Heath–Jarrow–Morton framework

Key takeaways

Heath–Jarrow–Morton framework belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Heath–Jarrow–Morton framework to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Heath–Jarrow–Morton framework from memory before moving on to harder problems.

Reference excerpt

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. 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 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".

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

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

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

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

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

d f ( t , s ) = μ ( t , s ) d t + σ ( t , s ) d W t {\displaystyle df(t,s)=\mu (t,s)dt+{\boldsymbol {\sigma }}(t,s)dW_{t}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Heath–Jarrow–Morton framework

Start with the simplest possible case. Write down what Heath–Jarrow–Morton framework claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Heath–Jarrow–Morton framework before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Heath–Jarrow–Morton framework ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Heath–Jarrow–Morton framework

In research: Heath–Jarrow–Morton framework appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Heath–Jarrow–Morton framework in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Heath–Jarrow–Morton framework is common in secondary-school and first-year university syllabi. It links to neighbouring topics Financial models, Fixed income analysis, Heath–Jarrow–Morton framework, so understanding it makes those chapters shorter.
In everyday life: Look for Heath–Jarrow–Morton framework outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Heath–Jarrow–Morton framework in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Heath–Jarrow–Morton framework means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Heath–Jarrow–Morton framework out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heath–Jarrow–Morton framework in simple terms?

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…

Why does Heath–Jarrow–Morton framework matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Heath–Jarrow–Morton framework?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Heath–Jarrow–Morton framework.