Ross Model Research Articles

Valuation of Floaters and Options on Floaters under Special Repo Rates

This paper studies the valuation of floaters and options on floaters under the assumption of a square root interest rate model. By incorporating the extension proposed by Duffie (1995) into the Cox, Ingersoll and Ross model (1985) in order to accommodate the presence of special repo rates, we obtain closed-form formulas for pricing debt claims with floating-rate coupons and options written on the price of these bonds. Using data from the Italian secondary market, the model has been estimated on a cross-section of fixed and floating-rate Treasuries, futures on Treasury bonds and futures options. Our results show that the model fits the market prices well.

Estimating term structure of interest rates: neural network vs one factor parametric models

The aim of this paper is twofold; first we concentrate on the work of Vasicek (1977) and Cox, Ingersoll and Ross (1985). We examine and test empirically each model and discuss its performance in predicting the term structure of interest rates using a parametric estimating approach GMM (Generalized Moments Method). Second we estimate the term structure of interest rate dynamics using a nonparametric approach ANN (Artificial Neural Network). Two neural network models are performed. The first model uses spreads between interest rates of 10 different maturities as the only explanatory variable of interest rate changes. The second model introduces two factors, spreads and interest rates' levels. Using historical U.S. Treasury bill rates and Treasury bond yields, we compare the ability of each model to predict the term structure of interest rates. Data are daily and cover the period from 3 January 1995 to 29 December 2000. Results suggest that neural network, Vasicek (1977) and Cox, Ingersoll and Ross (1985) models generate different yield curves. Neural network models outperform the parametric standard models. The most successful forecast is obtained with a two factors neural network model.

Interest rate swaps under CIR

We consider fixed-for-floating interest rate swaps under the assumption that interest rates are given by the mean-reverting Cox–Ingersoll–Ross model. By using a Green's function approach, we derive analytical expressions for the values of both a vanilla swap and an in-arrears swap.

Fair Valuation of a Guaranteed Life Insurance Participating Contract Embedding a Surrender Option

AbstractIn this article we deal with the problem of pricing a guaranteed life insurance participating policy, sold in the Italian market, which embeds a surrender option. This feature is an American‐style put option that enables the policyholder to sell back the contract to the insurer at the cash surrender value. Employing a recursive binomial formula patterned after the Cox, Ross, and Rubinstein (1979) discrete option pricing model we compute, first of all, the total price of the contract, which also includes a compensation for the participation feature (“participation option,” henceforth). Then this price is split into the value of three components: the basic contract, the participation option, and the surrender option. The numerical implementation of the model allows us to catch some comparative statics properties and to tackle the problem of suitably fixing the contractual parameters in order to obtain the premium computed by insurance companies according to standard actuarial practice.

Optimal investment strategies in the presence of a minimum guarantee

In a continuous-time framework, we consider the problem of a Defined Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution flow into the financial market, in order to maximize the expected utility function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee. We assume that the stochastic interest rates follow the affine dynamics, including the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985) 385] and the Vasiček model. The optimal investment strategies are obtained by assuming the completeness of financial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for different examples of guarantees and contributions.

Simulated Likelihood Approximations for Stochastic Volatility Models

Abstract. This paper deals with parametric inference for continuous‐time stochastic volatility models observed at discrete points in time. We consider approximate maximum likelihood estimation: for the kth‐order approximation, we pretend that the observations form a kth‐order Markov chain, find the corresponding approximate log‐likelihood function, and maximize it with respect to θ. The approximate log‐likelihood function is not known analytically, but can easily be calculated by simulation. For each k, the method yields consistent and asymptotically normal estimators. Simulations from a model based on the Cox–Ingersoll–Ross model are used for illustration.

Single factor models with Markovian spot interest rate: an analytical treatment

In the spirit of the Heath–Jarrow–Morton methodology, we provide an analytical characterization of bond prices within the context of single factor term structure models in which the spot rate follows a Markov process and the volatility structure of zero coupon bond returns is stochastic. Also, a perturbative analysis of the extended Cox–Ingersoll–Ross model is proposed.

Robust GMM analysis of models for the short rate process

We re-examine the empirical evidence concerning a well-known class of one-factor models for the short rate process (cf. Chan et al. [Journal of Finance 47 (1992) 1209] (CKLS)) and some recent extensions allowing for a nonlinear drift and for changing parameters with a new statistical methodology based on robust statistics, the Robust Generalized Method of Moments (RGMM). We find that standard GMM model selection procedures are highly unstable in these applications. When testing the CKLS models with the RGMM we find that they are all clearly misspecified and we identify a clustering of influential observations in the 1979–1982 subperiod, a time span that is well known to coincide with a temporary change in the monetary policy of the Federal Reserve. This clustering of influential observations does not disappear when we introduce a non-linearity in the drift and allow for a parameter shift during the 1979–1982 period. Moreover, a Cox–Ingersoll–Ross model (selected by the RGMM) might offer a satisfactory data description for the period after 1982, since there only a few isolated outliers are found. Comparable results are obtained for the Euro-mark case.

THE COX, INGERSOLL AND ROSS EXTENDED MODEL

This paper presents a simple construction of the extended Cox, Ingersoll & Ross model (ECIR) far term structure of interest rate, and a simple way of pricing general interest rate derivatives with this model. To price zero-cupon bonds , we calculate the Laplace transform of functionals of an "elementary process", which is taken as a squared Bessel process. This approach takes advantage of martingales and the Girsanov's theorem providing a presentation of results in a "user friendly" form. Furthermore, we propose and apply a method for the calibration of the one factor ECIR model using data of bonds prices.

Parameter Estimation of Nonlinear Stochastic Differential Equations: Simulated Maximum Likelihood versus Extended Kalman Filter and Itô-Taylor Expansion

This article compares several estimation methods for nonlinear stochastic differential equations with discrete time measurements. The likelihood function is computed by Monte Carlo simulations of the transition probability (simulated maximum likelihood SML) using kernel density estimators and functional integrals and by using the extended Kalman filter (EKF and second-order nonlinear filter SNF). The relation with a local linearization method is discussed. A simulation study for a diffusion process in a double well potential (Ginzburg–Landau equation) shows that, for large sampling intervals, the SML methods lead to better estimation results than the likelihood approach via EKF and SNF. A second study using a nonlinear diffusion coefficient (generalized Cox–Ingersoll–Ross model) demonstrates that the EKF type estimators may serve as efficient alternatives to simple maximum quasilikelihood approaches and Monte Carlo methods.

GL&V WRAPS UP C$80M EIMCO PURCHASE

The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t−μ∫0tX(s)ds for some μ≥1, in which X(t) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class of two-factor model, we also investigate the almost sure convergence of the long-term return t−μ∫0tY(s)ds for some μ≥1, where Y(t) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X(t)/(2β) in which X(t) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on.

Effects of Natural Rubber on Microwave Absorption Characteristics of Some Li–Ni–Zn Ferrite–Thermoplastic Natural Rubber Composites

A microwave (Li0.5Fe0.5)0.4Ni0.3Zn0.3Fe2O4 (LNZ) ferrite was prepared by a conventional sintering method in air. Thermoplastic natural rubber (TPNR) was prepared from polypropylene (PP) and natural rubber (NR) in the ratios of 80:20, 70:30, 60:40, 50:50 and 40:60 with liquid natural rubber as a compatibilizer by a melt blending technique. LNZ ferrite-TPNR composites with 20 wt% ferrite filler were prepared using a Brabender plasticorder internal mixer. The microwave electromagnetic properties of the composites were studied in the frequency range of 0.3–13.5 GHz using a microwave vector network analyzer (MVNA). The real and imaginary components of the relative complex dielectric permittivity (εr*=εr′-jεr′′) and magnetic permeability (µr*=µr′-jµr′′) were calculated from the measured complex scattering parameters (S11* and S12*) using the Nicolson–Ross model. The dielectric and magnetic properties were found to depend on the NR and PP content in the composites. The minimum reflection loss (RL) under the matching conditions increases with increasing NR content.

Estimating Term Structure of Interest Rates: Neural Network Vs one Factor Parametric Models

The aim of this paper is twofold; first we concentrate on the work of Vasicek (1977) and Cox, Ingersoll and Ross (1985). We examine and test empirically each model and discuss its performance in predicting the term structure of interest rates using a parametric estimating approach GMM (Generalized Moments Method). Second we estimate the term structure of interest rate dynamics using a nonparametric approach ANN (Artificial Neural Network). Two neural network models are performed. The first model uses spreads between interest rates of 10 different maturities as the only explanatory variable of interest rate changes. The second model introduces two factors, spreads and interest rates' levels. Using historical U.S. Treasury bill rates and Treasury bond yields, we compare the ability of each model to predict the term structure of interest rates. Data are daily and cover the period from 3 January 1995 to 29 December 2000. Results suggest that, neural network; Vasicek (1977) and Cox, Ingersoll and Ross (1985) models generate different yield curves. Neural network models outperform the parametric standard models. The most successful forecast is obtained with two factors neural network model.

Empirical Analysis of the Yield Curve: The Information in the Data Viewed through the Window of Cox, Ingersoll, and Ross

ABSTRACTThis paper uses recent advances in Bayesian estimation methods to exploit fully and efficiently the time‐series and cross‐sectional empirical restrictions of the Cox, Ingersoll, and Ross model of the term structure. We examine the extent to which the cross‐sectional data (five different instruments) provide information about the model. We find that the time‐series restrictions of the two‐factor model are generally consistent with the data. However, the model's cross‐sectional restrictions are not. We show that adding a third factor produces a significant statistical improvement, but causes the average time‐series fit to the yields themselves to deteriorate.

Pricing Claims Under GARCH-Level Dependent Interest Rate Processes

This article considers the pricing of interest-rate-sensitive claims when the underlying interest rate is driven by a two-state-variable GARCH process. Analytical solutions are established for the case when the innovations in the short rate are normal and/or chi-squared random variables and the volatility of rates take on a special GARCH form. GARCH models that nest level-dependent interest rate models, including the Cox, Ingersoll, and Ross model, are also considered. Algorithms are provided that permit the efficient pricing of American-style interest rate claims under a rather broad array of GARCH-Level dependent processes.

A Jump‐diffusion Model for Exchange Rates in a Target Zone

We propose a simple jump‐diffusion model for an exchange rate target zone. The model captures most stylized facts from the existing target zone models while remaining analytically tractable. The model is based on a modified two‐limit version of the COX, INGERSOLL and ROSS (1985) model. In the model the exchange rate is kept within the band because the variance decreases as the exchange rate approaches the upper or lower limits of the band. We also consider an extension of the model with parity adjustments, which are modeled as Poisson jumps. Estimation of the model is by GMM based on conditional moments. We derive prices of currency options in our model, assuming that realignment jump risk is idiosyncratic. Throughout, we apply the theory to EMS exchange rate data. We show that, after the EMS crisis of 1993, currencies remain in an implicit target zone which is narrower than the officially announced target zones.

Two‐State Option Pricing: Binomial Models Revisited

AbstractThis article revisits the topic of two‐state option pricing. It examines the models developed by Cox, Ross, and Rubinstein (1979), Rendleman and Bartter (1979), and Trigeorgis (1991) and presents two alternative binomial models based on the continuous‐time and discrete‐time geometric Brownian motion processes, respectively. This work generalizes the standard binomial approach, incorporating the main existing models as particular cases. The proposed models are straightforward and flexible, accommodate any drift condition, and afford additional insights into binomial trees and lattice models in general. Furthermore, the alternative parameterizations are free of the negative aspects associated with the Cox, Ross, and Rubinstein model. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:987–1001, 2001

Birefringence evaluation of helically wound optical fibres

A simple, non-destructive method is presented for the characterization of the birefringence properties of a helically wound mono-mode optical fibre. It is shown that the Jones matrix model developed by J.N. Ross can be applied to describe them, if the polarization evolution of light is measured with respect to a local reference frame (Serret-Frenet frame). In this case the Ross model is equal to the matrix model developed by R.C. Jones for a twisted crystal. In order to use a fixed reference frame (laboratory reference frame) it is necessary to consider the rotation of the reference frame at the fibre input face produced by the fibre topology. The orientation of the helical fibre axes at the input is defined following the geometrical criteria proposed by Ross. The use of Poincaré's method and Mueller calculus to rewrite the Ross model simplifies the physical interpretation of the results. The method presented provides an easy way to measure the total linear retardation induced by the fibre curvature, and to separate the circular retardation associated with the photo-elastic response of the fibre, from the topological rotation of the reference frame defined only by the geometry of the helically wound fibre.

SW—Soil and Water: Field Evaluation of Furrow Irrigation Models

In this study, four furrow irrigation models, referred to as the Ross, Walker, Strelkoff and Elliott models are evaluated for their prediction of advance and recession times and runoff, and for their computational time per simulation run and volume balance error under three field conditions in south-east Australia. The models simulated irrigation events based on hydrodynamic (HD), zero-inertia (ZI) and kinematic-wave (KW) approaches. Eighty-one irrigation events were monitored over a 3-year period. The models were applied to all the events, and it was found that the Walker HD, Walker ZI, Ross ZI and Elliott ZI models performed best in predicting advance times and the Strelkoff ZI and Strelkoff KW models performed best in predicting recession times. Considering the overall suitability of the models, both the Walker hydrodynamic and Walker zero inertia models are the most suitable for the field conditions encountered in the study.

A Binomial Basis for the Cox, Ingersoll and Ross Model of the Term Structure of Interest Rates

The technical demands of the Cox, Ingersoll and Ross (1985a and 1985b) papers are such that they can only be mastered by those who have a good understanding of some deep mathematics and statistical concepts, including the techniques of continuous time stochastic calculus and the measure theory upon which it is based, the Kuhn‐Tucker theory surrounding non‐linear optimisation techniques as well as variational methods founded on solutions of non‐linear differential equations. Hence, our purpose here is to formalise both investor preferences and the supply side which underscores the Cox, Ingersoll and Ross (1985b) ‘square root’ model of the term structure of interest rates in terms of some simple binomial filtration processes, thereby avoiding most of the intricate technical detail contained in the original papers. These procedures not only allow for a more focused evaluation of the model's underlying strengths and weaknesses but also provide a framework for assessing some of the strategies which the model makes available for hedging exposure against adverse interest rate movements.