Limiting connection between discrete and continuous time forward interest rate curve models
type: Item
instance of: scholarly article
title: Limiting connection between discrete and continuous time forward interest rate curve models
zbMATH Open document ID: 1035.60018
DOI: 10.1023/A:1025752526404
author: Gyula Pap
author: József Gáll
author: Martien C. A. van Zuijlen
published in: Acta Applicandae Mathematicae
publication date: 2003-12-09T00:00:00Z
review text: The authors study limiting connections between discrete time forward interest rate models introduced by themselves and corresponding generalized well-known continuous time models. The main aim is to derive a functional limit theorem for a sequence of discrete time forward interest rate models driven by random fields. To explain the sense of generalization let \(f(t,x)\) denote the instantaneous forward rate at time \(t\) with time to maturity \(x\) and in the famous Heath-Jarrow-Morton model be assumed to follow the dynamics \(df(t,x)=\alpha(t,x)\,dt+\sigma(t,x)\,dW(t),\) where \(W(t)_{t\in R_+}\) is a standard Wiener process. Then the price of zero coupon bond \(P(t,s)\) at time \(t\) with maturity date \(s\) may be defined by \[ P(t,s)=\exp\left\{-\int_0^{s-t}f(t,u)\,du\right\}, \quad 0\leq t\leq s. \] And having this equation one can set up and solve different problems in finance in general and in derivative securities in particular, for example. In this classical HJM model the forward rate process \(\{f(t,x)\}_{t\in R_+}\) is driven by the same Wiener process for any value \(x\geq 0.\) Therefore it is natural to generalize the model by incorporating a random driving field instead of a single driving process as follows. Let \(\{Z(t, s)\}_{t, s\in R_+}\) be a random field and suppose that for each fixed \(x\in R_+\) the forward rate dynamics is given by \[ df(t,x)=\alpha(t,x)\,dt+\sigma(t,x)\,dZ(dt,x), \] where \(\{Z(t,s)\}_{t\in R_+}\) is a semimartingale for any \(s\geq 0.\) In this way forward rates with different time to maturity can be driven by different processes and one can clearly get a much larger range of forward rate families than in the classical setup. Such generalization of the continuous time model has been proposed by Kennedy, and the corresponding discrete time forward rate models driven by random fields were proposed by the authors.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 91B24
Mathematics Subject Classification ID: 91B70
zbMATH DE Number: 2015154
reviewed by: Serguey Victorovich Zhulenev
MaRDI profile type: Publication
full work available at URL: a:1025752526404
OpenAlex ID: W1504965947
description: scientific article; zbMATH DE number 2015154