Heath-Jarrow-Morton framework

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Heath-Jarrow-Morton framework is a general framework to model the evolution of interest rates (forward rates in particular). The HJM framework originates from the work of D. Heath, R.A. Jarrow and A. 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.

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 (e.g. the Ho-Lee model) 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. One example is the work by P. Ritchken and L. Sankarasubramanian in "Volatility Structures of Forward Rates and the Dynamics of Term Structure", Mathematical Finance, 5, No. 1 (Jan 1995). They show that if the volatility structure of the forward rates satisfy certain conditions, then a HJM model can be expressed entirely by a finite state Markovian system, making it computationally feasible.

The mathematics is as follows

The class of models developed by Heath, Jarrow and Morton (1992) is based on modeling the forward rates, yet it does not capture all of the complexities of an evolving term structure. The forward rate $f \left(t,T\right),t\leq T,$ is the continuous compounding rate available at time $T$ as seen from time $t$ . It is defined by $f\left(t,T\right)=-\frac{1}{P\left(t,T\right)}\frac{\partial}{\partial T}P\left(t,T\right)=-\frac{\partial \textrm{log} P\left(t,T\right)}{\partial T},$ and the basic relation between the rates and the bond prices is given by $P\left(t,T\right)=e^{-\int_t^T f\left(t,s\right)\,ds}$ Consequently the bank account $\beta\left(t\right)$ grows according to $\beta\left(t\right)=e^{\int_0^t f\left(s,s\right)\,ds}$ The assumption of the HJM model is that the forward rates $f\left(t,T\right)$ satisfy for any $T$ $df\left(t,T\right)=\mu\left(t,T\right)dt+\xi\left(t,T\right)dW\left(t\right)$ where the processes $\mu\left(t,T\right),\xi\left(t,T\right)$ are continuous and adapted. For this assumption to be compatible with the assumption of the existence of martingale measures we need relation $\frac{dP\left(t,T\right)}{P\left(t,T\right)}=\left(r\left(t\right)-\alpha\left(t,T\right)\theta\left(t\right)\right)dt +\alpha\left(t,T\right)dW\left( t\right)*********(1).$ to hold.

We find the return on the bond in the HJM model and compare it (1) to obtain models that do not allow for arbitrage. Let $X\left(t\right)=\textrm{log} P\left(t,T\right),$ then $X\left(t\right)=-\int_t^T f\left(t,s\right)\,ds.$ Using Leibniz rule for differentiating under the integral sign we have that $dX=-d\left(\int_t^T f\left(t,s\right)\right)=-A\left(t,T\right)dt-\tau\left(t,T\right)dW\left(t\right),$ where $A\left(t,T\right)=-r\left(t\right)+\int_t^T\mu\left(t,s\right)\,ds~~~\textrm{and}~~\tau\left(t,T\right)=\int_t^T\xi\left(t,s\right)\,ds$ By Ito's formula $\frac{dP\left(t,T\right)}{P\left(t,T\right)}=dX+\frac{1}{2}(dX)^2*************(2)$ It follows from (2) and (1) we must have that $\alpha(t,T)=\int_t^T\xi\left(t,s\right)\,ds,$ $\alpha(t,T)\cdot\theta(t)=\int_t^T\mu(t,s)\,ds-\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2.$

Rearranging the terms we get that $\int_t^T\mu(t,s) \,ds=\frac{1}{2}\left(\int_t^T\xi\left(t,s\right)\,ds\right)^2-\theta\int_t^T\xi\left(t,s\right)\,ds.$ Differentiating wrt $$ T $,$ we have that $\mu(t,T)=\xi\left(t,s\right)\left(\int_t^T\xi\left(t,s\right)\,ds-\theta\right)********(3).$ Equation (3) is known as the no-arbitrage condition in the HJM model. Under the martingale probability measure $θ = 0$ and the equation for the forward rates becomes $df=\xi\left(t,T\right)\left(\int_t^T\xi\left(t,s\right)\,ds\right)dt+\xi\left(t,T\right)d\tilde{W}.$ This equation is used in pricing of bonds and it's derivatives.