Neither 'Normal' nor 'Lognormal': Modeling Interest Rates Across All Regimes

21 Pages Posted: 24 Nov 2013 Last revised: 20 Apr 2016

See all articles by Attilio Meucci

Attilio Meucci

ARPM - Advanced Risk and Portfolio Management

Angela Loregian


Date Written: April 1, 2016


We introduce a simple approach to managing portfolio interest rate risk that is consistent and performs well across different interest rate regimes, including when interest rates are low or even negative. Inspired by Black (1995), this approach uses a novel inverse-call transformation methodology to convert interest rates into shadow rates. We show that this methodology is more appropriate than the standard normal and lognormal models for forecasting and managing the distribution of the profits and losses of portfolios affected by the term structure of interest rates, producing more reliable forecasts and thus risk estimates for purposes of both internal and regulatory risk management.

Note: Documented code is available for download.

Keywords: rates as options, Black rates model, shadow rates, American option, perpetual option, call option, Bachelier process, VAR(1), term structure, yield curve, risk drivers, quest for invariance, projection

JEL Classification: C1, G11

Suggested Citation

Meucci, Attilio and Loregian, Angela, Neither 'Normal' nor 'Lognormal': Modeling Interest Rates Across All Regimes (April 1, 2016). Financial Analysts Journal, Vol. 72, No. 3, 2016. Available at SSRN: or

Attilio Meucci (Contact Author)

ARPM - Advanced Risk and Portfolio Management ( email )


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