Observable Forward Rates and Instantaneous Forward Rates

Summary

The explanation relates a simple forward rate for borrowing or lending over a finite accrual period to the ratio of two discount bond prices. For a period of length τ starting at maturity T, the rate is the bond-price ratio’s excess over one, divided by τ. Such finite-period rates correspond to market conventions such as three- or six-month LIBOR in the example.

Taking the period length toward zero yields the instantaneous forward rate, expressed as the negative maturity derivative of the logarithm of the discount bond price. This makes the instantaneous rate a curve-derived theoretical quantity, while a finite-tenor forward refers to a specific accrual interval and is closer to an observable market quote. The note gives definitions and intuition, but does not discuss conventions, day-count adjustments, credit effects, or how forward rates enter a particular pricing model.

Key ideas

• A finite-tenor simple forward rate can be derived from two discount bond prices.
• The accrual period length determines the interval over which the forward rate applies.
• The instantaneous forward rate is the limiting case as the interval shrinks to zero.
• It equals the negative derivative of the log discount bond price with respect to maturity.
• Instantaneous forwards are useful curve constructs even though they do not represent ordinary lending periods.

Tags

# What's the difference between instantaneous forward rates and observable forward rates?


# What's the difference between instantaneous forward rates and observable forward rates?












Source:

http://docs.fincad.com/support/developerFunc/mathref/LIBORMarketModel.htm

"In contrast to models that evolve the instantaneous short rate (Hull-White, Black-Karasinski models) or instantaneous forward rates (Heath-Jarrow-Morton model), which are not directly observable in the market, the objects modeled using LMM are market-observable quantities (LIBOR forward rates)."

## Answer by Magic is in the chain (score 3)

https://quant.stackexchange.com/a/45273

Recall that the simple forward rate as at time t for lending/borrowing between time T and $T+\tau$ can be written in terms of the discount factors as follows:

$F(t,T, T+\tau)= \frac{1}{\tau}\left( \frac{B(t,T)}{B(t,T+\tau)}-1\right)$

Think of $\tau$ as 6 months or 3 months, and simple forward rate as LIBOR. You can also write it as follows:

$F(t,T, T+\tau)= \frac{1}{\tau}\left( \frac{B(t,T)-B(t,T+\tau)}{B(t,T+\tau)} \right)$

If you let $\tau$ tends to zero, then you get instantaneous forward rate.

$F(t,T, T)=-\frac{\partial \ln B(t,T) }{\partial T}$

It’s theoretical in the sense that you won’t borrow or lend for such a short period of time, but it is quite a useful construct.

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.