The HJM Framework

Tutorial Contents

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

In this section, we assign dynamics to the major quantities we have defined thus far. Specifically, we look at the ZCB prices $P(t,T)$ and the instantaneous forward rate $f(t,T)$. By ensuring that the dynamics of these quantities do not admit arbitrage, we find that, when these values are driven by a finite number of Brownian Motions, the dynamics of our various quantities must satisfy certain conditions. This is the crux of the Heath-Jarrow-Morton (HJM) framework. The HJM framework prescribes the way in which the drift terms of our dynamics must relate to the diffusion terms in order to satisfy no-arbitrage.

Bond Price Dynamics

Within the HJM framework, we focus on modeling an entire continuum of $T$-indexed bond prices $P(t,T)$ for $t \in [0,T]$, starting from the known initial conditions $P(0,T)$. We consider ZCB prices over a finite time-horizon, i.e. $T \in [0, \mathcal{T}]$, $\mathcal{T} < \infty$, and assume that all models are generated by a $d$-dimensional Brownian motion. Let $W(t)$ represent a $d$-dimensional Brownian motion under the risk-neutral measure $\mathbb{Q}$, with numeraire $\beta(t)$, the continuously compounded money market account. Define the discounted ZCB prices $\tilde{P}(t,T) = \tfrac{P(t,T)}{\beta(t)}$. In the absence of arbitrage $\tilde{P}(t,T)$ is a martingale under $\mathbb{Q}$, and the martingale representation theorem then implies that

where $\sigma_P(t,T)$ is a $d$-dimensional stochastic process adapted to the filtration generated by $W$. We assume that $\sigma_P(t,T)$ is regular enough for $\tilde{P}(t,T)$ to be a square-integrable martingale, and since we have the boundary conditions $P(T,T) = 1$ for $T \in [0,\mathcal{T}]$, we impose the extra pull to par condition

Considering the SDE for $\beta(t)$, using Ito's lemma shows that

where $r(t)$ is the short rate. Equation

defines the class of $d$-dimensional HJM models.

Another application of Ito's lemma shows that forward bond prices $P(t,T,T+\tau) = \tfrac{P(t,T+\tau)}{P(t,T)}$ must satisfy

Hence, under the risk-neutral measure, forward bond prices satisfy the SDE

Since, under the $T$-forward measure $\mathbb{Q}^T$ $P(t,T,T+\tau)$ is a martingale, it must be driftless under the $\mathbb{Q}^T$ measure, which immediately shows that the $\mathbb{Q}^T$-Brownian motion is related to $W(t)$ via

so that by Girsanov's theorem, the density process $\eta(t) = \mathbb{E}_t^\mathbb{Q}\left[\frac{\text{d}\mathbb{Q}^T}{\text{d}\mathbb{Q}} \right]$ for the measure shift between $\mathbb{Q}^T$ and $\mathbb{Q}$ satisfies

Instantaneous Forward Rate Dynamics

While the above fully specifies the HJM dynamics of the ZCB and forward bond prices, the HJM framework is normally expressed in terms of the instantaneous forward rates. Doing so eliminates the need to consider the short rate $r$ and also reveals a number of the fundamental properties of HJM models. By Ito's lemma, under the risk-neutral measure, the quantity $Y(t,T) := \log P(t,T)$ satisfies

for some function $g$ that is irrelevant for our purposes. Differentiating both sides of the above SDE by $T$ and taking the partial differential inside the differential term, we see that $f(t,T) = \tfrac{\partial Y(t,T)}{\partial T}$ satisfies SDE

where

Using the relationship (4.1), we find that

Since $f(t,T)$ is a $\mathbb{Q}^T$ martingale, the drift term must vanish. Furthermore, integrating (4.2) with respect to T gives $\sigma_P(t,T) = \int_t^T \sigma_f(t,u)\text{d}u$. These two results together show that

This is the main result of Heath, Jarrow and Morton. Once the forward rate diffusion coefficients $\sigma_f(t,T)$ have been specified, the HJM model is fully specified. The HJM models take initial forward rates $f(0,T)$ as inputs, so that the models are automatically consistent with discount bond prices at time $0$. This is true irrespective of the choice of $\sigma_f(t,T)$, which can be set freely.

This automatic calibration to bond prices is the main selling point of HJM models. However, there are a number of less desirable features of HJM models. Particularly problematic is the dimensionality of the model. To describe bond prices at some future time $t$, an entire continuum of forward rates $\{f(t,u): u \in [t,T]\}$. In practice, to make this tractable requires either making assumptions on $\sigma_f$ that permits a finite-dimensional Markovian representation of the forward rate curve, or moving to a finite state space model that tracks forwards rates spanning time-buckets of finite length. This second approach is explored in much more depth in the Libor market model (LMM).

It's important to note that any arbitrage-free IR model set in a finite dimensional Brownian motion world must be a special case of an HJM model. In particular, any such model corresponds to a particular choice of $\sigma_f(t,T)$. So, in some sense, the HJM framework underlies all of the models that we will cover.

Short Rate Processes

In principle specification of the short-rate process is sufficient to completely specify a full yield curve model. Under the HJM framework, it follows that the short-rate $r(t)$ under $\mathbb{Q}$ satisfies

Clearly the process $r(t)$ is not Markovian in general, as can be seen by focusing on the term

for which we have

Unless the square-bracketed terms are either non-random, or a deterministic function of $D(t)$, we find that

Examples of HJM Models

The Gaussian Model

In the Gaussian model, we assume that $\sigma_P(t,T)$ is abounded deterministic function of $t$ and $T$. It follows that forward bond prices are then log-normal under both $\mathbb{Q}$ and $\mathbb{Q}^T$. The forward rate process under $\mathbb{Q}$ is

This implies that $r(T)$ is Gaussian with $\mathbb{Q}$ moments

The Gaussian short-rate model benefits from being tractable and permitting analytical price formulas for a number of European options and futures. It is often a good starting point for the development of closed-form approximations in other models. An example of a derivative admitting a closed-form solution under the Gaussian model is an option on a ZCB:

Example (Option on a ZCB)

Consider a European call option with maturity $S$ and strike price $K$ on a $T$-bond. The payoff at time $S$ is given by

Under the $S$-forward measure $\mathbb{Q}^S$ we know that

Applying the dynamics of $P(t, S, T)$ we find that

where

Gaussian Models with Markovian Short Rate

Although tractable, the Gaussian model does not generally allow for a finite-dimensional Markovian representation, and does not typically imply Markov behaviour of the short rate. By considering the special choice of forward rate volatility

where $h$ is a positive-valued real function and $g(t) \in \mathbb{R}^d$, we can ensure that $r(t)$ is Markovian. In this case, we find that

and

We find that the term

is Markov, since

It quickly follows that $r(t)$ itself is Markovian by differentiating (4.4). In fact, we have the following result.

Proposition (Markovian short rate)

In the $d$-dimensional Gaussian HJM model, when $(4.3)$ holds, the short rate satisfies an SDE of the form

where $\xi(t) \in \mathbb{R}$ and $\sigma_r(t) \in \mathbb{R}^d$ are deterministic functions of time, and

Log-Normal HJM Models

To avoid the issue of negative forward rates that arises in Gaussian HJM models, it is tempting to consider forward rates of the form

where $\sigma(t,T)$ is deterministic and bounded. In the $T$-forward measure,

so that $f(t,T)$ is log-normal. This leads to severe issues. Under $\mathbb{Q}$ we find that forward rates explode to infinity with non-zero probability. Hence, all valuation formulas under $\mathbb{Q}$ lead to ZCB prices being zero, implying obvious arbitrage opportunities.