Heath–Jarrow–Morton framework

The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). 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.^[1]^: 394 For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example.

The HJM framework originates from the work of David Heath, Robert A. Jarrow, and 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".^[2]^[3]

Framework[]

The key to these techniques is the recognition that the drifts of the 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 short-rate models 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).

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.

Mathematical formulation[]

The class of models developed by Heath, Jarrow and Morton (1992) is based on modelling the forward rates, yet it does not capture all of the complexities of an evolving term structure.

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

P(t,T)=e^{-\int _{t}^{T}f(t,s)ds}

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

\beta (t)=e^{\int _{0}^{t}f(u,u)du}

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

df(t,s)=\mu (t,s)dt+{\boldsymbol {\sigma }}(t,s)dW_{t}

Where $\textstyle W_{t}$ is a $\textstyle d$ -dimensional Wiener process and $\textstyle \mu (u,s)$ , $\textstyle {\boldsymbol {\sigma }}(u,s)$ are $\textstyle {\mathcal {F}}_{u}$ adapted processes. Now based on these dynamics for $\textstyle f$ , we'll attempt to find the dynamics for $\textstyle P(t,s)$ and find the conditions that need to be satisfied under risk-neutral pricing rules. Let's define the following process:

Y_{t}\triangleq \log P(t,s)=-\int _{t}^{s}f(t,u)du

The dynamics of $\textstyle Y_{t}$ can be obtained through Leibniz's rule:

{\begin{aligned}dY_{t}&=f(t,t)dt-\int _{t}^{s}df(t,u)du\\&=r_{t}dt-\int _{t}^{s}\mu (t,u)dtdu-\int _{t}^{s}{\boldsymbol {\sigma }}(t,u)dW_{t}du\end{aligned}}

If we define $\textstyle \mu (t,s)^{*}=\int _{t}^{s}\mu (t,u)du$ , $\textstyle {\boldsymbol {\sigma }}(t,s)^{*}=\int _{t}^{s}{\boldsymbol {\sigma }}(t,u)du$ and assume that the conditions for Fubini's Theorem are satisfied in the formula for the dynamics of $\textstyle Y_{t}$ , we get:

dY_{t}=\left(r_{t}-\mu (t,s)^{*}\right)dt-{\boldsymbol {\sigma }}(t,s)^{*}dW_{t}

By Itō's lemma, the dynamics of $\textstyle P(t,T)$ are then:

{\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}

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

\mu (t,u)={\boldsymbol {\sigma }}(t,u)\int _{t}^{u}{\boldsymbol {\sigma }}(t,s)^{T}ds

Which finally tells us that the dynamics of $\textstyle f$ must be of the following form:

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}

Which allows us to price bonds and interest rate derivatives based on our choice of $\textstyle {\boldsymbol {\sigma }}$ .

References[]

Notes[]

^ M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling. 2nd ed. New York : Springer-Verlag, 2004. Print.
^ One math geek's plan to reform Wall Street, Newsweek, May 2009
^ Newsweek 2009

Sources[]

Heath, D., Jarrow, R. and Morton, A. (1990). Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation. Journal of Financial and Quantitative Analysis, 25:419-440.
Heath, D., Jarrow, R. and Morton, A. (1991). Contingent Claims Valuation with a Random Evolution of Interest Rates. , 9:54-76.
Heath, D., Jarrow, R. and Morton, A. (1992). 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
Robert Jarrow (2002). Modelling Fixed Income Securities and Interest Rate Options (2nd ed.). Stanford Economics and Finance. ISBN 0-8047-4438-6

v t Bond market
Bond Debenture Fixed income
Types of bonds by issuer	Agency bond Corporate bond Senior debt Subordinated debt Distressed debt Emerging market debt Government bond Infrastructure bond Municipal bond
Types of bonds by payout	Accrual bond Auction rate security Callable bond Commercial paper Consol Contingent convertible bond Convertible bond Exchangeable bond Extendible bond Floating rate note High-yield debt Inflation-indexed bond Inverse floating rate note Lottery bond Perpetual bond Puttable bond Reverse convertible securities Zero-coupon bond
Bond valuation	Clean price Convexity Coupon Credit spread Current yield Dirty price Duration I-spread Mortgage yield Nominal yield Option-adjusted spread Risk-free bond Weighted-average life Yield curve Yield spread Yield to maturity Z-spread
Securitized products	Asset-backed security Collateralized debt obligation Collateralized mortgage obligation Commercial mortgage-backed security Mortgage-backed security
Bond options	Callable bond Convertible bond Embedded option Exchangeable bond Extendible bond Puttable bond
Institutions	Commercial Mortgage Securities Association (CMSA) International Capital Market Association (ICMA) Securities Industry and Financial Markets Association (SIFMA)

v t Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

Heath–Jarrow–Morton framework

Contents

Framework[]

Mathematical formulation[]

See also[]

References[]

Notes[]

Sources[]

Further reading[]