Instantaneous interest rate model

A current interest rate model (English short rate model ) is a mathematical model that describes the dynamics of the current interest rate (English short rate ).

The aim is to obtain the values ​​of zero coupon bonds P (t, T) for any point in time t <T by describing the current interest rate - often abbreviated as r . The expansion of r (t) is given by one or more stochastic differential equations, whereby different models are differentiated depending on the exact form. The models differ from one another both in the complexity of the formulas, which makes an analytical formula for bond prices impossible for some models, and in the qualitative behavior of the interest rate itself: for example, r (t) in the Vasicek model can assume negative values.

The current rate

The instantaneous interest rate is the (annualized) interest rate at which a market participant can borrow money for an infinitesimal period . The course of the entire yield curve does not yet follow from the current instantaneous interest rate . However, one can by means of the usually assumed for the models arbitrage show that the price of a zero-coupon bond with maturity T at the time t by ${\displaystyle r_{t}\,}$ ${\displaystyle [t,t+dt]\,}$

${\displaystyle P(t,T)=\mathbb {E} \left[\left.\exp {\left(-\int _{t}^{T}r_{s}\,ds\right)}\right|{\mathcal {F}}_{t}\right]}$

given is. This is the natural filtration of the process. This means that a model for the future development of the current interest rate determines the prices of all bonds. ${\displaystyle {\mathcal {F}}_{t}\,}$

Examples of current interest models

In this section denotes a Wiener process under a risk-neutral probability measure and its differential. ${\displaystyle W_{t}}$ ${\displaystyle dW_{t}}$

Prominent examples of current interest models are:

• Vasicek (1977)
• Cox, Ingersoll and Ross (CIR) (1985)
• Hull and White (1990)

Vasicek model

In the Vasicek model, the dynamics of r (t) is described by an Ornstein-Uhlenbeck process :

${\displaystyle dr(t)=\kappa ({\bar {r}}-r(t))dt+\sigma dW_{t},\ r(0)=r_{0}.}$

This process always strives for its level of equilibrium . The model has attractive advantages: The differential equation can be solved explicitly and the instantaneous interest rate is normally distributed in this model . However, negative interest rates occur with a positive probability. ${\displaystyle {\bar {r}}}$

Other interest rate models

A second family of interest rate models is the Heath-Jarrow-Morton model framework (HJM). In doing so, it is not the current instantaneous interest rate, but the entire development of the instantaneous interest rate, i.e. all of the forward instantaneous interest rates, that is modeled. For some cash register models, such as the CIR and the Hull-White model, there is an equivalent description in the HJM model framework; other models do not have a dual HJM representation.

literature

• Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models - Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer publishing house. ISBN 978-3-540-22149-4 .

Individual evidence

1. ^ Vasicek, Oldrich (1977). "An Equilibrium Characterization of the Term Structure". Journal of Financial Economics 5 (2): 177-188. doi : 10.1016 / 0304-405X (77) 90016-2 .
2. ↑ Cox, JC, JE Ingersoll and SA Ross (1985). "A Theory of the Term Structure of Interest Rates". Econometrica 53: 385-407. doi : 10.2307 / 1911242 .
3. ↑ John Hull and Alan White (1990). "Pricing interest-rate derivative securities". The Review of Financial Studies 3 (4): 573-592.