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.     

Modeling interest rate volatility: A Realized GARCH approach

We propose using a Realized GARCH (RGARCH) model to estimate the daily volatility of the short-term interest rate in the euro–yen market. The model better fits the data and provides more accurate volatility forecasts by extracting additional information from realized measures. In addition, we propose using the ARMA–Realized GARCH (ARMA–RGARCH) model to capture the volatility clustering and the mean reversion effects of interest rate behavior. We find the ARMA–RGARCH model fits the data better than the simple RGARCH model does, but it does not provide superior volatility forecasts.     

Can interest rate volatility be extracted from the cross section of bond yields?

Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.     

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.     

The affine styled-facts price dynamics for the natural gas: evidence from daily returns and option prices

This study analyzes affine styled-facts price dynamics of Henry Hub natural gas price by incorporating the price features of jump risk, and seasonality within stochastic volatility framework. Affine styled-facts dynamics has the advantage of being able to incorporate mean reversion (MR), stochastic volatility (SV), seasonality trends (S), and jump diffusion (J) in a standardized inclusive framework. Our main finding is that models that incorporate jumps significantly improve overall out-of-sample option pricing performance. The combined MRSVJS model provides the best fit of both daily gas price returns and the related cross section of option prices. Incorporating seasonal effects tend to provide more stable pricing ability, especially for the long-term option contracts.     

REGIME-SWITCHING IN AUSTRALIAN SHORT-TERM INTEREST RATES

This paper applies a generalized regime-switching (GRS) model of the short-term interest rate to Australian data. The model allows the short rate to exhibit both mean reversion and conditional heteroscedasticity and nests the popular generalized autoregressive conditional heteroscedasticity (GARCH) and regime-switching specifications. It is shown that empirical estimates of many popular interest rate models provide curious results which imply that innovations to the short rate process are extremely persistent, and that the short rate is potentially non-stationary. The source of these curious results, which are also present in US and European interest rates, is identified in the context of the GRS model, which is shown, via specification and forecasting tests, to capture the features of Australian short-term interest rate data better than existing models. The stochastic process of short-term interest rates in Australia is compared with evidence from the US and Europe, highlighting a number of important differences.     

Option-implied volatility factors and the cross-section of market risk premia

The main goal of this paper is to study the cross-sectional pricing of market volatility. The paper proposes that the market return, diffusion volatility, and jump volatility are fundamental factors that change the investors’ investment opportunity set. Based on estimates of diffusion and jump volatility factors using an enriched dataset including S&P 500 index returns, index options, and VIX, the paper finds negative market prices for volatility factors in the cross-section of stock returns. The findings are consistent with risk-based interpretations of value and size premia and indicate that the value effect is mainly related to the persistent diffusion volatility factor, whereas the size effect is associated with both the diffusion volatility factor and the jump volatility factor. The paper also finds that the use of market index data alone may yield counter-intuitive results.     

Volatility clustering,leverage effects,and jump dynamics in the US and emerging Asian equity markets

This paper proposes asymmetric GARCH-Jump models that synthesize autoregressive jump intensities and volatility feedback in the jump component. Our results indicate that these models provide a better fit for the dynamics of the equity returns in the US and emerging Asian markets, irrespective whether the volatility feedback is generated through a common GARCH multiplier or a separate measure of volatility in the jump intensity function. We also find that they can capture several distinguishing features of the return dynamics in emerging markets, such as, more volatility persistence, less leverage effects, fatter tails, and greater contribution and variability of the jump component.     

Additional evidence on transparency and bank financial performance

This paper examines the relationship between volatility and the probability of occurrence of expected extreme returns in the Canadian market. Four measures of volatility are examined: implied volatility from firm option prices, conditional volatility calculated using an EGARCH model, idiosyncratic volatility, and expected shortfall. A significantly positive relationship is observed between a firm's idiosyncratic volatility and the probability of occurrence of an extreme return in the subsequent month for firms. A 10% increase in idiosyncratic volatility in a given month is associated with the probability of an extreme shock in the subsequent month (top or bottom 1.5% of the returns distribution) of 26.4%. Other firm characteristics, including firm age, price, volume and book‐to‐market ratio, are also shown to be significantly related to subsequent firm extreme returns. The effects of conditional and implied volatility are mixed. The E‐GARCH and expected shortfall measures of conditional volatility are consistent with mean reversion: high short term realizations of conditional volatility foreshadow a lower probability of extreme returns.     

Multivariate GARCH Modeling of Exchange Rate Volatility Transmission in the European Monetary System

We construct a series of 3‐, 4‐ and 5‐variable multivariate GARCH models of exchange rate volatility transmission across the important European Monetary System (EMS) currencies including the French franc, the German mark, the Italian lira, and the European Currency Unit. The models are estimated without imposing the common restriction of constant correlation on both daily and weekly data from April 1979–March 1997. Our results indicate the importance of checking for specification robustness in multivariate Generalized Autoregressive Conditional Heleroskedasticity (GARCH) modeling, we find that increased temporal aggregation reduces observed volatility transmission, and that the mark plays a dominant position in terms of volatility transmission.     

Modelling and forecasting short-term interest rate volatility: A semiparametric approach

This paper employs a semiparametric procedure to estimate the diffusion process of short-term interest rates. The Monte Carlo study shows that the semiparametric approach produces more accurate volatility estimates than models that accommodate asymmetry, level effect and serial dependence in the conditional variance. Moreover, the semiparametric approach yields robust volatility estimates even if the short rate drift function and the underlying innovation distribution are misspecified. Empirical investigation with the U.S. three-month Treasury bill rates suggests that the semiparametric procedure produces superior in-sample and out-of-sample forecast of short rate changes volatility compared with the widely used single-factor diffusion models. This forecast improvement has implications for pricing interest rate derivatives.     

Forecasting volatility of crude oil futures using a GARCH–RNN hybrid approach

Volatility is an important element for various financial instruments owing to its ability to measure the risk and reward value of a given financial asset. Owing to its importance, forecasting volatility has become a critical task in financial forecasting. In this paper, we propose a suite of hybrid models for forecasting volatility of crude oil under different forecasting horizons. Specifically, we combine the parameters of generalized autoregressive conditional heteroscedasticity (GARCH) and Glosten–Jagannathan–Runkle (GJR)-GARCH with long short-term memory (LSTM) to create three new forecasting models named GARCH–LSTM, GJR-LSTM, and GARCH-GJRGARCH LSTM in order to forecast crude oil volatility of West Texas Intermediate on different forecasting horizons and compare their performance with the classical volatility forecasting models. Specifically, we compare the performances against existing methodologies of forecasting volatility such as GARCH and found that the proposed hybrid models improve upon the forecasting accuracy of Crude Oil: West Texas Intermediate under various forecasting horizons and perform better than GARCH and GJR-GARCH, with GG–LSTM performing the best of the three proposed models at 7-, 14-, and 21-day-ahead forecasts in terms of heteroscedasticity-adjusted mean square error and heteroscedasticity-adjusted mean absolute error, with significance testing conducted through the model confidence set showing that GG–LSTM is a strong contender for forecasting crude oil volatility under different forecasting regimes and rolling-window schemes. The contribution of the paper is that it enhances the forecasting ability of crude oil futures volatility, which is essential for trading, hedging, and purposes of arbitrage, and that the proposed model dwells upon existing literature and enhances the forecasting accuracy of crude oil volatility by fusing a neural network model with multiple econometric models.     

The role of jump dynamics in the risk–return relationship

Surprisingly, a positive risk–return relationship has not been consistently observed for the traditional GARCH in the mean model in other studies. In this paper, we employ a combination of the jump diffusion and GARCH model in the mean equation to test the risk–return relationship for U.S. stock returns. The results suggest a statistically significant relationship between risk and return if the risk measure includes components of smoothly changing variance and jump events.     

Forecasting Commodity Market Returns Volatility: A Hybrid Ensemble Learning GARCH-LSTM based Approach

This study investigates the advantage of combining the forecasting abilities of multiple generalized autoregressive conditional heteroscedasticity (GARCH)-type models, such as the standard GARCH (GARCH), exponential GARCH (eGARCH), and threshold GARCH (tGARCH) models with advanced deep learning methods to predict the volatility of five important metals (nickel, copper, tin, lead, and gold) in the Indian commodity market. This paper proposes integrating the forecasts of one to three GARCH-type models into an ensemble learning-based hybrid long short-term memory (LSTM) model to forecast commodity price volatility. We further evaluate the forecasting performance of these models for standalone LSTM and GARCH-type models using the root mean squared error, mean absolute error, and mean fundamental percentage error. The results highlight that combining the information from the forecasts of multiple GARCH types into a hybrid LSTM model leads to superior volatility forecasting capability. The SET-LSTM, which represents the model that combines forecasts of the GARCH, eGARCH, and tGARCH into the LSTM hybrid, has shown the best overall results for all metals, barring a few exceptions. Moreover, the equivalence of forecasting accuracy is tested using the Diebold–Mariano and Wilcoxon signed-rank tests.     

Uncertainty in Value-at-risk Estimates under Parametric and Non-parametric Modeling

This study evaluates a set of parametric and non-parametric value-at-risk (VaR) models that quantify the uncertainty in VaR estimates in form of a VaR distribution. We propose a new VaR approach based on Bayesian statistics in a GARCH volatility modeling environment. This Bayesian approach is compared with other parametric VaR methods (quasi-maximum likelihood and bootstrap resampling on the basis of GARCH models) as well as with non-parametric historical simulation approaches (classical and volatility adjusted). All these methods are evaluated based on the frequency of failures and the uncertainty in VaR estimates.Within the parametric methods, the Bayesian approach is better able to produce adequate VaR estimates, and results mostly in a smaller VaR variability. The non-parametric methods imply more uncertain 99%-VaR estimates, but show good performance with respect to 95%-VaRs.     

The Impact of Jumps in Volatility and Returns

This paper examines continuous‐time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood‐based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we find strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.     

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.     

Modeling the Euro overnight rate

This paper describes the evolution of the daily Euro overnight interest rate (EONIA) by using several models containing the jump component, such as a single-regime ARCH-Poisson–Gaussian process, with either a piecewise function or an autoregressive conditional specification (ARJI) for the jump intensity, and a two-regime-switching process with jumps and time-varying transition probabilities. To model the jump intensity, we include the following effects which are significant for the occurrence of jumps: (1) the end of maintenance period effect because of reserve requirements, (2) the end of month effect, also known as the calendar day effect, caused mainly by accounting adjustments and finally, (3) the meeting effect caused by the meetings of the Governing Council of the European Central Bank (ECB). These effects lead to better performance and several of them are also included for the behavior of the transition probabilities. Since the target of the ECB is to maintain the EONIA rate close to the policy rate, we model the conditional mean of the overnight rate series as a reversion process to this policy rate, distinguishing two alternative speeds of reversion, specifically, a different speed if EONIA is higher or lower than the policy rate. We also study the jumps of the EONIA rate around the ECB's meetings by using the ex-post probabilities of the ARJI model. Finally, we develop a volatility forecasting analysis to measure the performance of the different candidate models.     

Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices

This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state-dependent arrival intensity. We discuss how to perform likelihood-based inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.     

Target zones and conditional volatility: The role of realignments

This paper examines the relationship between the conditional volatility of target zone exchange rates and realignments of the system. To investigate this question, modified jump-diffusion Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and absolute value GARCH models are fit to six exchange rates of the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS). Time-varying jump probability and absolute value GARCH models are effective in improving the fit of jump-diffusion models on target zone data. There is some evidence that conditional volatility is higher around the periods of realignments.