2637:
2630:
2626:
2622:
2611:
2610:
2609:
2608:
2604:
2600:
2599:
2598:
2596:
2592:
2588:
2578:
2564:
2544:
2519:
2496:
2490:
2487:
2482:
2478:
2474:
2469:
2465:
2455:
2450:
2446:
2442:
2438:
2430:
2426:
2420:
2417:
2414:
2405:
2401:
2397:
2394:
2388:
2385:
2380:
2376:
2372:
2364:
2360:
2356:
2353:
2335:
2331:
2327:
2323:
2320:
2317:
2314:
2308:
2304:
2300:
2292:
2288:
2281:
2278:
2275:
2267:
2263:
2246:
2242:
2238:
2232:
2228:
2224:
2221:
2218:
2208:
2202:
2199:
2194:
2190:
2186:
2181:
2177:
2170:
2164:
2158:
2155:
2152:
2144:
2140:
2123:
2119:
2115:
2111:
2108:
2105:
2102:
2096:
2092:
2088:
2082:
2076:
2073:
2070:
2048:
2044:
2040:
2034:
2030:
2026:
2023:
2020:
2011:
2005:
2002:
1995:
1991:
1986:
1982:
1978:
1972:
1967:
1963:
1956:
1950:
1944:
1941:
1938:
1930:
1926:
1923:
1919:
1901:
1897:
1875:
1855:
1835:
1815:
1812:
1809:
1787:
1783:
1779:
1773:
1769:
1765:
1762:
1759:
1750:
1746:
1740:
1736:
1732:
1727:
1723:
1716:
1711:
1707:
1703:
1695:
1691:
1674:
1670:
1647:
1643:
1639:
1635:
1632:
1629:
1626:
1620:
1616:
1612:
1607:
1603:
1599:
1591:
1587:
1586:
1585:
1565:
1561:
1552:
1533:
1529:
1525:
1522:
1519:
1516:
1492:
1488:
1482:
1460:
1456:
1452:
1448:
1440:
1436:
1430:
1427:
1424:
1415:
1411:
1407:
1404:
1398:
1395:
1390:
1386:
1382:
1360:
1356:
1352:
1348:
1340:
1336:
1330:
1327:
1324:
1315:
1311:
1307:
1304:
1301:
1295:
1290:
1286:
1282:
1274:
1270:
1267:
1249:
1245:
1241:
1235:
1231:
1227:
1224:
1221:
1218:
1212:
1208:
1204:
1201:
1196:
1192:
1188:
1180:
1176:
1173:
1168:
1152:
1132:
1112:
1090:
1086:
1082:
1078:
1075:
1072:
1069:
1060:
1056:
1052:
1049:
1043:
1040:
1035:
1031:
1027:
1005:
1001:
997:
993:
990:
987:
984:
975:
971:
967:
964:
961:
955:
950:
946:
942:
934:
933:Vasicek model
930:
927:
923:
905:
900:
896:
873:
868:
864:
860:
857:
854:
851:
848:
843:
839:
835:
830:
826:
817:
814:
813:
812:
810:
806:
802:
797:
793:
788:
784:
774:
756:
752:
747:
744:
741:
736:
732:
708:
704:
695:
677:
673:
665:, a variable
664:
660:
641:
637:
633:
625:
621:
602:
598:
583:
566:
563:
560:
554:
548:
525:
516:
513:
510:
504:
498:
495:
489:
477:
474:
468:
465:
462:
456:
449:
448:
447:
445:
444:forward rates
441:
437:
433:
390:
386:
380:
367:
361:
357:
354:
348:
344:
338:
333:
329:
325:
321:
316:
313:
305:
301:
296:
288:
282:
279:
276:
270:
263:
262:
261:
247:
239:
223:
203:
196:
192:
173:
169:
160:
156:
140:
132:
113:
109:
100:
97:
96:instantaneous
93:
90:
80:
63:
59:
50:
46:
42:
38:
34:
26:
22:
18:
5199:Forward rate
5110:Total return
4998:Real options
4901:Ratio spread
4881:Naked option
4841:Debit spread
4672:Fixed income
4614:Strike price
4508:
4478:
4464:the original
4459:
4438:(1): 69–96.
4435:
4431:
4408:
4382:
4363:
4349:
4326:
4295:
4266:
4238:
4215:
4196:
4173:
4151:
4142:
4138:
4132:
4115:
4111:
4091:
4075:
4070:
4058:
4049:
4030:
4019:
4002:
3996:
3982:(3): 35–46.
3979:
3975:
3962:
3950:
3945:
3931:(4): 52–59.
3928:
3924:
3910:
3899:the original
3892:
3868:
3841:
3837:
3824:
3807:
3801:
3784:
3759:
3753:
3740:
3726:cite journal
3701:
3697:Econometrica
3695:
3674:
3665:
3635:(1): 11–24.
3632:
3626:
3620:
3593:
3587:
3578:
3561:1721.1/49331
3543:
3539:
3530:
3522:
3475:
3470:
3447:
3442:
3415:
3390:
3385:
3349:
3338:
3331:
3323:
2584:
2536:
1590:Ho–Lee model
1583:
780:
659:differential
624:risk-neutral
589:
540:
405:
86:
48:
32:
30:
5130:Zero Coupon
5060:Correlation
5008:Vanna–Volga
4866:Iron condor
4652:Bond option
4360:; entry in
4263:; entry in
3915:Black, F.;
3356:shadow rate
1931:(1990) has
157:. However,
155:yield curve
5439:Categories
5404:Tax policy
5120:Volatility
5030:Amortising
4871:Jelly roll
4806:Box spread
4801:Backspread
4793:Strategies
4629:Volatility
4624:the Greeks
4589:Expiration
3878:Derman, E.
3794:Alan White
3378:References
2969:Chen model
783:stochastic
440:zero curve
89:stochastic
49:short rate
5095:Inflation
5045:Commodity
5003:Trinomial
4938:Bachelier
4930:Valuation
4811:Butterfly
4745:Commodore
4594:Moneyness
4312:cite book
3846:CiteSeerX
3790:John Hull
3746:T.S.Y. Ho
3688:S.A. Ross
3680:Cox, J.C.
3657:154495945
3598:CiteSeerX
3274:η
3261:σ
3236:σ
3232:−
3223:β
3203:σ
3169:σ
3156:α
3131:α
3127:−
3118:ζ
3098:α
3064:σ
3026:α
3022:−
3013:θ
2916:σ
2896:θ
2887:μ
2767:−
2656:−
2587:Longstaff
2545:ϕ
2520:ϕ
2491:ϕ
2439:σ
2398:−
2324:σ
2305:θ
2282:
2229:σ
2203:
2191:ϕ
2187:−
2178:θ
2159:
2112:σ
2093:θ
2077:
2031:σ
2006:
1992:σ
1979:σ
1964:θ
1945:
1922:trinomial
1898:θ
1876:σ
1856:α
1836:σ
1816:α
1810:θ
1770:σ
1737:α
1733:−
1724:θ
1671:θ
1636:σ
1617:θ
1530:σ
1483:σ
1449:σ
1408:−
1349:σ
1308:α
1305:−
1302:θ
1228:σ
1205:θ
1153:σ
1079:σ
1053:−
994:σ
968:α
965:−
962:θ
906:∗
874:∗
861:σ
796:parameter
748:
663:lognormal
552:↦
499:
487:∂
483:∂
478:−
330:∫
326:−
317:
302:
99:spot rate
5234:Slippage
5164:Contango
5148:Forwards
5115:Variance
5075:Dividend
5070:Currency
4983:Margrabe
4978:Lattices
4957:equation
4943:Binomial
4891:Strangle
4886:Straddle
4783:Swaption
4765:Lookback
4750:Compound
4692:Warrants
4667:European
4647:American
4639:Vanillas
4604:Pin risk
4584:Exercise
4477:(2002).
4407:(2002).
4331:Springer
4274:. 2004.
4103:(1992).
4080:Archived
4011:16187107
3919:(1991).
3884:(1990).
3796:(1990).
3750:S.B. Lee
3690:(1985).
3480:Archived
3458:, Prof.
3452:Archived
3366:See also
2591:Schwartz
2363:Mercurio
1987:′
888:: where
816:Merton's
801:binomial
622:under a
193:under a
5153:Futures
4773:Rainbow
4740:Cliquet
4735:Chooser
4715:Barrier
4702:Exotics
4564:Options
4145:: 1–88.
3882:Toy, W.
3776:2328161
3718:1911242
3649:2979016
3570:3003143
430:is the
39:, is a
5214:Margin
5080:Equity
4973:Heston
4876:Ladder
4826:Condor
4821:Collar
4778:Spread
4725:Binary
4720:Basket
4489:
4415:
4393:
4370:
4337:
4302:. The
4278:
4249:
4226:
4203:
4184:
4037:
4025:pg 218
4009:
3876:, F.;
3848:
3774:
3716:
3655:
3647:
3600:
3568:
3536:Merton
3430:
2512:where
2462:
2459:
1475:. The
406:where
5085:Forex
5040:Basis
5035:Asset
5022:Swaps
4948:Black
4851:Fence
4710:Asian
4572:Terms
4505:(PDF)
4467:(PDF)
4456:(PDF)
4300:(PDF)
4108:(PDF)
3902:(PDF)
3889:(PDF)
3874:Black
3834:(PDF)
3772:JSTOR
3714:JSTOR
3653:S2CID
3645:JSTOR
3566:JSTOR
2359:Brigo
236:of a
189:as a
4919:Bull
4915:Bear
4657:Call
4487:ISBN
4413:ISBN
4391:ISBN
4368:ISBN
4335:ISBN
4318:link
4276:ISBN
4247:ISBN
4224:ISBN
4201:ISBN
4182:ISBN
4035:ISBN
4023:See
4007:OCLC
3748:and
3732:link
3686:and
3428:ISBN
3343:and
2967:The
2601:The
2589:and
2361:and
2264:The
2141:The
1927:The
1868:and
1828:and
1692:The
1588:The
1526:>
1271:The
1177:The
931:The
807:and
696:and
657:its
21:Tree
4687:Put
4440:doi
4222:).
4120:doi
4027:in
3984:doi
3933:doi
3856:doi
3812:doi
3764:doi
3706:doi
3637:doi
3608:doi
3556:hdl
3548:doi
3420:doi
3396:NYU
3354:'s
745:exp
314:exp
25:OAS
5441::
4917:,
4677:FX
4507:.
4485:.
4481:.
4458:.
4434:.
4430:.
4389:.
4385:.
4333:.
4329:.
4314:}}
4310:{{
4270:.
4245:.
4241:.
4180:.
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4116:47
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4110:.
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3891:.
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3806:.
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3770:.
3760:41
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3712:.
3702:53
3700:.
3694:.
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3651:.
3643:.
3633:15
3631:.
3606:.
3592:.
3564:.
3554:.
3542:.
3521:,
3511:^
3494:^
3462:,
3426:.
3402:^
3362:.
2279:ln
2200:ln
2156:ln
2074:ln
2003:ln
1942:ln
1916:.
773:.
496:ln
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4959:)
4955:(
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4539:e
4532:t
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4190:.
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3990:.
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3862:.
3858::
3842:7
3818:.
3814::
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3766::
3734:)
3720:.
3708::
3659:.
3639::
3614:.
3610::
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3572:.
3558::
3550::
3544:4
3436:.
3422::
3297:.
3292:t
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3284:d
3278:t
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3216:=
3207:t
3199:d
3192:,
3187:t
3183:W
3179:d
3173:t
3160:t
3150:+
3147:t
3144:d
3140:)
3135:t
3122:t
3114:(
3111:=
3102:t
3094:d
3087:,
3082:t
3078:W
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3055:t
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3017:t
3009:(
3006:=
2997:t
2993:r
2989:d
2949:.
2944:t
2941:3
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2881:=
2876:t
2872:r
2868:d
2838:,
2833:t
2830:2
2826:W
2822:d
2816:t
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2803:t
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2778:t
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2754:(
2751:=
2742:t
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2640:=
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2386:=
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1957:=
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