Coping with Negative Short-Rates

Wilmott Magazine 2016(81) (2016) 58-68

30 Pages Posted: 10 Feb 2015 Last revised: 25 Jan 2016

See all articles by Zura Kakushadze

Zura Kakushadze

Quantigic Solutions LLC; Free University of Tbilisi

Date Written: August 7, 2015


We discuss a simple extension of the Ho and Lee model with generic time-dependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the short-rate process. Our extension amounts to introducing one or two reflecting barriers for the underlying Brownian motion (as opposed to the short-rate), which allows to have more realistic time-dependent drift (as opposed to constant drift). In our model the spectrum -- or, roughly, the set of short-rate values contributing to bond and other claim prices -- is discrete and positive. We discuss how to calibrate our model using empirical yield data by fitting three parameters and then read off the time-dependent drift.

Keywords: Short-rate models, Ho and Lee model, time-dependent drift, volatility, bond prices, bond option prices, reflecting barriers, Brownian motion, Neumann boundary conditions, Schrodinger equation, yield

JEL Classification: G00

Suggested Citation

Kakushadze, Zura, Coping with Negative Short-Rates (August 7, 2015). Wilmott Magazine 2016(81) (2016) 58-68, Available at SSRN: or

Zura Kakushadze (Contact Author)

Quantigic Solutions LLC ( email )

680 E Main St #543
Stamford, CT 06901
United States
6462210440 (Phone)
6467923264 (Fax)


Free University of Tbilisi ( email )

Business School and School of Physics
240, David Agmashenebeli Alley
Tbilisi, 0159

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