Modelling conditional heteroscedasticity and jumps in Australian short-term interest rates

The present paper explores a class of jump–diffusion models for the Australian short‐term interest rate. The proposed general model incorporates linear mean‐reverting drift, time‐varying volatility in the form of LEVELS (sensitivity of the volatility to the levels of the short‐rates) and generalized autoregressive conditional heteroscedasticity (GARCH), as well as jumps, to match the salient features of the short‐rate dynamics. Maximum likelihood estimation reveals that pure diffusion models that ignore the jump factor are mis‐specified in the sense that they imply a spuriously high speed of mean‐reversion in the level of short‐rate changes as well as a spuriously high degree of persistence in volatility. Once the jump factor is incorporated, the jump models that can also capture the GARCH‐induced volatility produce reasonable estimates of the speed of mean reversion. The introduction of the jump factor also yields reasonable estimates of the GARCH parameters. Overall, the LEVELS–GARCH–JUMP model fits the data best.     

Long term dependence in stock returns

We show, using the modified rescaled range statistic, that none of the return series of indices of five European countries, the United States and Japan exhibits long term dependence. This statistic — introduced by Lo (1991) — correct Hurst's (1951) ‘classical’ rescaled range statistic for short term dependence. We also report the classical rescaled range statistic after adjusting the series for short term dependence. This procedure shows, for cases where the results of the modified rescaled range statistic are mixed, that no long term dependence can be found. Simulations indicate reasonable power of this adjustment procedure. Furthermore, we find that estimates of the Hurst exponent, a related measure of long term dependence, are also biased by short term dependence. Simulations show that this measure — that has recently attracted growing interest — cannot distinguish between models with or without long term dependence.     

A model of discontinuous interest rate behavior,yield curves,and volatility

This paper develops an equilibrium model in which interest rates follow a discontinuous (generalized) gamma process. The gamma process has finite variation, takes an infinite number of “small” jumps in every interval, and includes the Wiener process as a limiting case. The gamma interest rate model produces yield curves that closely resemble those of diffusion models. But in contrast to diffusion models, the curvature of the yield curve does not directly depend on the true volatility of the interest rate process, but instead depends on a different risk-neutral volatility. The gamma model appears to fit the distribution of interest rates changes and the jump characteristics of interest rate paths. Empirical tests reject a diffusion model of interest rates in favor of the more general gamma model because daily interest rate innovations are highly leptokurtic. The author appreciates comments from George Constantinides, Jon Ingersoll, Herbert Johnson, Ray Rishel, and an anonymous referee, computational assistance from Kerry Back and Saikat Nandi, and support from Atlantic Asset Management. Any errors are the responsibility of the author.     

Central bank intervention and the volatility of foreign exchange rates: evidence from the options market

This paper tests the effects of central bank intervention on the ex ante volatility of $/DM and $/yen exchange rates between 1985 and 1991. In contrast to previous research which employed GARCH estimates of conditional volatility, we estimate ex ante volatility using the implied volatilities of currency option prices. We also control for the effects of other macroeconomic announcements. We find little support for the hypothesis that central bank intervention decreases expected exchange rate volatility. Instead, central bank intervention is generally associated with a positive change in ex ante exchange rate volatility, or with no change.     

A cyclical model of exchange rate volatility

In this paper, we investigate the long run dynamics of the intraday range of the GBP/USD, JPY/USD and CHF/USD exchange rates. We use a non-parametric filter to extract the low frequency component of the intraday range, and model the cyclical deviation of the range from the long run trend as a stationary autoregressive process. We use the cyclical volatility model to generate out-of-sample forecasts of exchange rate volatility for horizons of up to 1 year under the assumption that the long run trend is fully persistent. As a benchmark, we compare the forecasts of the cyclical volatility model with those of the range-based EGARCH and FIEGARCH models of Brandt and Jones (2006). Not only does the cyclical volatility model provide a very substantial computational advantage over the EGARCH and FIEGARCH models, but it also offers an improvement in out-of-sample forecast performance.     

Are there nonlinearities in short‐term interest rates?

The present paper investigates the characteristics of short‐term interest rates in several countries. We examine the importance of nonlinearities in the mean reversion and volatility of short‐term interest rates. We examine various models that allow the conditional mean (drift) and conditional variance (diffusion) to be functions of the current short rate. We find that different markets require different models. In particular, we find evidence of nonlinear mean reversion in some of the countries that we examine, linear mean reversion in others and no mean reversion in some countries. For all countries we examine, there is strong evidence of the need for the volatility of interest rate changes to be highly sensitive to the level of the short‐term interest rate. Out‐of‐sample forecasting performance of one‐factor short rate models is poor, stemming from the inability of the models to accommodate jumps and discontinuities in the time series data.     

Volatility transmission in emerging European foreign exchange markets

This paper studies the dynamics of volatility transmission between Central European (CE) currencies and the EUR/USD foreign exchange using model-free estimates of daily exchange rate volatility based on intraday data. We formulate a flexible yet parsimonious parametric model in which the daily realized volatility of a given exchange rate depends both on its own lags as well as on the lagged realized volatilities of the other exchange rates. We find evidence of statistically significant intra-regional volatility spillovers among the CE foreign exchange markets. With the exception of the Czech and, prior to the recent turbulent economic events, Polish currencies, we find no significant spillovers running from the EUR/USD to the CE foreign exchange markets. To measure the overall magnitude and evolution of volatility transmission over time, we construct a dynamic version of the Diebold–Yilmaz volatility spillover index and show that volatility spillovers tend to increase in periods characterized by market uncertainty.     

Bayesian comparison of several continuous time models of the Australian short rate

This paper provides an empirical analysis of a range of alternative single‐factor continuous time models for the Australian short‐term interest rate. The models are nested in a general single‐factor diffusion process for the short rate, with each alternative model indexed by the level effect parameter for the volatility. The inferential approach adopted is Bayesian, with estimation of the models proceeding through a Markov chain Monte Carlo simulation scheme. Discrimination between the alternative models is based on Bayes factors. A data augmentation approach is used to improve the accuracy of the discrete time approximation of the continuous time models. An empirical investigation is conducted using weekly observations on the Australian 90 day interest rate from January 1990 to July 2000. The Bayes factors indicate that the square root diffusion model has the highest posterior probability of all models considered.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

Explaining exchange rate volatility: an empirical analysis of ‘the holy trinity’ of monetary independence, fixed exchange rates, and capital mobility

This paper uses a panel of data from twenty-two countries between 1967 and 1992 to explain exchange rate volatility, focusing on potential tradeoffs between fixed exchange rates, independent monetary policy, and capital mobility. I use monetary models to parameterize monetary divergence and factor analysis to measure capital mobility. Exchange rate volatility is loosely linked to both monetary divergence and the degree of capital mobility. Interestingly, exchange rate volatility is significantly correlated with the width of the explicitly declared exchange rate band, even after taking monetary divergence and capital mobility into account.     

Is long memory necessary? An empirical investigation of nonnegative interest rate processes

This paper analyzes a class of nonnegative processes for the short-term interest rate. The dynamics of interest rates and yields are driven by the dynamics of the conditional volatility of the pricing kernel. We study Markovian interest rate processes as well as more general non-Markovian processes that display “short” and “long” memory. These processes also display heteroskedasticity patterns that are more general than those of existing models. We find that deviations from the Markovian structure significantly improve the empirical performance of the model. Certain aspects of the long memory effect can be captured with a (less parsimonious) short memory parameterization, but a simulation experiment suggests that the implied term structures corresponding to the estimated long- and short-memory specifications are very different. We also find that the choice of proxy for the short rate affects the estimates of heteroskedasticity patterns.     

Conditional beta pricing models: A nonparametric approach

We propose a two-stage procedure to estimate conditional beta pricing models that allows for flexibility in the dynamics of asset betas and market prices of risk (MPR). First, conditional betas are estimated nonparametrically for each asset and period using the time-series of previous data. Then, time-varying MPR are estimated from the cross-section of returns and betas. We prove the consistency and asymptotic normality of the estimators. We also perform Monte Carlo simulations for the conditional version of the three-factor model of Fama and French (1993) and show that nonparametrically estimated betas outperform rolling betas under different specifications of beta dynamics. Using return data on the 25 size and book-to-market sorted portfolios, we find that the nonparametric procedure produces a better fit of the three-factor model to the data, less biased estimates of MPR and lower pricing errors than the Fama–MacBeth procedure with betas estimated under several alternative parametric specifications.     

Option Prices, Implied Price Processes, and Stochastic Volatility

This paper characterizes all continuous price processes that are consistent with current option prices. This extends Derman and Kani (1994), Dupire (1994, 1997), and Rubinstein (1994), who only consider processes with deterministic volatility. Our characterization implies a volatility forecast that does not require a specific model, only current option prices. We show how arbitrary volatility processes can be adjusted to fit current option prices exactly, just as interest rate processes can be adjusted to fit bond prices exactly. The procedure works with many volatility models, is fast to calibrate, and can price exotic options efficiently using familiar lattice techniques.     

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.     

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.     

The Stochastic Volatility of Short-Term Interest Rates: Some International Evidence

This paper estimates a stochastic volatility model of short-term riskless interest rate dynamics. Estimated interest rate dynamics are broadly similar across a number of countries and reliable evidence of stochastic volatility is found throughout. In contrast to stock returns, interest rate volatility exhibits faster mean-reverting behavior and innovations in interest rate volatility are negligibly correlated with innovations in interest rates. The less persistent behavior of interest rate volatility reflects the fact that interest rate dynamics are impacted by transient economic shocks such as central bank announcements and other macroeconomic news.     

The incremental volatility information in one million foreign exchange quotations

The volatility information found in high-frequency exchange rate quotations and in implied volatilities is compared by estimating ARCH models for DM/$ returns. Reuters quotations are used to calculate five-minute returns and hence hourly and daily estimates of realised volatility that can be included in equations for the conditional variances of hourly and daily returns. The ARCH results show that there is a significant amount of information in five-minute returns that is incremental to options information when estimating hourly variances. The same conclusion is obtained by an out-of-sample comparison of forecasts of hourly realised volatility.     

Functional gradient descent for financial time series with an application to the measurement of market risk

The estimation and forecast of the volatility matrix are two of the main tasks of financial econometrics since they are essential ingredients in many practical applications. Unfortunately the use of classical multivariate methods in large dimensions is difficult because of the curse of dimensionality. We present a general semiparametric technique, based on functional gradient descent (FGD) and able to overcome most problems associated with a multivariate GARCH-type estimation. By testing the accuracy of the volatility estimates for the measurement of market risk on real data we provide empirical evidence of the strong predictive potential of the FGD approach, also in comparison to other standard methods.     

Estimating continuous-time stochastic volatility models of the short-term interest rate: a comparison of the generalized method of moments and the Kalman filter

This paper examines a model of short-term interest rates that incorporates stochastic volatility as an independent latent factor into the popular continuous-time mean-reverting model of Chan et al. (J Financ 47:1209–1227, 1992). I demonstrate that this two-factor specification can be efficiently estimated within a generalized method of moments (GMM) framework using a judicious choice of moment conditions. The GMM procedure is compared to a Kalman filter estimation approach. Empirical estimation is implemented on US Treasury bill yields using both techniques. A Monte Carlo study of the finite sample performance of the estimators shows that GMM produces more heavily biased estimates than does the Kalman filter, and with generally larger mean squared errors.     

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.