Spot Rate Models Research Articles

Modeling spot rate using a realized stochastic volatility model with level effect and dynamic drift☆

This paper proposes a class of realized stochastic volatility model based on both various realized volatility measures and spot rate. It applies the realized stochastic volatility model (Takahashi, Omori, & Watanabe, 2009, and Koopman & Scharth, 2013) to the spot rate model with dynamic drift and level effect setups (RSVL). A jointly approximated maximum likelihood procedure is used to estimate this model. The simulation results show that the RSVL model can be consistently estimated and noise-and-jump-robust realized volatility measures improve the accuracy of the estimation. This study empirically investigates the Chinese interbank repo market with RSVL model, which manifested the advantage of taking the level effect and nonlinear drift into consideration. The noise-and-jump-robust realized volatility measures (e.g. subsample realized volatility and threshold pre-average realized volatility) decrease the volatility fitting error. The nonparametric testing suggests that the RSVL model with noise-and-jump-robust realized volatility measures has more power on forecasting excess kurtosis and fat tails and predicting dynamics of higher order autocorrelations.

Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model

The discrete-time multifactor Vasiček model is a tractable Gaussian spot rate model. Typically, two- or three-factor versions allow one to capture the dependence structure between yields with different times to maturity in an appropriate way. In practice, re-calibration of the model to the prevailing market conditions leads to model parameters that change over time. Therefore, the model parameters should be understood as being time-dependent or even stochastic. Following the consistent re-calibration (CRC) approach, we construct models as concatenations of yield curve increments of Hull–White extended multifactor Vasiček models with different parameters. The CRC approach provides attractive tractable models that preserve the no-arbitrage premise. As a numerical example, we fit Swiss interest rates using CRC multifactor Vasiček models.

Continuous Time Spot Rate Models with a Nonstationary Mean

We present a model in which the interest rate reverts to a nonstationary stochastic mean. In our model, the stochastic mean is subject to a sequence of permanent shocks that can better capture the source of nonlinearity in the drift than extant models. We find that our model provides a better fit to observed interest rates relative to a large number of alternative models. In particular, the Vasicek model with a nonstationary mean outperforms constant elasticity of volatility models. The implied bond option price from our model accounts for the downward bias in existing bond option valuation models.

A Simple Stochastic Rate Model for Rate Equity Hybrid Products

A positive spot rate model driven by a gamma process and correlated to equity is introduced and calibrated via closed forms for the joint characteristic function for the rate r, its integral y and the logarithm of the stock price s under the T-forward measure. The law of the triple (r,y,s) is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics, doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described in Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk, 14, pp. 95-122).

Expectation Hypothesis Bias: Risk Aversion versus Stochastic Adjustments

For a long time the classical expectation hypothesis has been challenged from both empirical and theoretical perspective. Still no one could explain entirely the existent bias between expected future spot rates and forward rates, the so called puzzle in the expectation hypothesis In this work we will address this issue through arbitrage theory, in particular, focusing our attention in a connection of the classical expectation hypothesis to a certain probability measures where the relation between expected future spot rates and forward rates holds. We will approach this applying certain instantaneous spot rate models, verifying in these models that through changes from \real world probability measure to other probability measures, we will nd adjustments that will able us to explain this bias in the expectation hypothesis.

A Simple Stochastic Rate Model for Rate Equity Hybrid Products

A positive spot rate model driven by a gamma process and correlated with equity is introduced and calibrated via closed forms for the joint characteristic function for the rate r, its integral y and the logarithm of the stock price s under the T-forward measure. The law of the triple is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics, 46(1), doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described by Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk, 14(1), pp. 95–122).

Structure of Spot Rates and Duration Hedging*

The present study proposes a three-factor model using spot rates as proxies for the state variables of the term structure of interest rates. Empirical analysis is carried out on the in-sample explanatory power and the out-of-sample prediction ability of spot-rate models, and comparison is made between the modified Macaulay duration and spot-rate duration hedging for bond portfolios. The results not only show that the optimal three-spot-rate model outperforms the optimal two-spot-rate model proposed by Elton et al. (Journal of Finance, 45, 1990, 629–642) with respect to explanation ability of unexpected changes in the term structure of interest rates, but also illustrate the importance of capturing the curvature characteristic of the term structure of interest rates for spot-rate duration hedging methods. Moreover, the impressive performance of three-spot-rate duration hedging implies that it is feasible to reduce the dimensions of state variables to three for the purposes of risk exposure prediction and risk management of bond portfolios.

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 “horserace” 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 1982-2008. 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.

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

Abstract 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.

Modeling the dynamics of Chinese spot interest rates

Using the daily data of Chinese 7-day repo rates from January 1, 1997 to December 31, 2008, this paper tests a variety of popular spot rate models, including single-factor diffusion, GARCH, Markov regime-switching and jump-diffusion models. We document that Chinese spot rates are subject to both market forces and administrative forces. GARCH, regime-switching and jump-diffusion models capture some important features of the dynamics of Chinese spot rates, but all models under study are overwhelmingly rejected. We further explore possible sources of model misspecification using diagnostic tests.

The Normal Class of Arbitrage-Free Spot-Rate Models - Implication and Implementation

In this paper we first show how to determine the T-forward adjusted risk-measure using the concept of fundamental solution to linear PDE’s. After that, relying on Fourier transformation we derive bond-and bond-option prices for the Extended Vasicek model from Hull and White (1990) and the Quadratic Interest Rate model. With respect to the Quadratic Interest Rate model we succeed in carrying the analysis much further than Jamshidian (1996).A special discrete time model - which in some cases is appropriate for the Quadratic Interest Rate model - is also derived.The last part of the paper analyse Monte Carlo techniques in connection with spot-rate models with a time-dependent drift. We also introduce a method - using the concept of forward induction - that constrain the Monte Carlo simulated spot-rate process for the matching of the initial yield-curve. For the pricing of path-dependent contingent claim, we deduce that, even though Monte Carlo is the natural method to use, it might not be the most efficient one - at least not when the spot-rate is Markovian.

Estimation and inference in the yield curve model with an instantaneous error term

Many variations exist of yield curve modeling based on the exponential components framework, but most do not consider the generating process of the error term. In this paper, we propose a method of yield curve estimation using an instantaneous error term generated with a standard Brownian motion. First, we add an instantaneous error term to Nelson and Siegel’s instantaneous forward rate model [C.R. Nelson, A.F. Siegel, Parsimonious modeling of yield curves, Journal of Business 60 (1987) 473–489]. Second, after differencing multiperiod spot rate models transformed using Nelson and Siegel’s instantaneous forward rate model [C.R. Nelson, A.F. Siegel, Parsimonious modeling of yield curves, Journal of Business 60 (1987) 473–489], we obtain a model with serially uncorrelated error terms because of independent increment properties of Brownian motion. As the error term in this model is heteroskedastic and not serially correlated, we can apply weighted least squares estimation techniques. That is, this specification of the error term does not lead to incorrect estimation methods. In an empirical analysis, we compare the instantaneous forward rate curves estimated by the proposed method and an existing method. We find that the shape from the proposed estimation equation differ from the latter method when fluctuations in the interest rate data used for the estimation are volatile.

Modeling the Dynamics of Chinese Spot Interest Rates

Using the daily data of Chinese 7-day repo rates from January 1, 1997 to December 31, 2008, this paper tests a variety of popular spot rate models, including single-factor diffusion, GARCH, Markov regime-switching and jump-diffusion models. We document that Chinese spot rates are subject to both market forces and administrative forces. GARCH, regime-switching and jump-diffusion models capture some important features of the dynamics of Chinese spot rates, but all models under study are overwhelmingly rejected. We further explore possible sources of model misspecification using diagnostic tests.

A Comparison of Fixed Income Valuation Models: Pricing and Econometric Analysis of Interest Rate Derivatives

This study compares continuous-time stochastic interest rate and stochastic volatility models of interest rate derivatives, examining these models across several dimensions: different classes of models, factor structures, and pricing algorithms. We consider a broader universe of pricing models, using improved econometric and numerical methodologies. We establish several criteria for model quality that are motivated by financial theory as well as practice: realism of the assumed stochastic process for the term structure, consistency with no-arbitrage or financial market equilibrium, consistency with financial practice, parsimony, as well as computational efficiency. A model which scores well along these grounds will also exhibit superior pricing performance with regard to traded interest rate options. This helps resolve the controversies over the stochastic process for yield curve dynamics, the models that best manage and measure interest rate risk, and theories of the term structure that are supported by empirical results. We perform econometric experiments at three levels: the short rate, bond prices, as well as interest rate derivatives. We extend CKLS (1992) to a broader class of single factor spot rate models and international interest rates. We find that a single-factor general parametric model (1FGPM) of the term structure, with non-linearity in the drift function, better captures the time series dynamics of US 30 Day T-Bill rates. The 1FGPM not only forecasts interest rate changes out-of-sample better relative to other parametric models, but also relative to the non-parametric model of Jiang (1998). Finally, our results vary greatly across international markets. Building upon the work of Longstaff and Schwartz (1992), we perform a statistical analysis of the U.S. default-free term structure over the period 4:1964 to 10:1997. We utilize a constant correlation multivariate GARCH principal components analysis (CCM-PCA), and identify at least three factors associated with traditional measures of risk in the fixed income literature (level, slope, and curvature) that capture 98% of the variation in the default-free term structure. We perform tests of various term structure models on US Treasury bonds, comparing a two factor Cox-Ingersoll-Ross (2FCIR) model with a multi-layer perceptron neural network approach (MLP-ANN), in pricing and hedging discount bonds. We find that while the MLP-ANN can better fit bond prices in-sample, the 2F-CIR model is superior in hedging against unanticipated changes in the short rate and its volatility. Furthermore, we find the 2FCIR model to perform favorably in comparison to the CCM-PCA, MLP-ANN, as well as the 1FGPM in forecasting bond yield changes. Finally, we compare various interest rate bond option pricing models, in their ability to price interest rate derivatives and manage and interest rate risk. We compare three approaches to pricing interest rate derivatives: spot rate (e.g., CIR), forward-rate (i.e., HJM), and non-parametric models (e.g., multivariate kernel estimation.) This is extended to a broader factor structure. While the best model in terms of mean square error (MSE) is the non parametric (MNWK) model, the 3 factor jump diffusion (3FGJD) model performs best among parametric models. In hedging analysis, while these preferred models still outperform within each grouping, the non parametric model is no longer the best performing model, while the 2FCIR is the best model in hedging options in terms of MSE.

Bayesian, MLE, and GMM Estimation of a Spot Rate Model

ABSTRACT We develop Markov chain Monte Carlo algorithms for estimating the parameters of the short-term interest rate model. Using Monte Carlo experiments we compare the Bayes estimators with the maximum likelihood and generalized method of moments estimators. We estimate the model using the Japanese overnight call rate data.

Out-of-Sample Performance of Discrete-Time Spot Interest Rate Models

We provide a comprehensive analysis of the out-of-sample performance of a wide variety of spot rate models in forecasting the probability density of future interest rates. Although the most parsimonious models perform best in forecasting the conditional mean of many financial time series, we find that the spot rate models that incorporate conditional heteroscedasticity and excess kurtosis or heavy tails have better density forecasts. Generalized autoregressive conditional heteroscedasticity significantly improves the modeling of the conditional variance and kurtosis, whereas regime switching and jumps improve the modeling of the marginal density of interest rates. Our analysis shows that the sophisticated spot rate models in the existing literature are important for applications involving density forecasts of interest rates.

A Multifactor Spot Rate Model for the Pricing of Interest Rate Derivatives

We propose a multifactor model in which the spot rate, LIBOR, follows a lognormal process, with a stochastic conditional mean, under the risk-neutral measure. In addition to the spot rate factor, the second factor is related to the premium of the first futures rate over the spot LIBOR. Similarly, the third factor is related to the premium of the second futures rate over the first futures rate. We calibrate the model to the initial term structure of futures rates and to the implied volatilities of interest rate caplets. We then apply the model to price interest rate derivatives such as European- and Bermudan-style swaptions, and yieldspread options. The model can be employed to price more complex interest rate derivatives such as path-dependent derivatives or multi-currency-dependent derivatives because of its Markovian property.

Nonparametric Specification Testing for Continuous-Time Models with Application to Spot Interest Rates

We propose two nonparametric transition density-based speciþcation tests for continuous-time diffusion models. In contrast to marginal density as used in the literature, transition density can capture the full dynamics of a diffusion process, and in particular, can distinguish processes with the same marginal density but different transition densities. To address the concerns of the þnite sample performance of nonparametric methods in the literature, we introduce an appropriate data transformation and correct the boundary bias of kernel estimators. As a result, our tests are robust to persistent dependence in data and provide reliable inferences for sample sizes often encountered in empirical þnance. Simulation studies show that our tests have reasonable size and good power against a variety of alternatives in þnite samples even for data with highly persistent dependence. Besides the single-factor diffusion models, our tests can be applied to a broad class of dynamic economic models, such as discrete time series models, time-inhomogeneous diffusion models, stochastic volatility models, jump-diffusion models, and multi-factor term structure models. When applied to daily Eurodollar interest rates, our tests overwhelmingly reject some popular spot rate models, including those with nonlinear drifts that some existing tests can not reject after correcting size distortions. We þnd that models with nonlinear drifts do not signiþcantly improve the goodness-of-þt, and the main source of model inadequacy seems to be the violation of the Markov assumption. We also þnd that GARCH, regime switching and jump diffusion models perform signiþcantly better than single-factor diffusion models, although they are far from being adequate to fully capture the interest rate dynamics. Our study shows that nonparametric methods are a reliable and powerful tool for analyzing þnancial data.

The Valuation of Caps, Floors and Swaptions in a Multi-Factor Spot-Rate Model

We build a multi-factor, no-arbitrage model of the term structure of interest rates. The stochastic factors are the short-term interest rate and the premia of the futures rates over the short-term interest rate. In the three-factor version of the model, for example, the first factor is the three-month LIBOR, the second factor is the premium of the first futures LIBOR over spot LIBOR, and the third factor is the incremental premium of the second futures over the first. The model provides an extension of the lognormal interest rate model of Black and Karasinski (1991) to multiple factors, each of which can exhibit mean-reversion. The method is computationally efficient for several reasons. First, since our model is based on LIBOR futures prices, we can satisfy the no-arbitrage condition without resorting to iterative methods. Second, we modify and implement the binomial approximation methodology of Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrahmanyam (1995) to compute a multi-period tree of rates with the no-arbitrage property. The method uses a recombining two or three-dimensional binomial lattice of interest rates that minimizes the number of states and term structures over time. In addition to these computational advantages, a key feature of the model is that it is consistent with the observed term structure of futures rates as well as the term structure of volatilities implied by the prices of interest rate caps and floors. We use the model to price European-style, Bermudan-style, and American-style swaptions. To implement the methodology, we first calibrate the model to the caplet implied-volatility curve on a given day, and then use the model to price European-style swaptions. We find that the two-factor model, where the LIBOR mean reverts rapidly to a slowly mean-reverting second factor overprices the swaptions relative to market quotations. However, introducing a third factor significantly reduces the overpricing. Finally, we re-calibrate the two-factor model simultaneously to caplet and swaption prices and use the model output to price Bermudan-style swaptions.

Panel Data Analysis of Japanese Government Bond Market

The Cox-Ingersoll-Ross and Vasicek models are two broadly implemented spot rate models. There does not seem to be any comparison of the two models based on time series data of all maturity yields (panel data). It is difficult to compare them in the same circumstances because the non-central x-square distribution of the CIR model should be approximated by an appropriate normal distribution in application of a Kalman filter to panel data while the distribution of the Vasicek model follows a normal distribution without any approximation. Using a semiparametric estimation technique that does not assume any distribution for the models, the authors compare results for the panel data of the JGB market. They also examine whether two-factor extension of the models is effective and appropriate.