Why Forward-Rate Volatility Uses Logarithmic Changes

Summary

The document asks why the instantaneous volatility of a simply compounded forward rate is defined through the quadratic variation of its logarithm rather than the rate itself. The response interprets this as percentage volatility: changes in the log of a positive rate approximate proportional changes in the rate. This makes log-based volatility a measure of relative movement rather than an absolute cash-rate change.

The answer also points to modeling practice: some interest-rate frameworks model forward rates with lognormal dynamics, under which log rates have constant or deterministic volatility. That convention can make model specification and calculations more convenient. The explanation is brief and does not derive the relationship with Itô calculus or distinguish different modeling assumptions, so it should be read as intuition rather than a complete mathematical justification. The choice of volatility definition depends on the model and whether absolute or percentage changes are intended.

Key ideas

• Quadratic variation of the log rate represents percentage volatility of the underlying forward rate.
• Log changes approximate proportional changes for small movements in a positive rate.
• Lognormal interest-rate models often specify volatility in log terms for modeling convenience.
• The short explanation gives intuition but does not provide a formal derivation or cover alternative model conventions.

Tags

# Definition of instantaneous volatility of simply-compounded forward rate


# Definition of instantaneous volatility of simply-compounded forward rate












In section "3.6 Volatility Structures in One-Factor Short-Rate Models" in Brigo Mercurio, and in particular the subsection "Caplet Volatilities in the Market", the authors define the (percentage) instantaneous volatility at time $t$ of the simply-compounded forward rate $F(t; T, T+\tau)$ as: $$\sigma(t; T,T+\tau)^2 = d \ln F(t; T, T+\tau) d \ln F(t; T, T+\tau)$$ Can you help me understand where this definition comes from? I would expect the instantaneous volatility to be given by the quadratic covariation $\sigma(t; T,T+\tau)^2 = dF(t; T, T+\tau) dF(t; T, T+\tau)$ and don't understand why the $\ln$ appears in the definition.

## Answer by Sane (score 1, accepted)

https://quant.stackexchange.com/a/80579

Log Normality of Rates: In many interest rate models, the forward rates are modeled in such a way that the log of the forward rate follows a process with constant or deterministic volatility. This is often because the volatility of interest rates is more stable when expressed in log terms.

Percentage Changes: The volatility of the log of a variable reflects the percentage volatility of the original variable.

Mathematical Convenience: In practice, financial models often use the logarithm to make the differential equations governing the rates more manageable.

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