Forward Rate Curve Research Articles

PROJECTING THE FORWARD RATE FLOW ONTO A FINITE DIMENSIONAL MANIFOLD

Given a Heath–Jarrow–Morton (HJM) interest rate model [Formula: see text] and a parametrized family of finite dimensional forward rate curves [Formula: see text], this paper provides a technique for projecting the infinite dimensional forward rate curve rt given by [Formula: see text] onto the finite dimensional manifold [Formula: see text]. The Stratonovich dynamics of the projected finite dimensional forward curve are derived and it is shown that, under the regularity conditions, the given Stratonovich differential equation has a unique strong solution. Moreover, this projection leads to an efficient algorithm for implicit parametric estimation of the infinite dimensional HJM model. The feasibility of this method is demonstrated by applying the generalized method of moments.

Construction of Zero-Coupon Yield Curve From Coupon Bond Yield Using Australian Data

This paper briefly surveys the various approaches to modelling the zero coupon yield curve is the starting point for much finance research. The method adopted here for the Australian Treasury bond data is based upon polynomial spline fitting, but with the constraint that the long end of the term structure is stable. This approach has also been successfully applied to the Danish bond market (Tanggaard and Jakobsen (1988)). The forward rate curve then becomes the important input data for the modelling of the term structure of interest rates and pricing of interest rate contingent claims using the Heath-Jarrow-Morton (1992) model.

Estimation of the Zero Coupon Swap Yield Curve

The term structure of interest rates plays a central role in the valuation, pricing and management of interest rate dependent securities. In this paper I focus on the application of the B-Spline methodology to construct zero coupon and forward rate curves for the swap market. By allowing the placements of the knot points for the B-splines to be part of the optimisation process it is possible to construct smooth zero coupon curves that do not violate the bid-ask constraints of the market rates/prices observed.

AN ALTERNATIVE INTEREST RATE TERM STRUCTURE MODEL

This paper proposes an alternative approach to the modeling of the interest rate term structure. It suggests that the total market price for risk is an important factor that has to be modeled carefully. The growth optimal portfolio, which is characterized by this factor, is used as reference unit or benchmark for obtaining a consistent price system. Benchmarked derivative prices are taken as conditional expectations of future benchmarked prices under the real world probability measure. The inverse of the squared total market price for risk is modeled as a square root process and shown to influence the medium and long term forward rates. With constant parameters and constant short rate the model already generates a hump shaped mean for the forward rate curve and other empirical features typically observed.

Variance-in-mean effects of the long forward-rate slope

This paper contains an empirical analysis of the dependence of the long forward-rate slope on the long-rate variance. The long forward-rate slope and the long rate are described by a bivariate GARCH-in-mean model. In accordance with theory, a negative long-rate variance-in-mean effect for the long forward-rate slope is documented. Thus, the greater the long-rate variance, the steeper the long forward-rate curve slopes downward (the long forward-rate slope is negative). The variance-in-mean effect is both statistically and economically significant.

MODELING TERM STRUCTURE DYNAMICS: AN INFINITE DIMENSIONAL APPROACH

Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

Credit risk with infinite dimensional Lévy processes

The forward rate curve is assumed to follow a stochastic differential equation w.r.t. a Lévy process with infinite dimensions. Conditions under which the market is free of arbitrage are provided for both the interest rate case and for the case of credit risk with ratings. A simulation shows that typical movements of the yield curve are well captured by the model.

The Forward Rates for Multifactor Model of Term Structure “With Square Root”

The multi-factor model “with square root” is discussed in details. For such model, the representation of state variable process in the integral form is derived and its covariance matrix is found. The special attention to the problem connected with the tendency for the term structure of long-term forward rates to slope downwards is given. For multi-factor models with square root the following results are derived: representations of the forward rate curve through the volatility of the state variable process and through the volatility of zero coupon yield process are obtained; the expectations, variances and co-variances for the forward rates and the yield process volatility are calculated; the expectation and the variance for the derivative of forward rate are found; the Brown − Schaefer approximation for the spread of forward rate is examined. As examples two three-factor models (BDFS and Chen models) are examined. On the basis of the estimates of parameters of these models received by empirical way the numerical analysis including calculation of covariance matrices of process of state variables, calculation of an expectation of the local variance of yield process and calculation of the variance of zero coupon yield have been fulfilled.

Consistent calibration of HJM models to cap implied volatilities

This article proposes a calibration algorithm that fits multifactor Gaussian models to the implied volatilities of caps with the use of the respective minimal consistent family to infer the forward-rate curve. The algorithm is applied to three forward-rate volatility structures and their combination to form two-factor models. The efficiency of the consistent calibration is evaluated through comparisons with nonconsistent methods. The selection of the number of factors and of the volatility functions is supported by a principal-component analysis. Models are evaluated in terms of in-sample and out-of-sample data fitting as well as stability of parameter estimates. The results are analyzed mainly by focusing on the capability of fitting the market-implied volatility curve and, in particular, reproducing its characteristic humped shape. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:1093–1120, 2005

Modelling forward freight rate dynamics—empirical evidence from time charter rates

The purpose of this paper is to investigate the dynamics of forward freight rate dynamics. We specify our model in a Heath–Jarrow–Morton framework. This model was originally developed for interest rate markets and, in subsequent work, the model has been applied to various commodity markets. We analyse ten years of weekly time charter (TC) rates for a Panamax 65,000 dwt bulk carrier. Our data set consists of 6-, 12- and 36-month TC rates. We use this data to construct, each day, a forward rate function using a smoothing algorithm. We use the smooth data to investigate the factors governing the dynamics of the forward freight rate curve. We find a strange volatility structure in the data. Out results show that the volatility of the forward curve is bumped, with volatility reaching a peak for freight rates with roughly one year to maturity. Also, correlations between different parts of the term structure are in general low and even negative.

Kalman filtering of consistent forward rate curves: a tool to estimate and model dynamically the term structure

This work explores an original methodology to specify Gaussian dynamic term structure models and estimate them efficiently from time series data. The emphasis is on the cross-sectional dimension of the problem. The form of the term premium is chosen so as to take into account the recent contributions that provide evidence in favor of time-varying market prices of risk. A model based on a widely used family of forward curves is then estimated from time series of eight UK interest rates. An extensive diagnostic check is carried out to assess the fit of the model.

Finite dimensional affine realisations of HJM models in terms of forward rates and yields

Finite dimensional Markovian HJM term structure models provide ideal settings for the study of term structure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractability of Markovian models coexist. Consequently, these models became the focus of a series of papers including Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998), de Jong and Santa-Clara (1999), Bjork and Svensson (2001) and Chiarella and Kwon (2001a). However, these models usually required the introduction of a large number of state variables which, at first sight, did not appear to have clear links to the market observed quantities, and the explicit realisations of the forward rate curve in terms of the state variables were unclear. In this paper, it is shown that the forward rate curves for these models are affine functions of the state variables, and conversely that the state variables in these models can be expressed as affine functions of a finite number of forward rates or yields. This property is useful, for example, in the estimation of model parameters. The paper also provides explicit formulae for the bond prices in terms of the state variables that generalise the formulae given in Inui and Kijima (1998), and applies the framework to obtain affine representations for a number of popular interest rate models.

Limiting Connection between Discrete and Continuous Time Forward Interest Rate Curve Models

We study discrete time Heath–Jarrow–Morton (HJM) type of interest rate curve models, where the forward interest rates – in contrast to the classical HJM models – are driven by a random field. Our main aim is to investigate the relationship between the discrete time forward interest rate curve model and its continuous time counterpart. We derive a general result on the convergence of discrete time models and we give special focus on the nearly unit root spatial autoregression model.

On the construction of finite dimensional realizations for nonlinear forward rate models

We consider interest rate models of Heath-Jarrow-Morton type where the forward rates are driven by a multidimensional Wiener process, and where the volatility structure is allowed to be a smooth functional of the present forward rate curve. In a recent paper [3], Bjork and Svensson give necessary and sufficient conditions for the existence of a finite dimensional Markovian state space realization (FDR) for such a forward rate model, and in the present paper we provide a general method for the actual construction of an FDR.

Consistent Initial Curves for Interest Rate Models

It was a significant step forward in modeling the term structure of interest rates to require the model to be arbitrage-free, in the sense that applying the model’s dynamics to the current market yield curve did not produce any arbitrage opportunities, and therefore, internal inconsistency. The Heath-Jarrow-Morton (HJM) class of interest rate models leads to a large family of arbitrage-free term structure processes. Fitting HJM models empirically requires estimating a continuous forward rate curve from the discrete set of bond prices observed in the market. But, as the authors point out, unless this is done with care, the function that is fitted to market rates may not be consistent with the HJM dynamics for the forward curve. That is, current forward rates would be fitted to a functional form that cannot evolve to another forward curve of the same type in the next period under the assumed HJM model specification. This problem is generally overlooked in practice, since the forward rate curve is refitted every period anyway, but it may well show up in the form of parameter instability in the model. In this article, Angelini and Herzel examine this issue both theoretically and empirically, and show that requiring the forward curve to be fitted to a functional form that is consistent with the interest rate process can make a substantial difference in model performance and in parameter stability.

A Methodology to Analyze Model Risk with an Application to Discount Bond Options in a Heath-Jarrow-Morton Framework

In this paper, we propose a general methodology to analyse for discount bond options within a unified Heath, Jarrow, Morton (1992) framework. We illustrate its applicability by focusing on the hedging of discount bond options and options portfolios. We show how to decompose the agent's model risk profit and loss, and emphasize the importance of the position's gamma in order to control it. We further provide mathematical results on the distribution of the forward profit and loss function for specific Markov univariate term structure models. Finally, we run numerical simulations for naked and combined option's hedging strategies in order to quantify the sensitivity of the forward profit and loss function with respect to the volatility of the forward rate curve, the shape of the term structure, and the characteristics of the position being hedged.

On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite‐dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite‐dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.

WEIGHTED MONTE CARLO: A NEW TECHNIQUE FOR CALIBRATING ASSET-PRICING MODELS

A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given model for market dynamics (price diffusion, rate diffusion, etc.), the algorithm corrects price-misspecifications and finite-sample effects in the simulation by assigning "probability weights" to the simulated paths. The choice of weights is done by minimizing the Kullback–Leibler relative entropy distance of the posterior measure to the empirical measure. The resulting ensemble prices the given set of benchmark instruments exactly or in the sense of least-squares. We discuss pricing and hedging in the context of these weighted Monte Carlo models. A significant reduction of variance is demonstrated theoretically as well as numerically. Concrete applications to the calibration of stochastic volatility models and term-structure models with up to 40 benchmark instruments are presented. The construction of implied volatility surfaces and forward-rate curves and the pricing and hedging of exotic options are investigated through several examples.

The Dynamics of the Forward Interest Rate Curve with Stochastic String Shocks

This article offers a new class of models for the term structure of forward interest rates. We allow each instantaneous forward rate to be driven by a different stochastic shock, but constrain the shocks so that the forward rate curve is kept continuous. We term the shocks to the forward curve “stochastic string shocks,” and construct then as solutions to stochastic partial differential equations. This article offers a variety of parameterizations that can produce, with parsimony, any correlation pattern among forward rates of different maturities. We derive the no-arbitrage condition on the drift of forward rates shocked by stochastic strings and show how to price interest rate derivatives. Although derivatives can be easily priced, they can be perfectly hedged only by trading in an infinite number of bonds of all maturities. We show that the strings model is consistent with any panel dataset of bond prices, and does not require the addition of error terms in econometric models. Finally, we empirically calibrate some versions of the model and price the delivery option embedded in long bond futures. We show that the delivery option is much more valuable in string models than in a similar one-factor model. This is due to the greater variety of shapes of the term structure that string models can produce, which induces more changes in the cheapest-todeliver bond. In this article we develop a new class of bond pricing models. Our model is as parsimonious and tractable as the traditional Heath, Jarrow, and Morton (1992; hereafter HJM) model, but is capable of generating a much richer class of dynamics and shapes of the term structure of interest rates. Our main innovation consists in having each instantaneous forward rate driven by its own shock, while constraining these shocks in such a way as to keep the forward

Exponential-Polynomial Families and the Term Structure of Interest Rates

Exponential-polynomial families like the Nelson-Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with inter-temporal modelling. We characterize the consistent Ito processes which have the property to provide an arbitrage free interest rate model when representing the parameters of some bounded exponential-polynomial type function. This includes in particular diffusion processes. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families should rather not be used for modelling the term structure of interest rates.