Modeling the Volatility of the Heath-Jarrow-Morton Model

Modeling the Volatility of the Heath-Jarrow-Morton Model
Author: Anjun Zhou
Publisher:
Total Pages: 36
Release: 2000
Genre:
ISBN:

Based on the nonparametric study of Pearson and Zhou (1999), a parametric HJM model is developed for the forward rate volatility. It allows the volatility of the forward rate with different maturities to react in a different way with the level of forward rate and the forward spread. Specifically, the proposed forward rate volatility function is imbedded into GARCH family models and compared with several widely used HJM volatility specifications. It is shown that the proposed volatility specification performs the best. It is also confirmed that the volatility of forward rate with different maturities depends on the forward rate and the forward spread in a different way.

Multiple Time Scales Stochastic Volatility Modeling Method in Heath-jarrow-morton Model of Interest Rate Market

Multiple Time Scales Stochastic Volatility Modeling Method in Heath-jarrow-morton Model of Interest Rate Market
Author: Feiyue Di
Publisher:
Total Pages: 123
Release: 2011
Genre:
ISBN: 9781124773117

We utilize multiple time scales processes in consistent dynamic modeling to capture main time scales and heterogeneity features of the volatility process of Heath-Jarrow-Moton models in the fixed income market. The Black-Scholes type HJM models are prevailing in both industry and academy. However since these models assume that the volatility process of the underlying financial contract is constant during the term period, they are not able to incorporate some implied volatility phenomenons emerging after the Crash of 1987. Stochastic volatility modeling is one of the main approach to overcome the above defects of the Black-Scholes type models. By applying the time scale separation, that is, the singular perturbation method, we show that the stochastic volatility HJM model we proposed are parsimonious and robust effective models. In fact, we carry out this framework on the linear finite dimensional realizable HJM models, derive the explicit pricing formulas of floorlet contracts under this stochastic volatility HJM models and estimate the accuracy of the result. Meanwhile, as a specific example, we studied the stochastic volatility Hull-White model explicitly. Besides the pricing function of the floorlet contracts, we also obtain the explicit form of the pricing function of the swaption. Following the calibration procedures we proposed, we calibrated this model by a group of daily swaption data from PIMCO. The calibration result shows that the mutliple time scales stochastic volatility Hull-White model is able to capture the implied volatility smile and this model is stable statically.

Heath, Jarrow and Morton Implied Volatility Functions and Conditional Heteroskedasticity Models

Heath, Jarrow and Morton Implied Volatility Functions and Conditional Heteroskedasticity Models
Author: Kaushik I. Amin
Publisher:
Total Pages:
Release: 1998
Genre:
ISBN:

We evaluate various popular models of interest rate volatility and the Heath-Jarrow-Morton (HJM) approach to value interest rate derivatives by studying the information content and the forecast ability of HJM implied volatility in the Eurodollar futures options market. Implied volatility corresponding to the Ho-Lee, Courtadon, Cox-Ingersoll-Ross, Vasicek, and a linear proportional volatility model are examined within the HJM framework. The exercise compares these implied volatilities to a number of historical volatility benchmarks based on the GARCH model, the Glosten-Jagannathan-Runkle model, and several hybrid models combining the Cox-Ingersoll-Ross and Courtadon spot rate models and the GARCH and GJR approaches to model interest rate volatility. Our results show that there is a strong interaction effect between return shocks and the level of the interest rates in the volatility dynamics that none of the existing HJM volatility models and none of the GARCH type models can fully capture. Specifically, the impact of a shock to interest rate volatility is higher under a high interest rate than a low interest rate. The importance of implied volatility from the Ho-Lee, Courtadon, and Cox-Ingersoll-Ross models is significantly reduced after a term capturing the interaction effect is added to the volatility specification. The importance of implied volatility from the linear proportional and the Vasicek models is reduced but they can still explain a reasonably large portion of the time-variation in volatility.

Interest Rate Models - Theory and Practice

Interest Rate Models - Theory and Practice
Author: Damiano Brigo
Publisher: Springer Science & Business Media
Total Pages: 1016
Release: 2007-09-26
Genre: Mathematics
ISBN: 354034604X

The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.

Empirical Tests of Two-State-Variable Heath-Jarrow-Morton Models

Empirical Tests of Two-State-Variable Heath-Jarrow-Morton Models
Author: Robert R. Bliss
Publisher:
Total Pages:
Release: 1999
Genre:
ISBN:

Models for pricing interest rate claims, developed under the Heath-Jarrow-Morton paradigm, differ according to the volatility structure imposed on forward rates. For most general HJM structures the resultant path dependence creates implementation problems. Ritchken and Sankarasubramanian have recently identified necessary and sufficient conditions on the class of volatility structures of forward rates that enable the term structure dynamics to be captured by a finite set of state variables. The class is quite rich. The instantaneous spot rate volatility may be quite general, but the model curtails the structure of forward rate volatilities relative to this spot rate volatility. This article provides empirical tests for this class of volatility structures. Unlike other studies, the volatility structure is examined over a broad section of maturities in the yield curve. Using Treasury data over the period 1982-1994, we find support for this class. Furthermore, unlike other studies, no evidence of a quot;volatilityquot; hump is identified.

Modeling the Term Structure of Interest Rates

Modeling the Term Structure of Interest Rates
Author: Rajna Gibson
Publisher: Now Publishers Inc
Total Pages: 171
Release: 2010
Genre: Business & Economics
ISBN: 1601983727

Modeling the Term Structure of Interest Rates provides a comprehensive review of the continuous-time modeling techniques of the term structure applicable to value and hedge default-free bonds and other interest rate derivatives.

Handbook of Volatility Models and Their Applications

Handbook of Volatility Models and Their Applications
Author: Luc Bauwens
Publisher: John Wiley & Sons
Total Pages: 566
Release: 2012-03-22
Genre: Business & Economics
ISBN: 1118272056

A complete guide to the theory and practice of volatility models in financial engineering Volatility has become a hot topic in this era of instant communications, spawning a great deal of research in empirical finance and time series econometrics. Providing an overview of the most recent advances, Handbook of Volatility Models and Their Applications explores key concepts and topics essential for modeling the volatility of financial time series, both univariate and multivariate, parametric and non-parametric, high-frequency and low-frequency. Featuring contributions from international experts in the field, the book features numerous examples and applications from real-world projects and cutting-edge research, showing step by step how to use various methods accurately and efficiently when assessing volatility rates. Following a comprehensive introduction to the topic, readers are provided with three distinct sections that unify the statistical and practical aspects of volatility: Autoregressive Conditional Heteroskedasticity and Stochastic Volatility presents ARCH and stochastic volatility models, with a focus on recent research topics including mean, volatility, and skewness spillovers in equity markets Other Models and Methods presents alternative approaches, such as multiplicative error models, nonparametric and semi-parametric models, and copula-based models of (co)volatilities Realized Volatility explores issues of the measurement of volatility by realized variances and covariances, guiding readers on how to successfully model and forecast these measures Handbook of Volatility Models and Their Applications is an essential reference for academics and practitioners in finance, business, and econometrics who work with volatility models in their everyday work. The book also serves as a supplement for courses on risk management and volatility at the upper-undergraduate and graduate levels.

Consistency Problems for Heath-Jarrow-Morton Interest Rate Models

Consistency Problems for Heath-Jarrow-Morton Interest Rate Models
Author: Damir Filipovic
Publisher: Springer
Total Pages: 141
Release: 2004-11-02
Genre: Mathematics
ISBN: 354044548X

Bond markets differ in one fundamental aspect from standard stock markets. While the latter are built up to a finite number of trade assets, the underlying basis of a bond market is the entire term structure of interest rates: an infinite-dimensional variable which is not directly observable. On the empirical side, this necessitates curve-fitting methods for the daily estimation of the term structure. Pricing models, on the other hand, are usually built upon stochastic factors representing the term structure in a finite-dimensional state space. Written for readers with knowledge in mathematical finance (in particular interest rate theory) and elementary stochastic analysis, this research monograph has threefold aims: to bring together estimation methods and factor models for interest rates, to provide appropriate consistency conditions and to explore some important examples.