Instantaneous Forward Rates as the Limit of Forward LIBOR

Summary

The document distinguishes a tenor-based forward LIBOR rate from an instantaneous forward rate. A forward LIBOR rate applies over a finite accrual period, such as a three- or six-month interval, and is calculated from the relative prices of zero-coupon bonds maturing at the period’s endpoints.

The instantaneous forward rate is obtained by shrinking that accrual period toward zero. Taking the limit converts the bond-price difference into a maturity derivative: the rate equals the negative derivative of the log zero-coupon bond price with respect to maturity. This gives a concise connection between market-style forward rates over finite tenors and the maturity-specific rates modeled in frameworks such as HJM. The explanation is definitional and mathematical; it does not discuss calibration, empirical estimates, or how either rate behaves in practice.

Key ideas

• A forward LIBOR rate is defined over a finite accrual interval using two zero-coupon bond prices.
• The instantaneous forward rate is the limit of the finite-period forward rate as the interval shrinks to zero.
• It equals the negative maturity derivative of the log zero-coupon bond price.
• The distinction helps relate tenor-based market rates to instantaneous forward-rate models.

Tags

# instantaneous forward rates vs forward LIBOR rates


# instantaneous forward rates vs forward LIBOR rates












HJM describes the behavior of instantaneous forward rates while BGM describes the behavior of forward Libor rates. From concept perspective, I understand forward libor rate are like forward Libor rate with different tenor, e.g 3M. They are directly tradable in the market with quotes? But what is the instantenous forward rates?

## Answer by Gordon (score 3)

https://quant.stackexchange.com/a/24495

The forward Libor rate at time $t$ is the forward rate over a certain accrual period $[T, T+\Delta]$, where $\Delta$, in years, can be 3 months or 6 months, and is defined by \begin{align*} L(t, T, T+\Delta) = \frac{1}{\Delta}\left(\frac{P(t, T)}{P(t, T+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero coupon bond with unit face value and maturity $u$.

The instantaneous forward rate is the forward Libor rate over an infinitesimal accrual period. That is, \begin{align*} f(t, T) &=\lim_{\Delta\rightarrow 0}L(t, T, T+\Delta)\\ &=\lim_{\Delta\rightarrow 0}\frac{P(t, T) - P(t, T+\Delta)}{\Delta}\frac{1}{P(t, T+\Delta)}\\ &=-\frac{\partial P(t, T)}{\partial T}\frac{1}{P(t, T)}\\ &=-\frac{\partial }{\partial T}\ln P(t, T). \end{align*}

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.