Chapter 15: Calibration of the Heath, Jarrow and Morton framework

Overview

The HJM framework demonstrates that when the money market account is used as numeraire, forward rates are not martingales but have a non-zero drift term fully determined by the forward rate volatility. Hence, only the forward rate volatilities are needed to price and hedge interest rate contingent claims. To begin an implementation of HJM, the form of the forward rate volatility function must be specified. Consider the differential form of equation (11.32), the forward rate process under the martingale measure ^[1] :

Here the drift at time t of the T maturity instantaneous rate depends on the volatilities of the time t forward rates for all maturities from t to T . To price a claim contingent on forward rates over some interval, say { f ( t , x ), x ˆˆ [ T ₁ , T ₂ ]}, we require knowledge of the time t forward rate volatilities for all maturities t to T ₂ . To fully value the claim, these forward rates will need to be evolved from time t to expiry of the claim at time T , T ‰ T ₂ , T ₂ . Hence a full continuum of forward rate volatilities { ƒ ( x , y ), x ˆˆ [ t, T ], y ˆˆ [ t, T ₂ ]} is required. Such demanding data requirements quickly make the valuation impractical or intractable, hence constraints need to be imposed on the evolution of the forward rate volatilities.

This may be done by imposing a functional form on the volatility functions, where each unique functional form leads to a unique HJM model. The volatility functions may be estimated using two distinct approaches:

• Historical volatility estimation - historical time series of forward rates are used to determine the volatilities,

• Implied volatility estimation - current market prices of vanilla interest rate derivatives (caps and swaptions) are used to obtain volatility functions such that these market prices are as best possible reproduced by the model.

First, we consider possible forms of the volatility function within three broad categories. These groupings are not distinct, but rather depict various characteristics that influence choice of volatility function.

1. Well known volatility functions.

2. Gaussian volatility functions, i.e. those giving rise to Gaussian forward rate dynamics. The advantage of such forward rate dynamics is that there exist analytic formulae for prices of certain vanilla options.

3. Functional forms producing Markovian short rate dynamics. This means that ƒ ( ‰ , t , T ) depends only on the state space ( ‰ ) observable at time t . This property allows pricing by means of a recombining lattice where the number of nodes grows linearly with time. This contrasts to non-Markovian models, which need to be implemented by simulation or non- recombining lattices (bushy trees) where the number of nodes increases exponentially with time.

^[1] To make the notation less cumbersome, I assume a single source of uncertainty, hence n = 1, and ƒ ₁ ( t, T ) ‰ ƒ ( t, T ).