Implied interest rate pricing models
type: Item
instance of: scholarly article
title: Implied interest rate pricing models
zbMATH Open document ID: 0907.60046
author: P. J. Hunt
author: Joanne E. Kennedy
published in: Finance and Stochastics
publication date: 1999-02-18T00:00:00Z
review text: The main objective and result of this paper is to formulate a model for interest rates which is consistent with market prices of a set of vanilla swaptions (it is supposed that instruments in the market are divided into liquid vanilla products which are readily available at a known market price, and illiquid exotic products). It was done for swaptions of a common maturity date, \(T\), and under the assumption of a single factor model. But to describe the main result mathematically we need some notations. Let \(D_{tS}\) be the time-\(t\) value of a pure discount bond (vanilla pr.) paying unity at time \(S\) and consider a derivative which, at time \(T\), pays an amount \(V_T= V_T(D_{TS_0}, D_{TS_1},\dots, D_{TS_N})\) for a finite set of times \(T= S_0 < S_1 <\cdots <S_N\). As is known in the theory, given a suitable model for the evolution of the bond prices, the time-\(t\) value of the derivative (option, e.g.) is given by \(V_t= N_t\mathbb {E_N}[V_T N_T^{-1}\mid \mathcal F_t]\); here \(N_t\) is any suitable numeraire process, \(\{\mathcal F_t\}\) is the filtration generated by the processes \((N_u, D_{uS_i}, u\leq t, i= 1,\dots ,N)\), and \(\mathbb {E_N}\) denotes expectations in the measure \(\mathbb N\), in which all numeraire re-based assets, \(D_{tS_i}N_t^{-1}\), are \(\{\mathcal F_t\}\)-martingales. Then under some assumptions (all the discount bond prices at time \(T\) can be expressed as a monotone function of a univariate random variable \(r\), all interest rates are positive), and working with \(D_{tT}\) as numeraire, it was shown that the value of a European derivative is given by \[ V_t= D_{tT}\mathbb {E_F}[V_T(D_{TS_0}(r), D_{TS_1}(r),\dots,D_{TS_N}(r))\mid \mathcal F_t]; \] here \(\mathbb F\) is the forward measure, in which \(D_{TS}/D_{tT}\) is an \(\{\mathcal F_t\}\)-martingale for all \(S\). Now we are able to formulate the main result, using 1) the value of the fixed rate which gives the swap (first product considered) zero value and is given by \(…
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 91B28
zbMATH DE Number: 1160488
zbMATH Keywords: interest rate models
zbMATH Keywords: swaptions
zbMATH Keywords: martingale theory
reviewed by: Serguey Victorovich Zhulenev
MaRDI profile type: Publication
full work available at URL: s007800050041
OpenAlex ID: W1971173613
DOI: 10.1007/S007800050041
description: scientific article; zbMATH DE number 1160488