Heath–Jarrow–Morton 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle {\frac {dP(t,s)}{P(t,s)}}=\left(r_{t}-\mu (t,s)^{*}+{\frac {1}{2}}{\boldsymbol {\sigma }}(t,s)^{*}{\boldsymbol {\sigma }}(t,s)^{*T}\right)dt-{\boldsymbol {\sigma }}(t,s)^{*}dW_{t}}$

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

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

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

${\displaystyle df(t,u)=\left({\boldsymbol {\sigma }}(t,u)\int _{t}^{u}{\boldsymbol {\sigma }}(t,s)^{T}ds\right)dt+{\boldsymbol {\sigma }}(t,u)dW_{t}}$

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

• Black–Derman–Toy model
• Brace–Gatarek–Musiela model
• Chen model
• Cheyette model
• Ho–Lee model
• Hull–White model

• Non-Bushy Trees For Gaussian HJM And Lognormal Forward Models, Prof Alan Brace, University of Technology Sydney
• The Heath-Jarrow-Morton Term Structure Model Archived 2015-09-23 at the Wayback Machine, Prof. Don Chance E. J. Ourso College of Business, Louisiana State University
• Recombining Trees for One-Dimensional Forward Rate Models, Dariusz Gatarek, Wyższa Szkoła Biznesu – National-Louis University, and Jaroslaw Kolakowski
• 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
• Heath–Jarrow–Morton model and its application, Vladimir I Pozdynyakov, University of Pennsylvania
• 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
• Modeling Interest Rates with Heath, Jarrow and Morton. Dr Donald van Deventer, Kamakura Corporation:
  • With One Factor and Maturity-Dependent Volatility Archived 2020-08-09 at the Wayback Machine
  • With One Factor and Rate and Maturity-Dependent Volatility Archived 2016-03-05 at the Wayback Machine
  • With Two Factors and Rate and Maturity-Dependent Volatility Archived 2016-03-04 at the Wayback Machine
  • With Three Factors and Rate and Maturity-Dependent Volatility Archived 2020-09-20 at the Wayback Machine