# Bond (yield curve) dynamics in the Forward-LIBOR-market-model

The standard Libor-Forward-Market-Models provides a way of modelling the evolution of forward rates in time.

However the model does not seem to be well suited for the modelling of zero-bonds. But rather it seems to have been primarily developed to price swaptions and caps of different levels of complexity.

Assuming no counterparty and no liquidity risk - can one use the Libor-Forward-Model to model the evolution of the bond-yield curve ?

My approach:

If one defnes the forward rate via $$F(t,T_{k-1},T_k)=\frac{1}{T_k-T_{k-1}}[P(t,T_{k-1})/P(t,T_k)-1]$$ Note: This is not entirely correct - see Modern Pricing of Interest-Rate Derivatives (p. 32) - but is mostly assumed to hold in a theoretical context.

In the following I am going to use the shorter notations $F(t,T_{k-1},T_k)=F_k(t)$ and $\tau_k=T_k-T_{k-1}$

Now let us assume we have a set of co-terminal (or often called spanning) forward rates $F_1(t), \dots, F_n(t)$ with $T_1, \dots, T_n$

Using the above definition of the forward rate one can write (for $k>i$) $$\ln(P(t,T_k)/P(t,T_i))=\ln\left(1 /\left[\prod^k_{j=i+1}(1+\tau_k F_j(t))\right]\right)=-\sum^k_{j=i+1}\ln(1+\tau_j F_j(t))$$

Thus for $t=T_i$ the dynamics of the $(T_k-T_i)$-year-yield will be given by

$$\frac{1}{T_k-T_i}\ln(P(T_i,T_k)/P(T_i,T_i))=\frac{1}{T_k-T_i}\ln(P(T_i,T_k))=-\frac{1}{T_k-T_i}\sum^k_{j=i+1}\ln(1+\tau_j F_j(t))$$

Thus can I use the relationship $$\ln(P(T_i,T_k))=-\sum^k_{j=i+1}\ln(1+\tau_j F_j(t))$$

Question: Can I use above relationshp to describe the evolution of a zero-coupon-bond up ti it's Maturity $T_k$ ?

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