# 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

• In a negative rates environment, opting for a geometric Brownian motion to model short rates will clearly pose some problems... – Quantuple Jun 21 '17 at 14:55
• There is another problem with short rate models that have a geometric diffusion term: the expected value of the money market account is infinite for any time horizon, see e.g. Chapter 3.2.2 in Brigo and Mercurio (2007) who discuss this problem for the Dothan (1978) model. – LocalVolatility Jun 22 '17 at 7:23

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.

• What do u mean with "time depended" term exactly? can you specify that? Further, as far as I know, when it Comes to practice, models have to calibrated to market data. Is this possible under the HWM ? – Kosta S. Jun 22 '17 at 15:51
• $dr_t = (\mathbf{\theta(t)} - \lambda r_t) dt + \sigma(t) dt$. See for instance en.wikipedia.org/wiki/Hull%E2%80%93White_model – Antoine Conze Jun 23 '17 at 6:22
• Does someone have an illustration of an Driftless GBM ? What about that illustration: en.wikipedia.org/wiki/Martingale_(probability_theory)#/media/… Is it true that it is basically the same as a martingale ? – Kosta S. Jun 27 '17 at 8:51