A STATE-SPACE APPROACH TO ESTIMATE AND TEST MULTIFACTOR COX-INGERSOLL-ROSS MODELS OF THE TERM STRUCTURE

The objective of this paper is to estimate and test multifactor versions of the Cox-Ingersoll-Ross (CIR) model of the nominal term structure of interest rates. The proposed state-space approach integrates time series and cross-sectional aspects of the CIR model, is consistent with the underlying economic model, and can use information from all available points of the term structure. We recover estimates of the underlying factors that are consistent with the assumptions about the stochastic processes and compare them with factors obtained from standard factor analysis. We perform thorough diagnostic checking and thereby provide new evidence regarding conclusions about the adequacy of the CIR model. We present empirical results for U.S. Treasury market data. Although the specification of multifactor CIR models is sufficiently flexible for the shape of the term structure, we find strong evidence against the adequacy of the CIR model.     

最优多因子仿射利率期限结构模型选择

在利用NS模型估计出市场即期利率的基础上,采用卡尔曼滤波方法对多因子Vasieck和CIR模型进行参数估计,最后运用蒙特卡罗模拟方法对交易所国债价格进行模拟,并与实际价格进行比较,进而确定了符合我们国债市场的最优多因子仿射利率期限结构模型。研究结果表明：多因子CIR模型对数据的拟合效果及对国债价格模拟效果要明显优于多因子Vasicek模型;对于多因子CIR模型而言,因子个数增加并没有提高模型的价格模拟效果;两因子CIR模型具有最优的国债价格模拟效果。     

Gaussian Estimation and Forecasting of Multi-Factor Term Structure Models with an Application to Japan and the United Kingdom

In this paper we extend the exact discrete model of Bergstrom (1966) first used in empirical finance by Brennan and Schwartz (1979) to estimate their two-factor term structure model to estimate other two-factor term structure models using the recent assumption in Nowman (1997) for single factor models. Following Nowman (1997) we use the exact Gaussian estimation methods of Bergstrom (1983–1986, 1990) to estimate two-factor CKLS, Vasicek and CIR models. We estimate the models using monthly UK and Japanese interest rate data and our results indicate that the estimation method works well in practice.     

A general characterization of one factor affine term structure models

We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate processes with jumps. We provide a link to the theory of branching processes and show how CBI-processes naturally enter the field of term structure modelling. Using Markov semigroup theory we exploit the full structure behind an affine term structure model and provide a deeper understanding of some well-known properties of the CIR model. Based on these fundamental results we construct a new short rate model with jumps, which extends the CIR model and still gives closed form expressions for bond options. Manusript received: June 2000, final version received: October 2000     

Squared Bessel Processes and Their Applications to the Square Root Interest Rate Model

We study the Bessel processes withtime-varying dimension and their applications to the extended Cox-Ingersoll-Rossmodel with time-varying parameters. It is known that the classical CIR model is amodified Bessel process with deterministic time and scale change. We show thatthis relation can be generalized for the extended CIR model with time-varyingparameters, if we consider Bessel process with time-varying dimension. Thisenables us to evaluate the arbitrage free prices of discounted bonds and theircontingent claims applying the basic properties of Bessel processes. Furthermorewe study a special class of extended CIR models which not only enables us to fitevery arbitrage free initial term structure, but also to give the extended CIRcall option pricing formula.     

Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model

We introduce the two-dimensional shifted square-root diffusion (SSRD) model for interest-rate and credit derivatives with (positive) stochastic intensity. The SSRD is the unique explicit diffusion model allowing an automatic and separated calibration of the term structure of interest rates and of credit default swaps (CDSs), and retaining free dynamics parameters that can be used to calibrate option data. We propose a new positivity preserving implicit Euler scheme for Monte Carlo simulation. We discuss the impact of interest-rate and default-intensity correlation and develop an analytical approximation to price some basic credit derivatives terms involving correlated CIR processes. We hint at a formula for CDS options under CIR + + CDS-calibrated stochastic intensity.Received: March 2004, Mathematics Subject Classification (2000): 60H10, 60J60, 60J75, 91B70JEL Classification: G13     

Derivative prices from interest rate models: results for Canada, Hong Kong, and United States

In this paper, we compute implied bond and contingent claim prices from the CKLS, Vasicek, CIR, and BS interest rate models using historical estimates for Canada, Hong Kong, and the United States. We find that default-free bond prices and contingent claim prices are sensitive to the assumed model used for these currencies, and that for Canada the CIR is the best, for Hong Kong the Vasicek and CIR models, and for the US the BS model.     

基于套利理论与ICIR模型的债券市场发行定价偏离研究

基于套利定价理论与利率期限结构理论,运用Tobit多元线性回归模型,得出债券发行定价的主要影响因素为债券无风险利率、债券期限溢价、债项信用评级、债券主体信用评级和债券赎回风险溢价,在此基础上再通过改进的CIR定价模型(ICIR)对2006～2010年各债券定价偏离现象进行研究的结果表明,在1%的显著性水平上,ICIR模型测算的债券理论价格通过了二级市场的定价检验,ICIR模型对债券发行定价偏离进行检验具有较强的合理性;同时,从发行年份来看,近五年来,债券定价偏离总体呈逐年下降趋势,债券发行定价与ICIR定价与二级市场定价逐步接轨,市场化程度越来越高。     

Reduced-form Models with Regime Switching: An Empirical Analysis for Corporate Bonds

Empirical evidence shows that there is a close link between regime shifts and business cycle fluctuations. A standard term structure of interest rates, such as the Cox et al. (1985 Econometrica, 53, 385–407; CIR) model, is sharply rejected in the Treasury bond data. Only Markov regime-switching models on the entire yield curve of the Treasury bond data can account for the observed behavior of the yield curve. In this paper, we examine the impact of regime shifts on AAA-rated and BBB-rated corporate bonds through the use of a reduced-form model. The model is estimated by the Efficient Method of Moments (EMM). Our empirical results suggest that regime-switching risk has significant implications for corporate bond prices and hence has a material impact on the entire corporate bond yield curve, providing evidence for the approach of rating through the cycle employed by rating agencies.     

A Comparison of Forward and Futures Prices of an Interest Rate-Sensitive Financial Asset

This paper focuses on contractual distinctions as an explanation for the price divergence between futures and forward contracts. Specifically, it investigates the effect of marking-to-market on the observed price differences using the pricing model described in Cox, Ingersoll, and Ross (CIR) (1981, Journal of Financial Economics 9, 321–346). Using previously unavailable data, this paper employs Eurodollars, an interest rate-sensitive financial asset, to test the CIR model. Unlike prior empirical studies, test results support both the weak prediction concerning the sign of the average price difference and the stronger prediction that specific covariances explain the variation in the price differences.     

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACT

The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.     

Arbitrage-Based Estimation of Nonstationary Shifts in the Term Structure of Interest Rates

The purpose of this paper is to provide a test of a state-dependent multinomial model of intertemporal changes in the term structure of interest rates. The theoretical background for the model comes from Ho and Lee (1986) . The current paper extends their model in several significant ways. First, we perform diagnostic tests on the data to demonstrate that the empirical results reject a binomial model in favor of a trinomial one. After theoretically deriving the appropriate trinomial model, the current paper extends their model to allow for state-dependent shifts which are determined by the set of ex ante observable state variables. The methodology for the study utilizes OLS regressions to identify the exogenous explanatory variables which drive the hypothesized trinomial process of term structure evolution. The empirical tests indicate that the set of state variables explains a significant portion of the variability in the shifts of the term structure over time. The model also identifies and quantifies a set of variables which impact on changes in the term structure of interest rates.     

On the valuation of reverse mortgage insurance

This article presents a closed-form formula for calculating the loan-to-value (LTV) ratio in an adjusted-rate reverse mortgage (RM) with a lump sum payment. Previous literatures consider the pricing of RM in a constant interest rate assumption and price it on fixed-rate loans. This paper successfully considers the dynamic of interest rate and the adjustable-rate RM simultaneously. This paper also considers the housing price shock into the valuation model. Assuming that house prices follow a jump diffusion process with a stochastic interest rate and that the loan interest rate is adjusted instantaneously according to the short rate, we demonstrate that the LTV ratio is independent of the term structure of interest rates. This argument holds even when housing prices follow a general process: an exponential Lévy process. In addition, the HECM (Home Equity Conversion Mortgage) program may be not sustainable, especially for a higher level of housing price volatility. Finally, when the loan interest rate is adjusted periodically according to the LIBOR rate, our finding reveals that the LTV ratio is insensitive to the parameters characterizing the CIR model.     

Options and Bubbles

The Black-Scholes-Merton option valuation method involves derivingand solving a partial differential equation (PDE). But thismethod can generate multiple values for an option. We providenew solutions for the Cox-Ingersoll-Ross (CIR) term structuremodel, the constant elasticity of variance (CEV) model, andthe Heston stochastic volatility model. Multiple solutions reflectasset pricing bubbles, dominated investments, and (possiblyinfeasible) arbitrages. We provide conditions to rule out bubbleson underlying prices. If they are not satisfied, put-call paritymight not hold, American calls have no optimal exercise policy,and lookback calls have infinite value. We clarify a longstandingconjecture of Cox, Ingersoll, and Ross. (JEL G12 and G13)     

Bond yields and debt supply: new evidence through the lens of a preferred-habitat model

This paper examines the responsiveness of bond yields to changes in debt supply. The preferred-habitat theory predicts a positive relation between the term spread and relative supply of longer term debt, and that this relation is stronger when risk aversion is high. To capture this effect, a time-varying coefficient model is introduced and applied to German bond data. The results support the theoretical predictions and indicate substantial time variation: under high risk aversion, yield spreads react about three times more strongly than when risk aversion is low. The accumulated response of term spreads to a one standard deviation change in debt supply ranges between 4 and 46 basis points.     

改革以来我国经济波动与消费、投资及进出口关系的协整分析

根据协整检验和误差修正模型,利用我国1979～2003年的年度国民经济统计数据,分析了我国经济波动与消费、投资及进出口之间长期稳定的动态均衡关系和短期的波动关系.研究结果表明,消费和投资是影响我国经济波动的两大主要力量,进口和出口对我国经济波动的影响不大,净出口对我国经济波动的影响不明显.     

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

In- and out-of-sample specification analysis of spot rate models: Further evidence for the period 1982–2008

We review and construct consistent in-sample specification and out-of-sample model selection tests on conditional distributions and predictive densities associated with continuous multifactor (possibly with jumps) and (non)linear discrete models of the short term interest rate. The results of our empirical analysis are used to carry out a “horse-race” comparing discrete and continuous models across multiple sample periods, forecast horizons, and evaluation intervals. Our evaluation involves comparing models during two distinct historical periods, as well as across our entire weekly sample of Eurodollar deposit rates from 1982 to 2008. Interestingly, when our entire sample of data is used to estimate competing models, the “best” performer in terms of distributional “fit” as well as predictive density accuracy, both in-sample and out-of-sample, is the three factor Chen (Chen, 1996) model examined by Andersen, Benzoni and Lund (2004). Just as interestingly, a logistic type discrete smooth transition autoregression (STAR) model is preferred to the “best” continuous model (i.e. the one factor Cox, Ingersoll, and Ross (CIR: 1985) model) when comparing predictive accuracy for the “Stable 1990s” period that we examine. Moreover, an analogous result holds for the “Post 1990s” period that we examine, where the STAR model is preferred to a two factor stochastic mean model. Thus, when the STAR model is parameterized using only data corresponding to a particular sub-sample, it outperforms the “best” continuous alternative during that period. However, when models are estimated using the entire dataset, the continuous CHEN model is preferred, regardless of the variety of model specification (selection) test that is carried out. Given that it is very difficult to ascertain the particular future regime that will ensue when constructing ex ante predictions, thus, the CHEN model is our overall “winning” model, regardless of sample period.