Forward Rate Model Research Articles

Defaultable Discrete Forward Rate Model with Covariance Structure Guaranteeing Positive Credit Spreads

Defaultable Discrete Forward Rate Model with Covariance Structure Guaranteeing Positive Credit Spreads

The Reform of Money Market Benchmarks Worldwide: Construction of a Forward Rate Model for the Moroccan Interbank Market

The purpose of this research is to determine the practicality of a prospective reform of the Moroccan money market's interbank rate within the context of the worldwide shift of reference indices. By analyzing 351 days of quotations, we were able to acquire an optimistic result using the chosen approach, which includes regulated variations that keep the compound rate relatively stable in comparison to other rates. The findings can be used as a benchmark for future maturities. It is worth noting, however, that the compounding strategy does have certain drawbacks. This is the most often used strategy among market participants due to its simplicity in terms of theoretical foundations and openness. The primary shortcoming of the suggested model is its lack of predicting. The methodology is based on previous day's data, with an eye toward the past, which complicates the changeover given IBOR rates' future-oriented methodology. This research project is designed to act as a starting point for future revisions to the ideas made in the body of this research.

A pure-jump mean-reverting short rate model

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

The Reform of Money Market Benchmarks Worldwide: Construction of a Forward Rate Model for the Moroccan Interbank Market

The Reform of Money Market Benchmarks Worldwide: Construction of a Forward Rate Model for the Moroccan Interbank Market

Valor razonable de un swap: CVA y DVA. Una aproximación binomial

Según la NIIF 13, vigente en España desde el 1 de enero de 2013, se deben realizar ajustes por riesgo de crédito en el valor de los derivados financieros para obtener su valor razonable, necesario, desde un punto de vista contable, para las entidades financieras y aquellas que apliquen el PGC 1514/2007.
 Este trabajo obtiene el valor razonable de un swap genérico de tipos de interés. A su valor libre de riesgo se le deduce el CVA o ajuste negativo por riesgo de default de la contraparte y se le suma el DVA, que es la provisión positiva por el riesgo de default de la propia entidad.
 Para calcular el CVA/DVA es necesario conocer la exposición esperada del swap, que se obtiene a partir de un modelo binomial de tipos de interés forward, y determinar las probabilidades de default, que se obtienen a partir de spreads crediticios de bonos corporativos.

The Stochastic Leibniz Formula for Volterra Integrals under Enlarged Filtrations

We derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these settings, we compare the ordinary with the stochastic (Doleans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals.

A Lower-Bounded Short Rate Model

We present a new multi-factor short rate model which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein-Uhlenbeck processes such that the related bond price possesses an affine representation. We also provide the dynamics of the associated LIBOR and instantaneous forward rate and derive a condition under which the model can be market-consistently calibrated. We further establish customized probability measure changes to the respective forward measures. The analytical tractability of our model is illustrated by the derivation of explicit option price formulas. With view on practical applications, we propose suitable probability distributions for jump noise modeling. We conclude the paper by presenting multi-curve extensions of our short, forward and LIBOR rate models.

A multiple-curve Lévy forward rate model in a two-price economy

An advanced Heath–Jarrow–Morton forward rate model driven by time-inhomogeneous Lévy processes is presented which is able to handle the recent development to multiple curves and negative interest rates. It is also able to exploit bid and ask price data. In this approach in order to model spreads between curves for different tenors, credit as well as liquidity risk is taken into account. Deterministic conditions are derived to ensure the positivity of spreads and thus the monotonicity of the curves for the various tenors. Valuation formulas for standard interest rate derivatives such as caps, floors, swaptions and digital options are established. These formulas can be evaluated numerically very fast using Fourier-based valuation methods. In order to exploit bid and ask prices we develop this approach in the context of a two-price economy. Explicit formulas for bid as well as ask prices of the derivatives are stated. A specific model framework based on normal inverse Gaussian and Gamma processes is proposed which allows for calibration to market data. Calibration results are presented based on multiple-curve bootstrapping and cap market quotes. We use data from September 2013 as well as September 2016. The latter is of particular interest since rates were deep in negative territory at that time.

ON DYNAMIC FORWARD RATE MODELING AND PRINCIPAL COMPONENT ANALYSIS

In this paper, we show how to construct dynamic forward rate models in terms of exogenously specified eigenfunctions (or factor loadings). We also show how to link forward rate models with different number of driving Brownian motions to each other in a way consistent with the implied eigenfunctions. Finally, we discuss how to best parameterize the models in the sense of maximizing the number of free parameters for a given set of eigenfunctions.

Total Return Swap Valuation with Counterparty Risk and Interest Rate Risk

We study the pricing of total return swap (TRS) under the contagion models with counterparty risk and the interest rate risk. We assume that interest rate follows Heath-Jarrow-Morton (HJM) forward interest rate model and obtain the Libor market interest rate. The cases where default is related to the interest rate and independent of interest rate are considered. Using the methods of change of measure and the “total hazard construction,” the joint default probabilities are obtained. Furthermore, we obtain the closed-form formulas of TRS under different contagion models, respectively.

Model Calculation and Analysis of Geological Environmental Capacity in Qian'an Mining District

This paper puts forward environmental bearing capacity relative surplus rate and calculation model based on the theory of environmental bearing capacity, Through collecting and estimating the predictive value and ideal value of each index in different development stages and schemes, using analytic hierarchy process to established the target weight, environmental bearing capacity relative surplus rate of the rule layer and index level in Qian’an Mining area are calculated through the gradual convergence principle, accordingly the optimal solution to improve the environmental bearing capacity of Qian’an mining area is determined ,which provides scientific basis for promoting to realize the sustainable development of Qian’an Mining area.

Response of the term structure of forward exchange rate to jump in the interest rate

In this paper, we propose a dynamic model of the term structure of forward exchange rates and discuss the effects of jumps in interest rates on the term structure of forward exchange rates. First, we develop a dynamic three-factor model of forward exchange rates in continuous time that incorporates central bank policy. We model the policy-related events, especially the interest rate adjustment, as interest rate jumps. We apply the model framework to the pricing of USD/CAD and USD/JPY forward exchange rates. We use the Kalman filter approach to estimate the model and Markov Chain Monte Carlo (MCMC) algorithm to discriminate jumps in interest rates. The empirical results show strong evidence of jumps in interest rates related to macroeconomic announcements. Given these jumps, we conclude that our proposed model improves the performance of existing forward exchange rates models. Last but not least, when jumps in interest rates occur, whether the volatility curves of forward rates are asymmetric smile-shaped depends on the standard deviation of spot exchange rate, the speed of adjustment coefficient of foreign interest rate and the correlation coefficient between spot exchange rate and foreign interest rate.

Approximation and application of the Musiela stochastic PDE in forward rate models

In this article, we consider the Musiela equation of forward rates, which is a hyperbolic stochastic partial differential equation. A weak formulation of the problem using the streamline upwind Petrov–Galerkin method is analysed. Error analysis of the method yields estimates for the convergence rates. Computational examples are provided that illustrate not only the discretization methods used, but the type of results relevant to bond pricing that can be obtained from the equation.

Forward rate models with linear volatilities

The existence of solutions to the Heath–Jarrow–Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for the existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions, the Lévy process should necessarily be without a Gaussian part and without negative jumps. If this is the case, then necessary and sufficient conditions for the existence are formulated either in terms of the behavior of the Lévy measure of the noise near the origin or the behavior of the Laplace exponent of the noise at infinity.

Correlations in Lévy interest rate models

In a series of papers during the last ten years an interest rate theory with models which are driven by Lévy or more general processes has been developed. In this paper we derive explicit formulas for the correlations of interest rates as well as zero coupon bonds with different maturities. The models considered in this general setting are the forward rate (HJM), the forward process and the LIBOR model as well as the multicurrency extension of the latter. Specific subclasses of the class of generalized hyperbolic Lévy motions are studied as driving processes. Based on a data set of parametrized yield curves derived from German government bond prices we estimate correlations. In a second step the empirical correlations are used to calibrate the Lévy forward rate model. The superior performance of the Lévy driven models becomes obvious from the graphs.

A Wiener Chaos Approach to Hyperbolic SPDEs

We construct generalized weighted Wiener chaos solutions for hyperbolic linear SPDEs driven by a cylindrical Brownian motion. Explicit conditions for the existence, uniqueness, and regularity of generalized (Wiener Chaos) solutions are established in Sobolev spaces. An equivalence relation between the Wiener Chaos solution and the traditional one is established. The Heath–Jarrow–Morton (HJM) forward rate model is used as an example to illustrate the general construction.

A relaxed cutting plane algorithm for solving the Vasicek-type forward interest rate model

This work considers the solution of the Vasicek-type forward interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type forward interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.

Mean reversion for HJMM forward rate models

We examine the long-time behavior of forward rates in the framework of Heath-Jarrow-Morton-Musiela models with infinite-dimensional Lévy noise. We give an explicit condition under which the rates have a mean reversion property. In a special case we show that this condition is fulfilled for any Lévy process with variance smaller than a given constant, depending only on the state space and the volatility.

Mean reversion for HJMM forward rate models

We examine the long-time behavior of forward rates in the framework of Heath-Jarrow-Morton-Musiela models with infinite-dimensional Lévy noise. We give an explicit condition under which the rates have a mean reversion property. In a special case we show that this condition is fulfilled for any Lévy process with variance smaller than a given constant, depending only on the state space and the volatility.

On the valuation of compositions in Lévy term structure models

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Lévy process.