Current rate

The instantaneous interest (English short rate ) is currently valid for an infinitesimal (infinitely) small period of interest rate in the context of continuous interest rate models. In reality and with discrete interest rate modeling, on the other hand, the current interest rate is only determined as a spot rate for specific terms .

Current interest models ( short-rate models ) work with the current interest rate as a theoretical construct that cannot be observed on the capital market. With the instantaneous interest rate models, the instantaneous interest rate forms the basis for modeling the future development of the interest rate structure . The current interest rate models the interest rate of a "safe" investment, a risk-free money market account with continuously variable interest rates.

Modeling

The money market or bank account with constant interest is defined as a rolling investment: At the start time , a monetary unit is invested, which is always interest at the current interest rate. Describes the credit balance of the money market account, this follows - assuming the current interest rate is constant - the equation ${\displaystyle t=0}$ ${\displaystyle t\geq 0}$ ${\displaystyle r_{t}}$ ${\displaystyle B_{t}}$

${\displaystyle B_{t}=\exp \left(\int \limits _{0}^{t}r_{u}\mathrm {d} u\right)}$.

By deriving the equivalent form as a differential equation with starting condition:

${\displaystyle B_{0}=1}$

and

${\displaystyle {\frac {dB_{t}}{dt}}=r_{t}\cdot B_{t}}$.

The credit balance at time t (due to the normalization ) also represents the accumulation or compounding factor for a payment at time 0. ${\displaystyle B_{t}}$ ${\displaystyle B_{0}=1}$

application

The money market account is of particular importance in the context of risk-neutral valuation , in which the money market account is used as a numéraire .

literature

• Nicole Branger, Christian Schlag: Interest rate derivatives. Models and evaluation . Springer, Berlin 2004, ISBN 3-540-21228-0 .