Heath-Jarrow-Morton Research Articles

Parsimonious HJM Modelling for Multiple Yield-Curve Dynamics

For a long time interest-rate models were built on a single yield curve used both for discounting and forwarding. However, the crisis that has affected financial markets in the last years led market players to revise this assumption and accommodate basis-swap spreads, whose remarkable widening can no longer be neglected. In recent literature we find many proposals of multi-curve interest-rate models, whose calibration would typically require market quotes for all yield curves. At present this is not possible since most of the quotes are missing or extremely illiquid. Thanks to a suitable extension of the HJM framework, we propose a parsimonious model based on observed rates that deduces yield-curve dynamics from a single family of Markov processes. Furthermore, we detail a specification of the model reporting numerical examples of calibration to quoted market data.

Parsimonious HJM modelling for multiple yield curve dynamics

The opinions here expressed are solely those of the authors and do not represent in any way those of their employers. Classical interest-rate models were formulated to satisfy by construction, no-a...

Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model

We consider the valuation of European quanto call options in an incomplete market where the domestic and foreign forward interest rates are allowed to exhibit regime shifts under the Heath–Jarrow–Morton (HJM) framework, and the foreign price dynamics is exogenously driven by a regime switching jump-diffusion model with Markov-modulated Poisson processes. We derive closed-form solutions for four different types of quanto call options, which include: options struck in a foreign currency, a foreign equity call struck in domestic currency, a foreign equity call option with a guaranteed exchange rate, and an equity-linked foreign exchange-rate call.

Correct Numeraire/Bond Correlation in One-Factor HJM Models

HJM one-factor models (including Hull White) have many applications within finance. The risk neutral measure is one of the most common measures to use with HJM models. Since the risk neutral numeraire (money market account) and bond are driven by the same Brownian motion it is frequently assumed that the log correlation between these assets is perfect over an arbitrary interval. Our paper proves that the log correlation over is not perfect and we calculate the correct correlation for one factor HJM models. Finally using Monte Carlo Simulation we demonstrate our correlations are correct and using perfect correlation can lead to significant erroneous results.

Numerical Integration of Heath-Jarrow-Morton Model of Interest Rates

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are highly computationally efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.

Numerical integration of the Heath-Jarrow-Morton model of interest rates

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are computationally highly efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.

Efficient Simulation and Calibration of General HJM Models by Splitting Schemes

We introduce efficient numerical methods for generic Heath--Jarrow--Morton equations of interest rate theory by means of high-order weak approximation schemes. These schemes allow for quasi-Monte Carlo implementations due to the relatively low-dimensional integration space. The complexity of the resulting algorithm is considerably lower than the complexity of multilevel Monte Carlo algorithms as long as the optimal order of quasi-Monte Carlo convergence is guaranteed. In order to make the methods applicable to real world problems, we introduce and use the setting of weighted function spaces, such that unbounded payoffs and unbounded characteristics of the equations in question are still admissible. We also provide an implementation, where we efficiently calibrate a Heath--Jarrow--Morton equation to market caplet data.

Monte Carlo Euler approximations of HJM term structure financial models

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

Pricing Interest Rate Derivatives in a Multifactor HJM Model with Time

We investigate the partial differential equation (PDE) for pricing interest derivatives in the multi-factor Cheyette Model, which involves time-dependent volatility functions with a special structure. The high dimensional parabolic PDE that results is solved numerically via a modified sparse grid approach, that turns out to be accurate and efficient. In addition we study the corresponding Monte Carlo simulation, which is fast since the distribution of the state variables can be calculated explicitly. The results obtained from both methodologies are compared to the known analytical solutions for bonds and caplets. When there is no analytical solution, both European and Bermudan swaptions have been evaluated using the sparse grid PDE approach that is shown to outperform the Monte Carlo simulation.

A Unified Heath-Jarrow-Morton Approach to Gaussian Dynamic Term Structure Models

Motivated by an extensive literature showing that government bond yields exhibit a strong non-Markov property, in the sense that moving averages of long-lagged yields significantly improve the predictability of excess bond returns. We then develop a systematic approach of constructing non-Markov Gaussian dynamic term structure models (GDTSMs) under the Heath-Jarrow-Morton (HJM) framework. Compared to the current literature, our approach is more flexible and parsimonious, enabling us to estimate an economically significant non-Markov effect that helps predict excess bond returns both in-sample and out-of-sample.

Deliverable Interest Rate Swap Futures: Pricing in Gaussian HJM Model

CME will soon be proposing a new product: Deliverable Interest Rate Swap Futures. This note describes the product and analyses its pricing in the Gaussian multi-factor HJM model and multi-curves framework. We also provide numerical example of prices and hedging with those futures.

A Quadratic Volatility Cheyette Model

In this paper we present an extension of the one factor blended Cheyette model for pricing single currency exotics, allowing for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables. We show a practical application of this family of models by analyzing calibration and pricing in the case of a quadratic volatility function. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging techniques. We also improve the usual naive mean state estimation by the use of Gaussian approximations. Last we present an efficient large step Monte-Carlo simulation for strongly path dependent exotics.

A multiple-curve HJM model of interbank risk

In the aftermath of the 2007–2009 financial crisis, a variety of spreads have developed between quantities that had been essentially the same until then, notably LIBOR–OIS spreads, LIBOR–OIS swap spreads, and basis swap spreads. By the end of 2011, with the sovereign credit crisis, these spreads were again significant. In this paper we study the valuation of LIBOR interest rate derivatives in a multiple-curve setup, which accounts for the spreads between a risk-free discount curve and LIBOR curves. Towards this end we resort to a defaultable HJM methodology, in which these spreads are explained by an implied default intensity of the LIBOR contributing banks, possibly in conjunction with an additional liquidity factor. Markovian short rate specifications are given in the form of an extended CIR and a Levy Hull–White model for a risk-free short rate and a LIBOR short spread. The use of Levy drivers leads to the more parsimonious specification. Numerical values of the FRA spreads and the basis swap spreads computed with the latter largely cover the ranges of values observed even at the peak of the 2007–2009 crisis.

Non-Parametric Pricing of Interest Rates Options

The pricing models for interest rates derivatives largely used today employ, many times,excessively restrictive premises in regards to the underlying assets' volatility. The Black and Scholes and the Vasicek methods, for example, consider the variance of the series as constant in time and among different maturities, assumption that may not be themost adequate in all cases. In this paper we discuss the non-parametric estimation of the volatility function using a kernel regression technique and later the pricing of options in a Gaussian HJM model. We analyzed different possible specifications for the non-parametric estimation using the Monte Carlo simulations to price options on zero coupon bonds. We also carried out an empirical study using the proposed methodologyfor the pricing of IDI Index options in the Brazilian market..

An Accompaniment to a Course on Interest Rate Modeling: With Discussion of Black-76, Vasicek and HJM Models and a Gentle Introduction to the Multivariate LIBOR Market Model

The goal of this paper is to help the motivated students with their course on interest rate modeling and/or to help them learn the LIBOR model by themselves. It implies the reader knows what forward rates, caps, and swap[tion]s are and has some knowledge of the quantitative finance: at least Ito Calculus, Black-Scholes-[Merton] Formula, Girsanov’s Theorem (in one dimension is enough) and the risk-neutral pricing. This stuff is usually taught in the first course on continuous financial modeling and is relatively easy. The interest rate modeling is much more complicated. Still those, who carefully read the wonderful Steven Shreve’s book can learn the short-rate models, change of numeraire and Heath-Jarrow-Morton framework. But not the multivariate LIBOR Model (though there is a short section on the one factor LIBOR Model and its relation to the HJM). However, the Bond/IR market is essentially multivariate and the LIBOR Model can be introduced independently. But I could not find any tutorial, which would suit me. So I decided to write my own. It concerns theory only and not the calibration and computational aspects, which are the issues for the future papers.

Explicit Volatility Specification for the Linear Cheyette Model

In this paper we present an extension of the classical Hull-White framework for pricing single currency exotics, which allows for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables.We show a practical application of this family of models by analyzing calibration and pricing in the case where the forward rate volatility is a linear function of the short rate. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging technics. We also improve the usual naive mean state approximation by the use of Gaussian approximations. We give numerical examples.

LEVY-BASED HEATH-JARROW-MORTON INTEREST RATE DERIVATIVES: CHANGE OF TIME METHOD AND PIDES

In this paper, we show how to calculate the price of zero-coupon bonds Heath-Jarrow-Morton (HJM) L\'{e}vy model of forward interest rate $f(t,T)$ using the change of time method. We also derive partial integro-differential equations (PIDEs) for the values of swaps, caps, floors and options on them, swaptions, captions and floortions, respectively. We apply the change of time method to price the interest rate derivatives for the forward interest rates $f(t,T)$ described by the stochastic differential equation driven by $\alpha$-stable L\'{e}vy processes.

Geração de Cenários de Estresse para Curva de Juros

Este trabalho descreve uma aplicação da classe de
 modelos Heath-Jarrow-Morton na geração de cenários de estresse para a estrutura a termo da taxa de juros. Por meio da análise de componentes principais consegue-se reduzir a dimensão do problema e criar uma ponte entre a informação que um especialista possui para definir cenários, informação geralmente de dimensão baixa, e a robustez do modelo Heath-Jarrow-Morton. A metodologia é aplicada ao mercado brasileiro no auge da crise em 2008 e em outras oportunidades.

On Gaussian HJM framework for Eurodollar Futures

One of the standard tools for the theoretical analysis of fixed income securities and their associated derivatives is the term structure model of Heath, Jarrow and Morton. In this paper the question, what specific HJM model is consistent with the observed price of an Eurodollar Futures contract? is discussed. Eurodollar Futures, apart from being the most heavily traded futures are connected to London Inter Bank Offered Rate (LIBOR) and to domestic monetary conditions. The answer to the above question will help in pricing any new derivative on Eurodollar Futures or the one that is not heavily traded. A simple tool to measure the adequacy of different HJM structures that may be used to model Eurodollar Futures price process is suggested. Moreover, the question of estimation of parameters of these models by different methods—method of realized volatility, method of maximum likelihood (ML) and a two‐stage method that combines both the realized volatility and ML—is addressed. Although it sounds like a typical statistical procedure, one must be careful in applying standard statistical techniques that are not suitable under arbitrage theory, in particular, ML method. Copyright © 2010 John Wiley & Sons, Ltd.

Estimating value-at-risk under a Heath–Jarrow–Morton framework with jump

This article proposes a new methodology for measuring Value-at-Risk (hereafter VaR) using a model that incorporates both volatility and jumps. Heath–Jarrow–Morton (HJM) model has been used for the valuation of interest rate derivatives. This study extends the use of HJM model to the estimation VaR. This article specifically uses a two-factor HJM jump-diffusion model for the computation. The study models the Eurodollar futures prices using its derivatives. In addition, this article uses a new volatility specification of Ze-To (2002) to construct the HJM dynamics. The result indicates that the VaR model using HJM jump-diffusion framework performs well in capturing the nonnormality and in providing accurate VaR forecasts in the in-sample and out-sample tests.