How Numeraire Choice Creates Drift in the LIBOR Market Model

Summary

The document explains how the chosen numeraire affects drift in the LIBOR Market Model (LMM), which models several forward rates at once. Under a measure associated with a bond maturing at a particular date, the forward rate tied to that bond's payment period is driftless. Other modeled rates generally retain drift under that same measure, so one numeraire cannot make every rate driftless simultaneously.

Using the initial discount bond as numeraire typically gives the rates drift terms, while the cited answer says this choice generally produces lower variance. A bond numeraire can simplify pricing for a derivative tied to its maturity, such as a caplet, but the model's broader purpose is to evolve multiple rates together. The excerpt gives a conceptual explanation rather than equations, derivations, or detailed comparisons of pricing methods; its variance statement is qualitative and has no conditions or evidence elaborated here.

Key ideas

• A numeraire determines the probability measure and drift behavior of modeled forward rates.
• A bond maturing at a given date makes the forward rate associated with that bond's period driftless.
• Under that measure, other forward rates in a multi-rate LMM generally have drift.
• Using the initial discount bond as numeraire gives drift terms and is described as generally lowering variance.

Tags

# Why change numeraire for the LIBOR Market Model


# Why change numeraire for the LIBOR Market Model












There are two form of LIBOR Market Model that has a drift introduced. I would like to know in plain english explanation why do practitioners use these changes of measure. Are there any significance to the pricing of interest rate derivatives like Caps? I know that caplet is priced in the zero coupon bond associated with the measure that makes the LIBOR Model in the form of zero-drift. But why do we have the model with drifts?

## Answer by Mark Joshi (score 5)

https://quant.stackexchange.com/a/33568

the point of the LMM is to evolve several different rates simultaneously. If you have rates $f_i$ from $t_i$ to $t_{i+1}$ and take a bond expiring at $t_j$ as numeraire then only the rate $f_{j-1}$ is driftless.

Typically $P_{t_0}$ is used as numeraire which makes all the rates have drift. It generally gives lower variance.

(see my book More mathematical finance for extensive discussion of drifts.)

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.