Applications of Interest Rate Models

Main purpose of this paper is to describe the background of interest rate modeling, i.e. introduce main division of interest rate models, explain what is the instantaneous interest rate and what is its suitable representative. We will deeply focus on Vaš́ıček and Hull-White model and analyze problem of the calibration of these models. Introduction It is well known that interest rates and their structure modeling is very important for financial engineers, actuaries, etc. In recent decades there were developed many models which try to describe the behavior of yield curve and which are based on the theory of probability and of stochastic processes. After some general division of interest rate two of the one-factor models will be discussed in particulary. In next part we will focus on the instantaneous interest rate which is the modeled value in most of models. At the end we analyze the problem of model calibration because correct parameter estimation is crucial for model’s application and mention some open problems. Important variables In this chapter we define following important variables: • R(t, T ) the continuously compounded spot rate at time t for the maturity T • P (t, T ) the zero-bond price at time t for the maturity T The relationship between P (t, T ) and R(t, T ) is according to intuition as follows: P (t, T ) = e )(T−t) (1) • r(t) the instantaneous interest rate at time t, defined as limT→tR(t, T ) It can be derived that under some assumptions P (t, T ) is P (t, T ) = EQ  exp  − T ∫ t r(s)ds   |Ft   , (2) where Q is the risk-neutral probability measure. Models of interest rates Generally there are two basic types of processes that describe dynamic of interest rates: One-factor and two-factor models. One-factor models work with only one source of uncertainty which is represented by the interest for some infinitely short period often is also therefore called as an instantaneous interest rate. Such rate according to the model describes whole yield curve and therefore it is important to find its suitable representative in practical usage. Examples of these models are: Rendleman-Bartter model (which is actually identical with traditional model for stock), Vaš́ıček model (with mean reversion to one value), Hull-White model (extended WDS'07 Proceedings of Contributed Papers, Part I, 198–204, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS