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| The '''Heath–Jarrow–Morton (HJM) framework''' is a general framework to model the evolution of [[interest rate]] curve – instantaneous forward rate curve in particular (as opposed to simple [[forward rate]]s). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the '''Gaussian Heath–Jarrow–Morton (HJM) Model''' of forward rates <ref>M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling. 2nd ed. New York : Springer-Verlag, 2004. Print.</ref>{{rp|394}}. For direct modeling of simple forward rates the [[LIBOR Market Model|Brace–Gatarek–Musiela Model]] represents an example.
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| The HJM framework originates from the work of David Heath, [[Robert A. Jarrow]] and Andrew Morton in the late 1980s, especially ''Bond pricing and the term structure of interest rates: a new methodology'' (1987) – working paper, [[Cornell University]], and ''Bond pricing and the term structure of interest rates: a new methodology'' (1989) – working paper (revised ed.), Cornell University. It has its critics, however, with [[Paul Wilmott]] describing it as "...actually just a big rug for [mistakes] to be swept under".<ref>[http://www.thedailybeast.com/newsweek/2009/05/28/revenge-of-the-nerd.html ''Newsweek'' 2009]</ref>
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| ==Framework==
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| The key to these techniques is the recognition that the drifts of the [[rational pricing|no-arbitrage]] evolution of certain variables can be expressed as functions of their volatilities and the correlations among themselves. In other words, no drift estimation is needed.
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| Models developed according to the HJM framework are different from the so-called [[short-rate model]]s in the sense that HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve (the short rate).
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| However, models developed according to the general HJM framework are often non-[[Markovian]] and can even have infinite dimensions. A number of researchers have made great contributions to tackle this problem. They show that if the volatility structure of the forward rates satisfy certain conditions, then an HJM model can be expressed entirely by a finite state Markovian system, making it computationally feasible. Examples include a one-factor, two state model (O. Cheyette, "Term Structure Dynamics and Mortgage Valuation", ''Journal of Fixed Income,'' 1, 1992; P. Ritchken and L. Sankarasubramanian in "Volatility Structures of Forward Rates and the Dynamics of Term Structure", ''Mathematical Finance'', 5, No. 1, Jan 1995), and later multi-factor versions.
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| ==Mathematical formulation==
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| The class of models developed by Heath, Jarrow and Morton (1992) is based on modeling the forward rates, yet it does not capture all of the complexities of an evolving term structure.
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| The instantaneous forward rate <math>f \left(t,T\right),t\leq T</math> is the continuous compounding rate available at time <math>\, T</math> as seen from time <math>\, t</math>. It is defined by:
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| {{Equation|1=f\left(t,T\right)=-\frac{1}{P\left(t,T\right)}\frac{\partial}{\partial T}P\left(t,T\right)=-\frac{\partial \textrm{log} P\left(t,T\right)}{\partial T},|2=1}}
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| The basic relation between the rates and the bond prices is given by:
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| {{Equation|1=P\left(t,T\right)=e^{-\int_t^T f\left(t,s\right)\,ds}.|2=2}}
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| Consequently, the bank account <math>\beta\left(t\right)</math> grows according to:
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| {{Equation|1=\beta\left(t\right)=e^{\int_0^t f\left(s,s\right)\,ds}|2=3}}
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| since the spot rate at time <math> t </math> is <math> r(t)=f(t,t)</math>.
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| The assumption of the HJM model is that the forward rates <math>f\left(t,T\right)</math> satisfy for any <math>\, T</math>:
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| {{Equation|1=df\left(t,T\right)=\mu\left(t,T\right)dt+\xi\left(t,T\right)dW\left(t\right)|2=4}}
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| where the processes <math>\mu\left(t,T\right),\xi\left(t,T\right)</math> are continuous and adapted.
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| For this assumption to be compatible with the assumption of the existence of martingale measures we need the following relation to hold:
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| {{Equation|1=
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| \frac{dP\left(t,T\right)}{P\left(t,T\right)}=\left[r\left(t\right)-\alpha\left(t,T\right)\theta\left(t\right)\right]dt +\alpha\left(t,T\right)dW\left( t\right).|2=5}}
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| We find the return on the bond in the HJM model and compare it ([[#equation-5|5]]) to obtain models that do not allow for arbitrage.
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| Let {{Equation|1=X\left(t\right)=\textrm{log} P\left(t,T\right).|2=6}}
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| Then {{Equation|1=X\left(t\right)=-\int_t^T f\left(t,s\right)\,ds.|2=7}}
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| Using [[Leibniz integral rule|Leibniz's rule]] for differentiating under the integral sign we have that:
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| {{Equation|1=dX=-d\left(\int_t^T f\left(t,s\right) d s\right)=-A\left(t,T\right)dt-\tau\left(t,T\right)dW\left(t\right),|2=8}}
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| where
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| <math>A\left(t,T\right)=-r\left(t\right)+\int_t^T\mu\left(t,s\right)\,ds~~~\textrm{and}~~\tau\left(t,T\right)=\int_t^T\xi\left(t,s\right)\,ds.</math> | |
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| By [[Itō's lemma]],
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| {{Equation|1=\frac{dP\left(t,T\right)}{P\left(t,T\right)}=dX+\frac{1}{2}(dX)^2.|2=9}}
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| It follows from ([[#equation-5|5]]) and ([[#equation-9|9]]), we must have that
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| {{Equation|1=\alpha(t,T)=-\int_t^T\xi\left(t,s\right)\,ds,|2=10}}
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| {{Equation|1=\alpha(t,T)\cdot\theta(t)=\int_t^T\mu(t,s)\,ds-\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2.|2=11}}
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| Rearranging the terms we get that
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| {{Equation|1=\int_t^T\mu(t,s) \,ds=\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2-\theta(t)\, \int_t^T\xi\left(t,s\right)\,ds.|2=12}}
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| Differentiating both sides with respect to <math>\, T</math>, we have that
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| {{Equation|1=
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| \mu(t,T)=\xi\left(t,T\right)\left(\int_t^T\xi\left(t,s\right)\,ds-\theta(t)\right).|2=13}}
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| Equation ([[#equation-13|13]]) is known as the no-arbitrage condition in the HJM model. Under the martingale probability measure <math>\ , \theta=0</math> and the equation for the forward rates becomes:
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| {{Equation|1=df(t,T)=\xi\left(t,T\right)\left(\int_t^T\xi\left(t,s\right)\,ds\right)dt+\xi\left(t,T\right)d\tilde{W}.|2=14}}
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| This equation is used in pricing of bonds and its derivatives.
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| ==See also==
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| *[[Ho–Lee model]]
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| *[[Hull–White model]]
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| *[[Black–Derman–Toy model]]
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| *[[Chen model]]
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| *[[LIBOR Market Model|Brace–Gatarek–Musiela Model]]
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| ==External links and references==
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| '''Notes'''
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| {{reflist}}
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| '''Primary references'''
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| * Heath, D., Jarrow, R. and Morton, A. (1990). [http://www.defaultrisk.com/pa_price_14.htm Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation]. ''[[Journal of Financial and Quantitative Analysis]]'', 25:419-440.
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| * Heath, D., Jarrow, R. and Morton, A. (1991). [http://www.kamakuraco.com/LinkClick.aspx?fileticket=3pl0IkdYSrY%3D&tabid=208&mid=735 Contingent Claims Valuation with a Random Evolution of Interest Rates]. ''[[Review of Futures Markets]]'', 9:54-76.
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| * Heath, D., Jarrow, R. and Morton, A. (1992). [http://www.defaultrisk.com/pa_price_45.htm Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation]. ''[[Econometrica]]'', 60(1):77-105. {{doi|10.2307/2951677}}
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| * Robert Jarrow (2002). ''Modelling Fixed Income Securities and Interest Rate Options'' (2nd ed.). Stanford Economics and Finance. ISBN 0-8047-4438-6
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| '''Articles'''
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| *[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.9444 Non-Bushy Trees For Gaussian HJM And Lognormal Forward Models], Prof Alan Brace, [[University of Technology Sydney]]
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| * [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN02-01.pdf The Heath-Jarrow-Morton Term Structure Model], Prof. Don Chance [[E. J. Ourso College of Business]], [[Louisiana State University]]
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| *[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=359040 Recombining Trees for One-Dimensional Forward Rate Models], Dariusz Gatarek, [[Wyższa Szkoła Biznesu – National-Louis University]], and Jaroslaw Kolakowski
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| *[http://www.iijournals.com/doi/abs/10.3905/jfi.1999.319247#sthash.5R3a6VDr.dpbs Implementing No-Arbitrage Term Structure of Interest Rate Models in Discrete Time When Interest Rates Are Normally Distributed], Dwight M Grant and Gautam Vora. ''[[The Journal of Fixed Income]]'' March 1999, Vol. 8, No. 4: pp. 85-98
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| * [http://repository.upenn.edu/dissertations/AAI3015358/ Heath–Jarrow–Morton model and its application], Vladimir I Pozdynyakov, [[University of Pennsylvania]]
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| * [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=123170 An Empirical Study of the Convergence Properties of the Non-recombining HJM Forward Rate Tree in Pricing Interest Rate Derivatives], A.R. Radhakrishnan [[New York University]]
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| * Modeling Interest Rates with Heath, Jarrow and Morton. Dr Donald van Deventer, [[Kamakura Corporation]]:
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| **[http://www.kamakuraco.com/Blog/tabid/231/EntryId/384/Heath-Jarrow-and-Morton-Example-One-Modeling-Interest-Rates-with-One-Factor-and-Maturity-Dependent-Volatility.aspx With One Factor and Maturity-Dependent Volatility]
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| **[http://www.kamakuraco.com/Blog/tabid/231/EntryId/385/Heath-Jarrow-and-Morton-Example-Two-Modeling-Interest-Rates-with-One-Factor-and-Rate-and-Maturity-Dependent-Volatility.aspx With One Factor and Rate and Maturity-Dependent Volatility]
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| **[http://www.kamakuraco.com/Blog/tabid/231/EntryId/388/Heath-Jarrow-and-Morton-Example-Three-Modeling-Interest-Rates-with-Two-Factors-and-Rate-and-Maturity-Dependent-Volatility.aspx With Two Factors and Rate and Maturity-Dependent Volatility]
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| **[http://www.kamakuraco.com/Blog/tabid/231/EntryId/408/Heath-Jarrow-and-Morton-Example-Four-Modeling-Interest-Rates-with-Three-Factors-and-Rate-and-Maturity-Dependent-Volatility-Updated-June-26-2012.aspx With Three Factors and Rate and Maturity-Dependent Volatility]
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| {{Bond market}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Heath-Jarrow-Morton framework}}
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| [[Category:Mathematical finance]]
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| [[Category:Fixed income analysis]]
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