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| A '''short-rate model''', in the context of [[interest rate derivatives]], is a [[mathematical model]] that describes the future evolution of [[interest rate]]s by describing the future evolution of the '''short rate''', usually written <math>r_t \,</math>.
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| ==The short rate==
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| Under a short rate model, the [[stochastic]] [[state variable]] is taken to be the [[instantaneous]] [[spot rate]].<ref>[http://www.math.nyu.edu/~alberts/spring07/Lecture5.pdf ''Short rate models''], Prof. Andrew Lesniewski, [[NYU]]</ref> The short rate, <math>r_t \,</math>, then, is the ([[Compound_interest#Continuous_compounding|continuously compounded]], annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time <math>t</math>. Specifying the current short rate does not specify the entire [[yield curve]]. However [[arbitrage|no-arbitrage arguments]] show that, under some fairly relaxed technical conditions, if we model the evolution of <math>r_t \,</math> as a [[stochastic process]] under a [[risk-neutral measure]] <math>Q</math> then the price at time <math>t</math> of a [[zero-coupon bond]] maturing at time <math>T</math> is given by
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| :<math> P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] </math>
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| where <math>\mathcal{F}</math> is the [[natural filtration]] for the process. Thus specifying a model for the short rate specifies future bond prices. This means that instantaneous [[forward rate]]s are also specified by the usual formula
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| :<math> f(t,T) = - \frac{\partial}{\partial T} \ln(P(t,T)). </math>
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| ==Particular short-rate models==
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| Throughout this section <math>W_t\,</math> represents a standard [[Wiener process|Brownian motion]] under a [[Risk-neutral measure|risk-neutral]] probability measure and <math>dW_t\,</math> its [[differential (mathematics)|differential]]{{disambiguation needed|date=July 2013}}. Where the model is [[lognormal]], a variable <math>X_t \,</math>, is assumed to follow an [[Ornstein–Uhlenbeck process]] and <math>r_t \,</math> is assumed to follow <math>r_t = \exp{X_t}\,</math>.
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| ===One-factor short-rate models===
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| Following are the one-factor models, where a single [[stochastic]] factor – the short rate – determines the future evolution of all interest rates. Other than Rendleman–Bartter and Ho–Lee, which do not capture the [[mean reversion]] of interest rates, these models can be thought of as specific cases of Ornstein–Uhlenbeck processes. The Vasicek, Rendleman–Bartter and CIR models have only a finite number of [[free parameter]]s and so it is not possible to specify these [[parameter]] values in such a way that the model coincides with observed market prices ("calibration"). This problem is overcome by allowing the parameters to vary deterministically with time.<ref>[http://wwwhome.math.utwente.nl/~jamshidianf/pdf/Overview%20of%20interest%20%20rate%20modeling.pdf ''An Overview of Interest-Rate Option Models''], Prof. [[Farshid Jamshidian]], [[University of Twente]]</ref><ref>[http://www.columbia.edu/~mh2078/cts_shortrate_models.pdf ''Continuous-Time Short Rate Models''], Prof Martin Haugh, [[Columbia University]]</ref> In this way, 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. Here, the implementation is usually via a [[binomial options pricing model|binomial tree]] ([[Lattice model (finance)|lattice]]).<ref name="BenningaWiener">[http://simonbenninga.com/wiener/MiER73.pdf Binomial Term Structure Models], ''Mathematica in Education and Research'', Vol. 7 No. 3 1998.
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| Simon Benninga and Zvi Wiener.</ref>
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| #[[Robert C. Merton|Merton's]] model (1973) explains the short rate as <math>r_t = r_{0}+at+\sigma W^{*}_{t}</math>: where <math>W^{*}_{t}</math> is a one-dimensional Brownian motion under the spot [[martingale measure]].<ref>{{cite journal | title=Theory of Rational Option Pricing | last=[[Robert C. Merton|Merton]] |first=Robert C.| journal=Bell Journal of Economics and Management Science | year=1973| volume=4| issue=1 | pages=141–183 | doi=10.2307/3003143}}</ref>
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| #The [[Vasicek model]] (1977) models the short rate as <math>dr_t = (\theta-\alpha r_t)\,dt + \sigma \, dW_t</math>; it is often written <math>dr_t = a(b-r_t)\, dt + \sigma \, dW_t</math>.<ref>{{cite journal | author=[[Oldrich Vasicek|Vasicek, Oldrich]] | title=An Equilibrium Characterisation of the Term Structure | journal=[[Journal of Financial Economics]]| year=1977 | volume=5 | pages=177–188 | doi=10.1016/0304-405X(77)90016-2 | issue=2 }}</ref>
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| #The [[Rendleman–Bartter model]] (1980) explains the short rate as <math>dr_t = \theta r_t\, dt + \sigma r_t\, dW_t</math>.<ref>{{cite journal | last=Rendleman |first=R. |first2=B. |last2=Bartter | title=The Pricing of Options on Debt Securities | journal=[[Journal of Financial and Quantitative Analysis]]| year=1980 | volume=15 | pages=11–24 | doi=10.2307/2979016}}</ref>
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| #The [[Cox–Ingersoll–Ross model]] (1985) supposes <math>dr_t = (\theta-\alpha r_t)\,dt + \sqrt{r_t}\,\sigma\, dW_t</math>, it is often written <math>dr_t = a(b-r_t)\, dt + \sqrt{r_t}\,\sigma\, dW_t</math>. The <math>\sigma \sqrt{r_t}</math> factor precludes (generally) the possibility of negative interest rates.<ref>{{Cite journal | author=[[John Carrington Cox|Cox, J.C.]], [[Jonathan E. Ingersoll|J.E. Ingersoll]] and [[Stephen Ross (economist)|S.A. Ross]] | title=A Theory of the Term Structure of Interest Rates | journal=[[Econometrica]] | year=1985 | volume=53 | pages=385–407 | doi=10.2307/1911242}}</ref>
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| #The [[Ho–Lee model]] (1986) models the short rate as <math>dr_t = \theta_t\, dt + \sigma\, dW_t</math>.<ref>{{cite journal | author= [[Thomas Ho (finance)|T.S.Y. Ho]] and [[Sang Bin Lee|S.B. Lee]] | title=Term structure movements and pricing interest rate contingent claims| journal= [[Journal of Finance]] | year=1986 | volume=41 | doi=10.2307/2328161}}</ref>
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| #The [[Hull–White model]] (1990)—also called the extended Vasicek model—posits <math>dr_t = (\theta_t-\alpha r_t)\,dt + \sigma_t \, dW_t</math>. In many presentations one or more of the parameters <math>\theta, \alpha</math> and <math>\sigma</math> are not time-dependent. The model may also be applied as lognormal. [[Lattice model (finance)|Lattice-based implementation]] is usually [[trinomial tree|trinomial]].<ref>{{cite journal| authors= [[John C. Hull|John Hull]] and [[Alan White (economist)|Alan White]]|title= Pricing interest-rate derivative securities | year=1990 | volume=3| issue = 4 |pages=573–592|journal= [[Review of Financial Studies]] |url= http://www.defaultrisk.com/pa_related_24.htm}}</ref>
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| # The [[Black–Derman–Toy model]] (1990) has <math> d\ln(r) = [\theta_t + \frac{\sigma '_t}{\sigma_t}\ln(r)]dt + \sigma_t\, dW_t </math> for time-dependent short rate volatility and <math>d\ln(r) = \theta_t\, dt + \sigma \, dW_t </math> otherwise; the model is lognormal.<ref>{{cite journal|first=F.|last=[[Fischer Black|Black]]|coauthors=[[Emanuel Derman|Derman, E.]] and [[William Toy|Toy, W.]]|title=A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options|year=1990|pages=24–32|journal=[[Financial Analysts Journal]]|url=http://savage.wharton.upenn.edu/FNCE-934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf}}</ref>
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| #The [[Black–Karasinski model]] (1991), which is lognormal, has <math> d\ln(r) = [\theta_t-\phi_t \ln(r)] \, dt + \sigma_t\, dW_t </math>.<ref>{{cite journal|first=F.|last=Black|coauthors=[[Piotr Karasinski|Karasinski, P.]]|title=Bond and Option pricing when Short rates are Lognormal|year=1991|pages=52–59|journal=Financial Analysts Journal| url= http://www.defaultrisk.com/pa_related_29.htm}}</ref> The model may be seen as the lognormal application of Hull–White;<ref>[http://php.portals.mbs.ac.uk/Portals/49/docs/spoon/IRD/Ch5_ShortRateNOTE.pdf ''Short Rate Models''], Professor Ser-Huang Poon, [[Manchester Business School]]</ref> its lattice-based implementation is similarly trinomial (binomial requiring varying time-steps).<ref name="BenningaWiener"/>
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| #The [[Kalotay–Williams–Fabozzi model]] (1993) has the short rate as <math> d \ln(r_t) = \theta_t\, dt + \sigma\, dW_t</math>, a lognormal analogue to the Ho–Lee model, and a special case of the Black–Derman–Toy model.<ref>{{cite journal |last1=Kalotay |first1=Andrew J. |last2=Williams |first2=George O. |last3=[[Frank J. Fabozzi|Fabozzi]] |first3=Frank J. |authorlink=Andrew Kalotay |year=1993 |title=A Model for Valuing Bonds and Embedded Options |journal=Financial Analysts Journal |publisher=[[CFA Institute|CFA Institute Publications]] |volume=49 |issue=3 |pages=35–46 |url=http://www.cfapubs.org/doi/abs/10.2469/faj.v49.n3.35 |doi=10.2469/faj.v49.n3.35}}</ref>
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| ===Multi-factor short-rate models===
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| Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are the [[Francis Longstaff|Longstaff]] and [[Eduardo Schwartz|Schwartz]] 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 [[Monte_Carlo_methods_in_finance#Overview|interest rate simulations]]," 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".<ref>[http://www.kamakuraco.com/Blog/tabid/231/EntryId/347/Pitfalls-in-Asset-and-Liability-Management-One-Factor-Term-Structure-Models.aspx ''Pitfalls in Asset and Liability Management: One Factor Term Structure Models''], Dr. Donald R. van Deventer, Kamakura Corporation</ref>
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| # The [[Longstaff–Schwartz model]] (1992) supposes the short rate dynamics are given by: <math> dX_t = (a_t-b X_t)\,dt + \sqrt{X_t}\,c_t\, dW_{1t}</math>, <math>d Y_t = (d_t-e Y_t)\,dt + \sqrt{Y_t}\,f_t\, dW_{2t}</math>, where the short rate is defined as <math> dr_t = (\mu X + \theta Y)dt + \sigma_t \sqrt{Y} dW_{3t} </math>.<ref>{{cite journal | authors= [[Francis Longstaff|Longstaff, F.A.]] and [[Eduardo Schwartz|Schwartz, E.S.]]|year=1992 |title= Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model |journal= Journal of Finance |volume=47 |issue=4 |pages=1259–82|url=http://efinance.org.cn/cn/FEshuo/19920901Interest%20Rate%20Volatility%20and%20the%20Term%20Structure%20A%20Two-Factor%20General%20Equilibrium%20Model,%20pp.%201259-1282.pdf}}</ref>
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| #The [[Chen model]] (1996) which has a stochastic mean and volatility of the short rate, is given by : <math> dr_t = (\theta_t-\alpha_t)\,dt + \sqrt{r_t}\,\sigma_t\, dW_t</math>, <math>
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| d \alpha_t = (\zeta_t-\alpha_t)\,dt + \sqrt{\alpha_t}\,\sigma_t\, dW_t</math>, <math>
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| d \sigma_t = (\beta_t-\sigma_t)\,dt + \sqrt{\sigma_t}\,\eta_t\, dW_t</math>.<ref>{{cite journal| author = Lin Chen |year= 1996 | title= 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 | journal=Financial Markets, Institutions, and Instruments |volume=5|pages=1–88}}</ref>
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| ==Other interest rate models==
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| The other major framework for interest rate modelling is the [[Heath–Jarrow–Morton framework]] (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.
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| The HJM framework with multiple sources of randomness, including as it does the [[Brace–Gatarek–Musiela model]] and [[market model]]s, is often preferred for models of higher dimension.
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| ==References==
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| {{reflist|30em}}
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| ==Further reading==
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| * {{cite book | author = Martin Baxter and Andrew Rennie | year = 1996 | title = Financial Calculus | publisher = [[Cambridge University Press]] | isbn = 978-0-521-55289-9 }}
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| *{{cite book | title = Interest Rate Models – Theory and Practice with Smile, Inflation and Credit| author = Damiano Brigo, Fabio Mercurio | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
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| * {{cite book | author = Gerald Buetow and James Sochacki| year = 2001 | title = Term-Structure Models Using Binomial Trees | publisher = The Research Foundation of AIMR ([[CFA Institute]]) | isbn = 978-0-943205-53-3 }}
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| *{{cite book | title = Interest Rate Models – An Introduction | author = Andrew J.G. Cairns | publisher = [[Princeton University Press]] | year = 2004 | isbn = 0-691-11894-9 }}
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| *Andrew J.G. Cairns (2004). [http://www.ma.hw.ac.uk/~andrewc/papers/ajgc33.pdf Interest-Rate Models]; entry in {{cite book| title = [http://eu.wiley.com/legacy/wileychi/eoas/contents.html Encyclopaedia of Actuarial Science]|publisher = [[John Wiley and Sons]] | year = 2004 | isbn = 0-470-84676-3 }}
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| *{{cite book | author= K. C. Chan, G. Andrew Karolyi, [[Francis Longstaff]], and Anthony Sanders| year = 1992 |title= [http://personal.anderson.ucla.edu/francis.longstaff/empiricalcomparison.pdf An Empirical Comparison of Alternative Models of the Short-Term Interest Rate] |publisher= The [[Journal of Finance]], Vol. XLVII, No. 3 July 1992. }}
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| * {{cite book | author = Lin Chen | year = 1996 | title = Interest Rate Dynamics, Derivatives Pricing, and Risk Management | publisher = [[Springer Science+Business Media|Springer]] | isbn = 3-540-60814-1 }}
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| * {{cite book | author = Rajna Gibson, [[François-Serge Lhabitant]] and Denis Talay | year = 1999|title= Modeling the Term Structure of Interest Rates: An overview |publisher= The Journal of Risk, 1(3): 37–62, 1999}}
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| * Lane Hughston (2003). [http://www.mth.kcl.ac.uk/finmath/articles/LPH_risk.pdf The Past, Present and Future of Term Structure Modelling]; entry in {{cite book | author = Peter Field| year = 2003 | title = Modern Risk Management| publisher =Risk Books | isbn = 9781906348304}}
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| * {{cite book | author = Jessica James and Nick Webber | year = 2000 | title = Interest Rate Modelling| publisher = [[John Wiley & Sons|Wiley Finance]] | isbn = 0-471-97523-0 }}
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| *{{cite book | title = Modelling Fixed Income Securities and Interest Rate Options (2nd ed.)| author = [[Robert Jarrow]] | publisher = Stanford Economics and Finance | year = 2002 | isbn = 0-8047-4438-6}}
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| * {{cite book | author = Robert Jarrow| year = 2009|title= [http://econpapers.repec.org/article/anrrefeco/v_3a1_3ay_3a2009_3ap_3a69-96.htm The Term Structure of Interest Rates] |publisher= Annual Review of Financial Economics, 2009, vol. 1, issue 1, pages 69-96 }}
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| * {{cite journal | author = F.C. Park | year = 2004|title= Implementing Interest Rate Models: a Practical Guide | journal = CMPR Research Publication| url= http://www.cmpr.co.kr/asset/research_material/implementing_interest_rate_models.pdf}}
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| * {{cite book | author = [[Riccardo Rebonato]] | year = 2002 | title = Modern Pricing of Interest-Rate Derivatives | publisher = [[Princeton University Press]] | isbn = 0-691-08973-6}}
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| * {{cite journal | author = Riccardo Rebonato | year = 2003|title= Term-Structure Models: a Review| journal = Royal Bank of Scotland Quantitative Research Centre Working Paper| url=http://faculty.maxwell.syr.edu/cdkao/teaching/taiwan/2003/TSMRS.pdf}}
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| {{derivatives market}}
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| [[Category:Mathematical finance]]
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| [[Category:Interest rates]]
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| [[Category:Short-rate models|*]]
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