View source for Finance:Heath–Jarrow–Morton framework

The '''Heath–Jarrow–Morton''' ('''HJM''') '''framework''' is a general framework to model the evolution of [[Finance:Interest rate|interest rate]] curves&nbsp;– instantaneous [[Finance:Forward curve|forward rate curves]] in particular (as opposed to simple [[Finance:Forward rate|forward rate]]s). When the [[Finance:Volatility|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 [[Finance:LIBOR market model|Brace–Gatarek–Musiela model]] represents an example.
The HJM framework originates from the work of [[Biography:David Heath (probabilist)|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)&nbsp;– working paper, [[Organization:Cornell University|Cornell University]], and ''Bond pricing and the term structure of interest rates: a new methodology'' (1989)&nbsp;– 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.newsweek.com/one-math-geeks-plan-reform-wall-street-80135 One math geek's plan to reform Wall Street, Newsweek, May 2009]</ref><ref>[http://www.thedailybeast.com/newsweek/2009/05/28/revenge-of-the-nerd.html ''Newsweek'' 2009]</ref>

==Framework==
The key to these techniques is the recognition that the drifts of the [[Finance: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.

Models developed according to the HJM framework are different from the so-called [[Finance:Short-rate model|short-rate model]]s in the sense that HJM-type models capture the full dynamics of the entire [[Finance:Forward curve|forward rate curve]], while the short-rate models only capture the dynamics of a point on the curve (the short rate).

However, models developed according to the general HJM framework are often non-[[Markov property|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.

==Mathematical formulation==

The class of models developed by Heath, Jarrow and Morton (1992) is based on modelling the forward rates.

The model begins by introducing the instantaneous forward rate <math>\textstyle f(t,T)</math>, <math>\textstyle t \leq T</math>, which is defined as the continuous compounding rate available at time <math>\textstyle T</math> as seen from time <math>\textstyle t</math>. The relation between bond prices and the forward rate is also provided in the following way:

:<math>P(t,T) = e^{-\int_t^T f(t,s) ds}</math>

Here <math>\textstyle P(t,T)</math> is the price at time <math>\textstyle t</math> of a zero-coupon bond paying $1 at maturity <math>\textstyle T\geq t</math>. The risk-free money market account is also defined as

:<math>\beta(t) = e^{\int_0^t f(u,u)du}</math>

This last equation lets us define <math>\textstyle f(t,t) \triangleq r(t)</math>, the risk free short rate. The HJM framework assumes that the dynamics of <math>\textstyle f(t,s)</math> under a risk-neutral pricing measure <math>\textstyle \mathbb Q</math> are the following:

:<math>df(t,s) = \mu(t,s)dt + \boldsymbol \sigma(t,s) dW_t</math>

Where <math>\textstyle W_t</math> is a <math>\textstyle d</math>-dimensional [[Wiener process]] and <math>\textstyle \mu(u,s)</math>, <math>\textstyle \boldsymbol \sigma(u,s)</math> are <math>\textstyle \mathcal F_u</math> [[Adapted process|adapted process]]es. Now based on these dynamics for <math>\textstyle f</math>, we'll attempt to find the dynamics for <math>\textstyle P(t,s)</math> and find the conditions that need to be satisfied under risk-neutral pricing rules. Let's define the following process:

:<math>Y_t \triangleq \log P(t,s) = -\int_t^s f(t,u) du</math>

The dynamics of <math>\textstyle Y_t</math> can be obtained through [[Leibniz integral rule|Leibniz's rule]]:

:<math>\begin{align} dY_t &= f(t,t) dt - \int_t^s df(t,u) du \\ &= r_t dt - \int_t^s \mu(t,u) dt du - \int_t^s \boldsymbol \sigma(t,u) dW_t du \end{align}</math>

If we define <math>\textstyle \mu(t,s)^* = \int_t^s \mu(t,u) du</math>, <math>\textstyle \boldsymbol \sigma(t,s)^* = \int_t^s \boldsymbol \sigma(t,u) du</math> and assume that the conditions for Fubini's Theorem are satisfied in the formula for the dynamics of <math>\textstyle Y_t</math>, we get:

:<math> dY_t = \left(  r_t - \mu(t,s)^* \right)dt - \boldsymbol \sigma(t,s)^* dW_t </math>

By Itō's lemma, the dynamics of <math>\textstyle P(t,T)</math> are then:

:<math>\frac{dP(t,s)}{P(t,s)} = \left( r_t - \mu(t,s)^* + \frac{1}{2} \boldsymbol \sigma(t,s)^* \boldsymbol \sigma(t,s)^{*T} \right)dt - \boldsymbol \sigma(t,s)^* dW_t</math>

But <math>\textstyle \frac{P(t,s)}{\beta(t)}</math> must be a martingale under the pricing measure <math>\textstyle \mathbb Q</math>, so we require that <math>\textstyle  \mu(t,s)^* = \frac{1}{2} \boldsymbol \sigma(t,s)^* \boldsymbol \sigma(t,s)^{*T} </math>. Differentiating this with respect to <math>\textstyle s</math> we get:

:<math>\mu(t,u) = \boldsymbol \sigma(t,u) \int_t^u \boldsymbol \sigma(t,s)^{T} ds</math>

Which finally tells us that the dynamics of <math>\textstyle f</math> must be of the following form:

:<math>df(t,u) =  \left( \boldsymbol \sigma(t,u) \int_t^u \boldsymbol \sigma(t,s)^{T} ds \right) dt + \boldsymbol \sigma(t,u) dW_t</math>

Which allows us to price bonds and interest rate derivatives based on our choice of <math>\textstyle \boldsymbol \sigma </math>.

==See also==
*[[Finance:Black–Derman–Toy model|Black–Derman–Toy model]]
*[[Finance:LIBOR market model|Brace–Gatarek–Musiela model]]
*[[Finance:Chen model|Chen model]]
*[[Finance:Cheyette model|Cheyette model]]
*[[Finance:Ho–Lee model|Ho–Lee model]]
*[[Finance:Hull–White model|Hull–White model]]

==References==
===Notes===
{{reflist}}

===Sources===
* 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]. ''[[Finance:Journal of Financial and Quantitative Analysis|Journal of Financial and Quantitative Analysis]]'', 25:419-440.
* 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] {{Webarchive|url=https://web.archive.org/web/20170428115423/http://www.kamakuraco.com/LinkClick.aspx?fileticket=3pl0IkdYSrY=&tabid=208&mid=735 |date=2017-04-28 }}. ''Review of Futures Markets'', 9:54-76.
* 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]. ''[[Finance:Econometrica|Econometrica]]'', 60(1):77-105. {{doi|10.2307/2951677}}
* Robert Jarrow (2002). ''Modelling Fixed Income Securities and Interest Rate Options'' (2nd ed.). Stanford Economics and Finance. {{ISBN|0-8047-4438-6}}

==Further reading==
*[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
* [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN02-01.pdf The Heath-Jarrow-Morton Term Structure Model] {{Webarchive|url=https://web.archive.org/web/20150923195333/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN02-01.pdf |date=2015-09-23 }}, Prof. Don Chance E. J. Ourso College of Business, [[Organization:Louisiana State University|Louisiana State University]]
*[https://ssrn.com/abstract=359040 Recombining Trees for One-Dimensional Forward Rate Models], Dariusz Gatarek, Wyższa Szkoła Biznesu – National-Louis University, and Jaroslaw Kolakowski
*[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. ''[[Finance:The Journal of Fixed Income|The Journal of Fixed Income]]'' March 1999, Vol. 8, No. 4: pp.&nbsp;85–98
* [http://repository.upenn.edu/dissertations/AAI3015358/ Heath–Jarrow–Morton model and its application], Vladimir I Pozdynyakov, [[Organization:University of Pennsylvania|University of Pennsylvania]]
* [https://ssrn.com/abstract=123170 An Empirical Study of the Convergence Properties of the Non-recombining HJM Forward Rate Tree in Pricing Interest Rate Derivatives], A.R. Radhakrishnan [[Organization:New York University|New York University]]
* Modeling Interest Rates with Heath, Jarrow and Morton. Dr Donald van Deventer, [[Company:Kamakura Corporation|Kamakura Corporation]]:
**[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] {{Webarchive|url=https://web.archive.org/web/20200809002440/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 |date=2020-08-09 }}
**[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] {{Webarchive|url=https://web.archive.org/web/20160305162906/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 |date=2016-03-05 }}
**[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] {{Webarchive|url=https://web.archive.org/web/20160304174512/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 |date=2016-03-04 }}
**[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] {{Webarchive|url=https://web.archive.org/web/20200920104602/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 |date=2020-09-20 }}

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