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Deriving LIBOR Rates from Short-Rate Models in Negative-Rate Settings

Article Quant Q&A · Author: Michael
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Summary

The document explains how to obtain a LIBOR rate from a simulated short-rate model. Such models specify zero-coupon bond prices as functions of the current short rate; in an affine model, the bond price has an exponential form involving time-dependent coefficients and the short rate.

Once the bond price for the relevant accrual period is known, the document applies the standard simple-compounding relation between that bond price and LIBOR to solve for the rate. This construction allows the resulting LIBOR to be negative when the bond price exceeds one, so it can represent a negative-rate environment. The answer gives the relationship but does not compare short-rate models, calibration choices, or simulation performance, and it does not develop a separate model tailored to negative rates.

Key ideas

  • A short-rate model can produce zero-coupon bond prices from simulated short-rate values.
  • An affine model expresses bond prices through time-dependent coefficients and the short rate.
  • LIBOR for an accrual period can be calculated from the corresponding zero-coupon bond price.
  • The bond-price relationship can yield negative LIBOR when the bond price is above one.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# Negative Libor Simulation


# Negative Libor Simulation












Can LIBOR rates be simulated using short rate models? If no, what is the reason behind it?

What is a simple model to simulate LIBOR rates? Especially in a negative rate environment.

## Answer by byouness (score 3, accepted)

https://quant.stackexchange.com/a/46768

Yes, LIBOR rates can be simulated using short rate models. Or rather, Libor rates can be obtained from simulated short rate values.

Usually, you have formulas giving you the zero-coupon bond price as a function of the short rate. For affine models for example, this would be of the form: $$P(t, T) = e^{A(t, T) - r(t)B(t,T)}$$ (for example, for the one-factor Hull-White model, see: https://quant.stackexchange.com/a/31998/26242)

Then, the Libor is deduced from the zero bond could be done using its definition:

$$P(t, t + \delta) = \frac{1}{1 + \delta L(t, \delta)} \iff L(t, \delta) = \frac{1}{\delta}\left(\frac{1}{P(t, t+\delta)} - 1\right)$$

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.