Deriving Zero-Coupon Bond Yields from Forward LIBOR Rates
Summary
The document asks whether a forward LIBOR market model can describe zero-coupon bond and yield-curve evolution. It starts from a simple-period forward-rate relation to adjacent bond prices, then expresses a longer-maturity bond price as a product of forward-rate terms. Taking logarithms gives the bond’s continuously compounded yield as an average of those terms across the remaining periods.
This is a proposed derivation and an open question, not a validated modeling result. The author notes that the forward-rate definition is an approximation in theoretical treatments and asks whether the resulting identity can describe a bond through maturity. The text offers no calibration, numerical evidence, or discussion of how the model’s measure, dynamics, or assumptions affect bond-price behavior, so those issues remain unresolved.
Key ideas
- Adjacent bond prices can be related to simple-period forward rates under the stated convention.
- A longer-maturity bond price can be written as a product across the intervening forward periods.
- Taking logarithms converts that product into a sum that can be used to express a yield.
- The document presents the relationship as a question and does not establish that it fully models bond evolution.
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Full text
# Bond (yield curve) dynamics in the Forward-LIBOR-market-model
# 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 $T_k$ ?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.