The Heath, Jarrow & Morton approach to the modeling of the whole forward rate curve was a major breakthrough in the pricing of fixed-income products, from Paul Wilmott.
I will follow Paul's steps in deriving the model and will give a simulation of the forward curve for your reference.
As mentioned at the beginning, the concept of HJM model is to model the whole forward curve instead of just the short end. Let's start from the price of the zero-coupon bond at time t and maturing at time T:
$$Z(t;T) = e^{-\int_t^TF(t;s)ds}$$
From which, we can find that:
$$F(t;T)=-\frac{\partial}{\partial T}logZ(t;T)$$ Let's assume that the zero-coupon bond follows such a stock like evolution:
$$\frac{dZ(t;T)}{Z(t;T)}=\mu(t,T)dt+\sigma(t,T)dX$$ From the above three equations, we can find the evolution of the forward rate:
$$dF(t;T)=\frac{\partial}{\partial T}(\frac{1}{2}\sigma^2(t,T)-\mu(t,T))dt-\frac{\partial}{\partial T}\sigma(t,T)dX$$ if we denote the coefficient of the volatility as v(t,T), we can get this following equation:
$$dF(t;T)=v(t,T)(\int_t^Tv(t,s)ds)dt+v(t,T)dX$$ and this is the final model.
MUSIELA PARAMETERIZATION
In practise, the model for the volatility structure of the forward rate curve will be of the form
$$v(t,T) = \bar{v}(t,T-t)$$ then the model will transform to:
$$d\bar{F}(t;\tau)=\bar{v}(t,\tau)(\int_0^{\tau}\bar{v}(t,s)ds+\frac{\partial}{\partial \tau}\bar{F}(t,\tau))dt+\bar{v}(t,\tau)dX$$ MULTI-FACTOR HJM
The N-dimensional HJM satisfies the following equation:
$$d\bar{F}(t;\tau)=(\sum_{i=1}^N\bar{v}_i(t,\tau)\int_0^{\tau}\bar{v}_i(t,s)ds)dt+\sum_{i=1}^N\bar{v}_i(t,\tau)dX_i+\frac{\partial}{\partial \tau}\bar{F}(t,\tau)$$
SPREADSHEET IMPLEMENTATION
The following is an spreadsheet example implementing the HJM model, this is a two factor model, and a relative small time from 0 to 10 will be shown later. For simplicity, the two volatility will be chosen as one constant and the other be linear with maturity. Another assumption is that the time evolution is relatively small with respect to the maturity and hence as time passes during this evolution, maturity can be treated as constant. You can find the spreadsheet here.
$$F(t;T)=-\frac{\partial}{\partial T}logZ(t;T)$$ Let's assume that the zero-coupon bond follows such a stock like evolution:
$$\frac{dZ(t;T)}{Z(t;T)}=\mu(t,T)dt+\sigma(t,T)dX$$ From the above three equations, we can find the evolution of the forward rate:
$$dF(t;T)=\frac{\partial}{\partial T}(\frac{1}{2}\sigma^2(t,T)-\mu(t,T))dt-\frac{\partial}{\partial T}\sigma(t,T)dX$$ if we denote the coefficient of the volatility as v(t,T), we can get this following equation:
$$dF(t;T)=v(t,T)(\int_t^Tv(t,s)ds)dt+v(t,T)dX$$ and this is the final model.
MUSIELA PARAMETERIZATION
In practise, the model for the volatility structure of the forward rate curve will be of the form
$$v(t,T) = \bar{v}(t,T-t)$$ then the model will transform to:
$$d\bar{F}(t;\tau)=\bar{v}(t,\tau)(\int_0^{\tau}\bar{v}(t,s)ds+\frac{\partial}{\partial \tau}\bar{F}(t,\tau))dt+\bar{v}(t,\tau)dX$$ MULTI-FACTOR HJM
The N-dimensional HJM satisfies the following equation:
$$d\bar{F}(t;\tau)=(\sum_{i=1}^N\bar{v}_i(t,\tau)\int_0^{\tau}\bar{v}_i(t,s)ds)dt+\sum_{i=1}^N\bar{v}_i(t,\tau)dX_i+\frac{\partial}{\partial \tau}\bar{F}(t,\tau)$$
SPREADSHEET IMPLEMENTATION
The following is an spreadsheet example implementing the HJM model, this is a two factor model, and a relative small time from 0 to 10 will be shown later. For simplicity, the two volatility will be chosen as one constant and the other be linear with maturity. Another assumption is that the time evolution is relatively small with respect to the maturity and hence as time passes during this evolution, maturity can be treated as constant. You can find the spreadsheet here.
