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Short-Rate GBM, the Expectations Hypothesis, and Curve-Fitting Limits

Article Quant Q&A · Author: Kosta S.
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Summary

The document discusses the use of geometric Brownian motion for short-term interest rates in a model of embedded bond options, and asks how that assumption relates to the Pure Expectations Hypothesis. The response interprets the hypothesis as specifying risk-neutral rate dynamics so that default-free securities earn the instantaneous short rate in expectation. It notes a limitation: the cited model does not fit the initial discount curve, which conflicts with that interpretation in practice.

The response describes two desirable features of short-rate models: a time-dependent drift calibrated to the initial curve and mean reversion to limit implausibly large rates over long horizons. It names the Hull-White model as a simple framework combining those features. The excerpt offers a brief conceptual comparison rather than a full survey of current models or a detailed assessment of geometric Brownian motion’s advantages and disadvantages; it also does not provide calibration steps or empirical evidence.

Key ideas

  • The Pure Expectations Hypothesis is discussed in terms of risk-neutral short-rate dynamics.
  • The cited geometric Brownian motion model does not match the initial discount curve.
  • A time-dependent drift can be calibrated to fit the initial curve.
  • Mean reversion helps limit extreme rate behavior over long horizons.
  • Hull-White is presented as a simple model with curve fitting and mean reversion.

Tags

Fixed incomeStatisticsDerivatives pricing
Full text
# Short-Interest Rates Models - Geometric Brownian Motion?


# Short-Interest Rates Models - Geometric Brownian Motion?












in a paper of Brennon&Schwartz (1977), they model embedded bond options by using an stochastic interest rate model which follows a geometric Brownian Motion. Now they claim that this assumption does hold when we assume that the Pure Expectation Hypothesis holds? I do not get the link to that.

Further, what are Pros & Cos of using a Geometric Browninan Motion as an Interest Rate process, in general? What are state-of-the art models which are applied nowadays?

Greetings,

KS

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/34805

By "Pure Expectation Hypothesis" they mean that they are specifying the stochastic dynamics for the instantaneous interest rate directly under the risk neutral measure so that "the expected instantaneous rate of return on any default free security is the instantaneous interest rate".

However their model does not match the initial curve so that in fact contradicts this "pure expectation hypothesis".

In order for a short rate model to match the initial discount curve you generally need to have a time dependent term in the drift which is calibrated to the initial curve, if possible in closed form. Also you generally want short rate models to have a mean reversion feature so that you do not get very large rates over long time horizons. The Hull & White model is the simplest model that combines these features.

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.