Evaluating an Exponential Limit as Mean Reversion Vanishes

Summary

The document asks why an expression involving an exponential decay parameter converges to the time interval when that parameter approaches zero. The response applies l’Hôpital’s rule because both the numerator and denominator tend to zero. Differentiating with respect to the parameter reduces the ratio to the time interval multiplied by an exponential term, which converges to the interval as the parameter vanishes.

This limit is useful when simplifying formulas that contain a mean-reversion parameter, including expressions in interest-rate and stochastic-process models. It shows that the apparent singularity at zero can be replaced by a finite limiting value. The explanation is concise and assumes the time variables are fixed as the parameter tends to zero; it does not discuss the larger model or provide alternative derivations such as a Taylor expansion.

Key ideas

• The numerator and denominator both approach zero as the decay parameter vanishes.
• L’Hôpital’s rule converts the ratio into a simpler exponential expression.
• For fixed times, the limiting value is the elapsed time interval.
• The result helps interpret formulas at a zero mean-reversion parameter.

Tags

# How to understand the following limits when kapa limits to Zero


# How to understand the following limits when kapa limits to Zero












The equation is quite simple, however it is not very obvious to me to have the following relationship: $$\begin{equation} \frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \kappa \rightarrow 0 \end{equation}$$

Thanks in advance!

## Answer by user16651 (score 1, accepted)

https://quant.stackexchange.com/a/28017

:D Is it a joke? $$\underset{\kappa \to 0 }{\mathop{\lim }}\,\frac{1-e^{-\kappa(T-t)}}{\kappa}=\underset{\kappa \to 0 }{\mathop{\lim }}\,\frac{\frac{d}{d\kappa}\left(1-e^{-\kappa(T-t)}\right)}{\frac{d}{d\kappa}\kappa}=\underset{\kappa \to 0 }{\mathop{\lim }}\,(T-t) e^{-\kappa(T-t)}=T-t$$

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