Pricing in HJM framework

Question

As mentioned in earlier question, I am a math student, who attained a course in interest rate theory. However I have some question how these things actually work in reality.

So assume we are working with Heath-Jarrow-Morton (HJM) framework. We assume that the forward curve is given by $$f(t,T)=f(0,T) +\int_0^t\alpha(s,T)ds+\int_0^t\sigma(s,T)dW(s)$$ We know that under the drift condition, we have absence of arbitrage. Hence under the risk neutral measure $Q$, with $Q$ Brownian Motion $B$, we have $$f(t,T)=f(0,T)+\int_0^t[\sigma(s,T)\int_0^t\sigma(s,u)du]ds+\int_0^t\sigma(s,T)dB(s)\tag{1}$$ So modelling in the HJM case, you start at $(1)$ and try to choose $\sigma(s,T)$ to fit the forward curve from the market? For me, that seems quite hard, since your $\sigma$ depends on two parameters. Or do you fix one of these and try to fit the curve with the other?

So, it would be appreciated if someone could explain how you actually do a "martingale modelling" in the HJM framework.

If you use a short rate model, i.e. write down the dynamics of your short rate depending on a family of parameters, you would do the following:

1. calculate bond prices under this parametric family
2. choose parameter such that bond prices match with market prices
3. do pricing

is here a similar way? What is the main advantage of the HJM framework, beside you get a perfect fit for the initial forward curve and modelling the entire forward curve? Is this framework used in reality?

Answer 1

I have hardly ever seen HJM in action - I have to admit this. Short rate models and LIBOR market models are more widespread in my experience.

But let me share the following thoughts:

1. In my mind the drift condition is the central statement of HJM. It says that given today's term-structure and the volatilities what the only arbitrage-free drift is.

2. A very good paper on the continuous time and for pedagogical reasons (and applications too) the discrete time version can be found here: The Heath-Jarrow-Morton Framework by Martin Haugh You also find remarks for sepcifying $\sigma(s,t)$ there on page 7.

3. Application of the HJM framework to yield-curve prediction can be found here. There we see the drift conditon again: Consistent Long-Term Yield Curve Prediction by Josef Teichmann, Mario V. Wüthrich

Answer 2

The HJM model should be rather called HJM framework. Because: For specific choices of the forward rate's instantaneous volatility function $\sigma(t,T)$ you obtain other models from the HJM. The HJM is a super-set.

• The rate curve is calibrate via the specification of $f(0,T)$.
• The drift $\alpha(t,T)$ is - as always - confined by risk neutrality, i.e., it is given by the assumption that you cannot create arbitrage from a bond-portfolio. The $Q$-average relative performance of a bond w.r.t. the bank account numéraire is zero. (Note that for short rate models the drift is also restricted via this assumption, but since in a short rate model $f(0,T)$ appears in the drift, its drift is to some extend a free parameter (used to calibrate the rate curve). In that sense short rate models are a bit special. Due to this I wouldn't start a lecture on interest rates models with short rate models. I would start with LMM or HJM and then consider short rate models (going from general to special)).
• The dynamic of the rate curve is calibrate via $\sigma(t,T)$.

For example you obtain Hull-Whilte short rate model is given by $\sigma(t,T) = \sigma_r(t) \exp(-a (T-t))$ where $\sigma_r(t)$ is the short rate volatility and $a$ is the short rate mean reversion. For another choice of $\sigma(t,T)$ you obtain the smaller familiy of Cheyette models (Ritchken-Sankarasubramanian-Framework). There is also a special choice of the HJM volatlity $\sigma(t,T)$ which results in the LIBOR Market Model (see Section 24.2 of http://www.amazon.com/dp/0470047224 (Amazon Reader allows to look inside)).

The LIBOR Market Model framework can also be seen as a discrete version of the HJM, where you choose $\sigma(t,T)$ picewise constant. You may then calibrate this covariance matrix to products like swaptions etc. I believe that this is the closest candidate to what "calibrating HJM" could mean. Because for the unrestricted calibration of $\sigma(t,T)$ you would need a continuum of financial products, which you don't have. Thus you have to restrict $\sigma$ by either

• a functional form (e.g. getting a short rate model)
• a discretization (e.g. getting a LIBOR market model)

See also " Derive a short rate model from HJM "

Answer 3

I had to go and dig in one of the books I worked with in my grad studies which is particularly useful for Fixed Income: Term Structure Models by Filipovic.

The Chapter 6 is dedicated to the HJM models, and the most important theorem states the equation you mentioned in $(1)$ and that the discounted bond price go as follows:

$$\frac{P(t,T)}{B(t)} = P(0,T) \mathcal{E}_t(v(\cdot,T) \bullet W^*)$$

Where

• $B(t)$ is the discount of a risk-free bond
• $W^*$ is a $\mathbb{Q}$-brownian motion
• $(h \bullet X)_t=\int_0^th(s)dX_s$
• $\mathcal{E}_t = e^{X_t - \frac{1}{2} \langle X,X \rangle_t}$ is the stochastic exponential
• $v(s,T) = -\int_s^T \sigma(s,u)du$

So, the price of the bond (computed with the risk-neutral measure $\mathbb{Q}$) does not depend on the drift of the real world forward curve $\alpha(\cdot,T)$ at all: it only depends on the volatility of the forward curve $\sigma(\cdot,T)$.

From what I remember and could quickly read again, this is the main advantage of the HJM framework.