It is time to shift log-normal

Ren Raw Chen, Pei Lin Hsieh, Jeffrey Huang

Research output: Contribution to journalJournal Article peer-review

3 Scopus citations


In this article, we show how to go from a log-normally distributed LIBOR to a log-normally distributed 1 + LIBOR (or LIBOR being shifted log-normal) can resolve the severe problems of the existing LMM that has failed since the 2008 crisis. With this alternative modeling, the drift adjustment between any two forward measures is exact and requires no approximation. The derivation of the drift adjustment (Theorem 3) is done by a novel discovery that price and rate are martingales under the forward measures with one-period lead/lag of each other. Also amazing (but not surprising) is that the two martingale forward measures differ by the bond volatility. Furthermore, a shifted log-normally distributed LIBOR implies that LIBOR behaves more like a normal distribution when rates are low and price volatility (i.e., basis point volatility or normal volatility) should be used in cap and swaption valuations. This coincides with the recent observation that interest rates have not only been negative but distributed more like a normal distribution. Lastly, we demonstrate the approximation error of the swap measure. We should note that the swap measure exists only under continuous payoff. The error is presented numerically using the Vasicek model.

Original languageEnglish
Pages (from-to)89-103
Number of pages15
JournalJournal of Derivatives
Issue number3
StatePublished - 01 03 2018
Externally publishedYes

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