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Deriving Bachelier Volatility in a Displaced Libor Market Model

Article Quant Q&A · Author: Richi Wa
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Summary

The document asks whether normal (Bachelier) implied volatilities can be derived directly for caps or swaptions under a Libor market model, rather than first deriving Black-76 volatility and converting it. The proposed forward-rate dynamics use a displacement parameter, deterministic time-dependent volatility, and an instantaneous volatility scale that follows a CIR process.

No derivation, cited paper, numerical evidence, or answer is provided; the text is a research question that identifies a modeling setup and a choice of valuation approach. It therefore serves as a prompt for further literature review rather than a usable pricing method. Any comparison between direct normal-volatility derivation and conversion from Black-76 would need to account for the chosen model assumptions and instrument structure, which the document does not develop.

Key ideas

  • The question concerns normal implied volatility for caps and swaptions in a Libor market model.
  • The proposed forward-rate model includes displacement and deterministic time-dependent volatility.
  • The instantaneous volatility scale is modeled as a CIR process.
  • The document does not supply a derivation or recommend a method for converting Black-76 volatility.

Tags

Fixed incomeDerivatives pricingVolatility
Full text
# Bachelier implied volatilities in the Libor market model


# Bachelier implied volatilities in the Libor market model












There are various papers on how to derive a Black76 implied volatlity for a given specification of a Libor market model (especially for deterministic but time dependent volatility of forward rates and more envolved for the case of a stochastic volatility scaling).

My setting is the following: $$ dF_k(t)/(F_k(t)+\delta) = drift + v_t \sigma(t) dW_t $$ where $\delta$ is a displacement parameter, $\sigma(t)$ is deterministic and $v_t$ is the instantaneous volatility scaling that follows a CIR-process.

Are there any papers on a (direct not via Black76) derivation of Bachelier/normal vols for caps or even for swaptions? Or would it be best to first consider Black76 vols and then approximate normal vols?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.