Estimation for factor models of term structure of interest rates with jumps: the case of the Taiwanese government bond market

This paper examines the Ornstein–Uhlenbeck (O–U) process used by Vasicek, J. Financial Econ. 5 (1977) 177, and a jump-diffusion process used by Baz and Das, J. Fixed Income (Jnue, 1996) 78, for the Taiwanese Government Bond (TGB) term structure of interest rates. We first obtain the TGB term structures by applying the B-spline approximation, and then use the estimated interest rates to estimate parameters for the one-factor and two-factor Vasicek and jump-diffusion models. The results show that both the one-factor and two-factor Vasicek and jump-diffusion models are statistically significant, with the two-factor models fitting better. For two-factor models, compared with the second factor, the first factor exhibits characteristics of stronger mean reversion, higher volatility, and more frequent and significant jumps in the case of the jump-diffusion process. This is because the first factor is more associated with short-term interest rates, and the second factor is associated with both short-term and long-term interest rates. The jump-diffusion model, which can incorporate jump risks, provides more insight in explaining the term structure as well as the pricing of interest rate derivatives.     

Pricing Contingent Claims on a Risky Asset and Stochastic Interest Rates: A New Discrete Time Approach

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage-free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.     

Taylor rules, McCallum rules and the term structure of interest rates

Recent empirical research shows that a reasonable characterization of federal-funds-rate targeting behavior is that the change in the target rate depends on the maturity structure of interest rates and exhibits little dependence on lagged target rates. See, for example, Cochrane and Piazzesi [2002. The Fed and interest rates—a high-frequency identification. American Economic Review 92, 90-95.]. The result echoes the policy rule used by McCallum [1994a. Monetary policy and the term structure of interest rates. NBER Working Paper No. 4938.] to rationalize the empirical failure of the ‘expectations hypothesis’ applied to the term structure of interest rates. That is, rather than forward rates acting as unbiased predictors of future short rates, the historical evidence suggests that the correlation between forward rates and future short rates is surprisingly low. McCallum showed that a desire by the monetary authority to adjust short rates in response to exogenous shocks to the term premiums imbedded in long rates (i.e. “yield-curve smoothing”), along with a desire for smoothing interest rates across time, can generate term structures that account for the puzzling regression results of Fama and Bliss [1987. The information in long-maturity forward rates. The American Economic Review 77, 680-392.]. McCallum also clearly pointed out that this reduced-form approach to the policy rule, although naturally forward looking, needed to be studied further in the context of other response functions such as the now standard Taylor [1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195-214.] Rule. We explore both the robustness of McCallum's result to endogenous models of the term premium and also its connections to the Taylor Rule. We model the term premium endogenously using two different models in the class of affine term-structure models studied in Duffie and Kan [1996. A yield-factor model of interest rates. Mathematical Finance 57, 405-443.]: a stochastic volatility model and a stochastic price-of-risk model. We then solve for equilibrium term structures in environments in which interest rate targeting follows a rule such as the one suggested by McCallum (i.e., the “McCallum Rule”). We demonstrate that McCallum's original result generalizes in a natural way to this broader class of models. To understand the connection to the Taylor Rule, we then consider two structural macroeconomic models which have reduced forms that correspond to the two affine models and provide a macroeconomic interpretation of abstract state variables (as in Ang and Piazzessi [2003. A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables. Journal of Monetary Economics 50, 745-787.]). Moreover, such structural models allow us to interpret the parameters of the term-structure model in terms of the parameters governing preferences, technologies, and policy rules. We show how a monetary policy rule will manifest itself in the equilibrium asset-pricing kernel and, hence, the equilibrium term structure. We then show how this policy can be implemented with an interest-rate targeting rule. This provides us with a set of restrictions under which the Taylor and McCallum Rules are equivalent in the sense if implementing the same monetary policy. We conclude with some numerical examples that explore the quantitative link between these two models of monetary policy.     

Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

Using 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

Junp-Diffusion Interest Rate Process: An Empirical Examination

We investigate a jump-diffusion process, which is a mixture of an O-U process used by Vasicek (1977) and a compound Poisson jump process, for the term structure of interest rates. We develop a methodology for estimating the jump-diffusion model and complete an empirical study in comparing the model with the Vasicek model, for the US money market interest rates. The results show that when the short-term interest rate is low, both models predict an upward sloping term structure, with the jump-diffusion model fitting the actual term structure quite well and the Vasicek model overestimating significantly. When the short-term interest rate is high, both models predict a downward sloping term structure, with the jump-diffusion model underestimating the actual term structure more significantly than the Vasicek model.     

Yield curve modeling and forecasting using semiparametric factor dynamics

Using a dynamic semiparametric factor model (DSFM) we investigate the term structure of interest rates. The proposed methodology is applied to monthly interest rates for four southern European countries: Greece, Italy, Portugal and Spain from the introduction of the Euro to the recent European sovereign-debt crisis. Analyzing this extraordinary period, we compare our approach with the standard market method – dynamic Nelson–Siegel model. Our findings show that two nonparametric factors capture the spatial structure of the yield curve for each of the bond markets separately. We attributed both factors to the slope of the yield curve. For panel term structure data, three nonparametric factors are necessary to explain 95% variation. The estimated factor loadings are unit root processes and reveal high persistency. In comparison with the benchmark model, the DSFM technique shows superior short-term forecasting in times of financial distress.     

A comparison of diffusion models of the term structure

A number of different continuous time approaches that have been developed to model the term structure of interest rates are examined. These techniques span the interest rate literature over the last 20 years or so, and are the most commonly used among both academics and practitioners. We view this paper as a reference for the different term structure models, aiming to bring together the three most commonly used approaches, emphasizing their differences, analysing their respective advantages and disadvantages, and with explicit representations where they exist for prices of discount bonds.     

On the equivalence of a class of affine term structure models

In specifying a finite factor model for the term structure of interest rates, one usually begins by modeling the dynamics of the underlying factors. In most cases, this is sufficient to completely determine the term structure model. However, a point that is often overlooked is that seemingly different specifications of the factor dynamics may generate indistinguishable term structure models, in the sense that they produce pathwise identical bond prices. Consequently, it is important to be able to determine, at the level of factor dynamics, the conditions under which the models they generate are indistinguishable. In the case of time-homogeneous affine term structure models (ATSMs), such conditions were first described in Dai and Singleton (J Finance 55:1943–1978, 2000). In this paper, we formalize and extend their results to a class of time-inhomogeneous ATSMs, and obtain a simple method for determining the indistinguishability of these models in terms of the underlying factor dynamics.      

An examination of causality and predictability between Australian domestic and offshore interest rates

This paper studies the causality and predictability between Australian domestic and offshore short term interest rates in both the first and second moments during the period 1987 to 1996. Causality flow is observed to be stronger from the domestic to the offshore market in the earlier sub periods but characterised by significant two-way causality flow in the latter sub-periods. Volatility tests show that the volatility in one market spills over to the other market simultaneously, which is consistent with Australian markets being well integrated with global markets. The predictability across the two markets in the first moments is examined through an error correction model, whose forecasting performance is assessed relative to a benchmark random walk model. To test the predictability of volatility, four different models are compared: A GARCH model, A GARCH model incorporating contemporaneous spillover effects, a GARCH model with lagged spillover effects, and a benchmark random walk model. Results indicate that the error correction model and the GARCH model with contemporaneous volatility spillover are the superior models for forecasting changes in interest rates and for forecasting volatility, respectively.     

Yield-factor volatility models

The term structure of interest rates is often summarized using a handful of yield factors that capture shifts in the shape of the yield curve. In this paper, we develop a comprehensive model for volatility dynamics in the level, slope, and curvature of the yield curve that simultaneously includes level and GARCH effects along with regime shifts. We show that the level of the short rate is useful in modeling the volatility of the three yield factors and that there are significant GARCH effects present even after including a level effect. Further, we find that allowing for regime shifts in the factor volatilities dramatically improves the model’s fit and strengthens the level effect. We also show that a regime-switching model with level and GARCH effects provides the best out-of-sample forecasting performance of yield volatility. We argue that the auxiliary models often used to estimate term structure models with simulation-based estimation techniques should be consistent with the main features of the yield curve that are identified by our model.     

公开市场操作与利率期限结构行为——基于混频数据信息的研究视角

在混频数据信息环境中,精准识别公开市场操作(央行政策利率)和国债收益率曲线(基准利率体系)之间的关联机制至关重要,其影响了货币政策期限结构传导的有效性。本文在混频Nelson-Siegel(N-S)利率期限结构模型框架下,引入央行政策利率,揭示公开市场操作与利率期限结构(水平、斜率、曲度)因子之间的作用机制。实证结果表明:混频数据信息条件下,引入的公开市场操作信息显著改进国债收益率曲线的拟合效果;斜率因子冲击对公开市场操作具有显著的正向影响,而利率期限结构因子对政策调控的反应不敏感。进一步研究表明,2015年以来,公开市场操作对斜率因子的影响逐渐扩大,政策利率向国债收益率曲线的传导效率得到显著提高,我国现代货币政策框架日益健全。     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

基于参数模型的国债市场利率曲线的估计

即期利率和远期利率曲线是金融行业中最为基本和重要的工具。在对利率期限结构参数模型中被广泛运用的NS模型和Svensson模型进行比较分析的基础上,估计了我国国债市场的即期利率和远期利率曲线。实证研究表明,Svensson模型在以最小化收益率误差的估计方法下,能够较理想地构造中国国债市场的即期利率曲线和远期利率曲线。     

Are Indexed Bonds Really Inflation Proof? A Model of Real and Nominal Term Structures when Money has Real Effects

This paper studies the general behavior of the nominal and real term structures of interest rates in a general equilibrium framework. A central bank is introduced in the model as an agent facing a tradeoff between inflation and output and choosing a monetary policy variable. Prices and output are jointly determined in our model endogenously. Two multi-factor nominal and real term structure models are given as examples to illustrate the general model. In our economies, inflation indexed bonds are not completely inflation proof, but are still subject to the influence of inflation uncertainties. The models offer us an empirical framework that can be studied with indexed bond data and nominal bond data together in a single estimation.     

Accounting for a Shift in Term Structure Behavior with No-Arbitrage and Macro-Finance Models

This paper examines a shift in the dynamics of the term structure of interest rates in the United States during the mid-1980s. We document this shift using standard interest rate regressions and using dynamic, affine, no-arbitrage models estimated for the pre- and post-shift subsamples. The term structure shift largely appears to be the result of changes in the pricing of risk associated with a "level" factor. Using a macro-finance model, we suggest a link between this shift in term structure behavior and changes in the dynamics and risk pricing of the Federal Reserve's inflation target as perceived by investors.     

Factor models and the correlation structure of interest rates: Some evidence for USD,GBP, DEM and JPY

The aim of this paper is to develop models for producing accurate forecasts for the correlation of spot and forward interest rates. Correlation forecasts generated from factor models, where the correlations are expressed as a function of few underlying factors, are compared to forecasts produced by less sophisticated models like the full historical and constant correlation model. Contrary to previous studies, where the selection of factors is rather arbitrary, we test two factor identification methodologies. The first is based on factor analysis and the second is based on minimising the residual cross-correlations. We show that a three-factor model, with factors identified by minimising the residual cross-correlation criterion is sufficient to describe the correlation structure of interest rates. Furthermore, we show that it might be preferable to assume that all forward rate correlations are equal rather than using a misspecified factor model and that the spot factor model developed, consistently outperformed long established models like the “Barbell” model.     

An Unobserved Components Model of the Yield Curve

We develop an unobserved component model in which the short‐term interest rate is composed of a stochastic trend and a stationary cycle. Using the Nelson–Siegel model of the yield curve as inspiration, we estimate an extremely parsimonious state‐space model of interest rates across time and maturity. The time‐series process suggests a specific functional form for the yield curve. We use the Kalman filter to estimate the time‐series process jointly with observed yield curves, greatly improving empirical identification. Our stochastic process generates a three‐factor model for the term structure. At the estimated parameters, trend and slope factors matter while the third factor is empirically unimportant. Our baseline model fits the yield curve well. Model generated estimates of uncertainty are positively correlated with estimated term premia. An extension of the model with regime switching identifies a high‐variance regime and a low‐variance regime, where the high‐variance regime occurs rarely after the mid‐1980s. The term premium is higher, and more so for yields of short maturities, in the high‐variance regime than in the low‐variance regime. The estimation results support our model as a simple and yet reliable framework for modeling the term structure.     

人口学特征与利率期限结构:老年社会平缓的收益率曲线

本文将刻画人口结构的生命周期模型引入消费-资本资产定价模型,考察人口结构对利率期限结构的影响。模型表明,人口结构及其家庭生命周期特征不仅决定利率水平,而且将通过人对债券期限的不同偏好,影响利率期限结构。少年人口占比对利率期限结构的影响为正,中年和老年人口占比的影响为负。相比少年人口,中老年人口更偏好长期债券,使长期收益率下降,期限结构的斜率更为平缓。基于全球数据的经验研究验证了这一结论。少年人口占比增加期限利差,中老年人口占比则起反向作用。因此,在年长的经济体中,期限利差更小,呈现更平缓的收益率曲线特征。在更换人口结构变量、期限利差变量、估计方法、赋权样本和处理遗漏变量后,结果表现稳健。本文从人口学视角拓宽了利率期限结构的决定因素,揭示了老年经济体可能面临一个平缓的收益率曲线,而这说明老龄化还可能通过抑制短期投机和促进长期投资来提高长期经济发展质量。