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How LIBOR Market Model Drifts Support No-Arbitrage

Article Quant Q&A · Author: Quanti
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Summary

The document discusses how no-arbitrage is represented in the LIBOR market model (LMM). It raises the question of whether a numeraire can make bond-price processes martingales and where the model’s arbitrage-free construction is justified. The replies clarify that the standard setup models discrete bonds tied to forward-rate accrual-period ends, rather than every bond maturity.

Forward-rate drifts are selected so those modeled rates are driftless under their associated forward measures. The answers point to an extension addressing a broader set of bonds and mention an accessible treatment of the LMM under spot and forward measures. The material is a brief conceptual clarification, not a full derivation: it does not show the drift calculations, state model assumptions in detail, or establish that every bond process is a martingale under one common measure.

Key ideas

  • The standard LMM models bonds associated with the ends of forward-rate periods.
  • Forward-rate drifts are chosen to make the modeled rates driftless under suitable forward measures.
  • The no-arbitrage discussion depends on the choice of numeraire and associated measure.
  • Extending the construction to a wider set of bond maturities requires additional treatment.
  • The document points readers toward references but does not provide the derivation or detailed assumptions.

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Fixed incomeDerivatives pricingStatistics
Full text
# Why is the LIBOR-market model free of arbitrage?


# Why is the LIBOR-market model free of arbitrage?












Recently I have been reading a lot on the market models.

One thing that keeps escaping me - why is the Libor-market model (LMM) assumed to e free of aritrage in continuous time ?

To me this means that there must be a Numeraire so that all the bond-price-processes $P(t,T^*)$ with $T^* >t$ and $T^* \in [0,T]$ (with $T>T^*$ ) are martingales und the measure associated with that numeraire.

To state that the LMM is free of aritrage someone must have determines the relevant numeraire and done all the necessary checks. Is there a book or paper where those resuls can be found?

## Answer by Mark Joshi (score 3)

https://quant.stackexchange.com/a/15952

well generally only the discrete bonds associated to the ends of the forward rates are modelled. to make these be martingales the drifts of the rates are chosen to make them driftless.

for an extension to all bonds, see http://ssrn.com/abstract=1461285

## Answer by Kumar (score 1)

https://quant.stackexchange.com/a/17062

Glasserman on pg 166 has a very accessible introduction to LMM under spot and forward measures

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.