Modeling the conditional mean and variance of the short rate using diffusion,GARCH, and moving average models

This article introduces a two‐factor‐discrete‐time‐stochastic‐volatility model that allows for departures from linearity in the conditional mean and incorporates serially correlated unexpected news, asymmetry, and level effects into the definition of conditional volatility of the short rate. The new class of econometric specifications nests many popular existing symmetric and asymmetric GARCH as well as diffusion models of the short‐term interest rate. This study attempts to determine the correct specification of conditional mean and variance of the short rate by developing a more general econometric framework that allows for nonlinear effects in the drift of the short rate, and that defines the conditional volatility as a nonlinear function of unexpected information shocks and interest rate levels. The existing and alternative models are compared in terms of their ability to capture the stochastic behavior of the short‐term riskless rate. The empirical results indicate that the relative performance of the two‐factor models in predicting the future level and variance of interest‐rate changes is superior to the nested models. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:717–751, 2000     

Nonlinear asymmetric models of the short‐term interest rate

This study introduces a generalized discrete time framework to evaluate the empirical performance of a wide variety of well‐known models in capturing the dynamic behavior of short‐term interest rates. A new class of models that displays nonlinearity and asymmetry in the drift, and incorporates the level effect and stochastic volatility in the diffusion function is introduced in discrete time and tested against the popular diffusion, GARCH, and level‐GARCH models. Based on the statistical test results, the existing models are strongly rejected in favor of the newly proposed models because of the nonlinear asymmetric drift of the short rate, and the presence of nonlinearity, GARCH, and level effects in its volatility. The empirical results indicate that the nonlinear asymmetric models are better than the existing models in forecasting the future level and volatility of interest rate changes. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:869–894, 2006     

Short Rate Dynamics and Regime Shifts*

We characterize the dynamics of the US short‐term interest rate using a Markov regime‐switching model. Using a test developed by Garcia, we show that there are two regimes in the data: In one regime, the short rate behaves like a random walk with low volatility; in another regime, it exhibits strong mean reversion and high volatility. In our model, the sensitivity of interest rate volatility to the level of interest rate is much lower than what is commonly found in the literature. We also show that the findings of nonlinear drift in Aït‐Sahalia and Stanton, using nonparametric methods, are consistent with our regime‐switching model.     

Modeling discontinuous periodic conditional volatility: Evidence from the commodity futures market

This paper examines a wide variety of models that allow for complex and discontinuous periodic variation in conditional volatility. The value of these models (including augmented versions of existing models) is demonstrated with an application to high frequency commodity futures return data. Their use is necessary, in this context, because commodity futures returns exhibit discontinuous intraday and interday periodicities in conditional volatility. The former of these effects is well documented for various asset returns; however, the latter is unique amongst commodity futures returns, where contract delivery and climate are driving forces. Using six years of high‐frequency cocoa futures data, the results show that these characteristics of conditional return volatility are most adequately captured by a spline‐version of the periodic generalized autoregressive conditional heteroscedastic (PGARCH) model. This model also provides superior forecasts of future return volatility that are robust to variation in the loss function assumed by the user, and are shown to be beneficial to users of Value‐at‐Risk (VaR) models. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:805–834, 2004     

A Note on the Nelson–Siegel Family

We study a problem posed in Bj"ork and Christensen (1999): Does there exist any nontrivial interest rate model that is consistent with the Nelson–Siegel family? They show that within the Heath–Jarrow–Morton framework with deterministic volatility structure the answer is no. In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegel family. We characterize the consistent state space Itô processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Itô process.     

Monetary policy and long‐term interest rates: Evidence from the U.S. economy

This paper addresses the ability of central banks to affect the structure of interest rates. We assess the causal relationship between the short‐term Effective Federal Funds Rate (FF) and long‐term interest rates associated with both public and private bonds and specifically, the 10‐Year Treasury Bond (GB10Y) and the Moody's Aaa Corporate Bond (AAA). To do this, we apply Structural Vector Autoregressive models to U.S. monthly data for the 1954–2018 period. Based on results derived from impulse response functions and forecast error variance decomposition, we find: a bidirectional relationship when GB10Y is considered as the long‐term rate and a unidirectional relationship that moves from short‐ to long‐term interest rates when AAA is considered. These conclusions show that monetary policy is able to permanently affect long‐term interest rates and the central bank has a certain degree of freedom in setting the levels of the short‐term policy rate.     

Low‐Frequency Volatility of Yen Interest Rate Swap Market in Relation to Macroeconomic Risk*

Using ‘low‐frequency’ volatility extracted from aggregate volatility shocks in interest rate swap (hereafter, IRS) market, this paper investigates whether Japanese yen IRS volatility can be explained by macroeconomic risks. The analysis suggests that this low‐frequency yen IRS volatility has strong and positive association with most of the macroeconomic risk proxies (e.g., volatility of consumer price index, industrial production volatility, foreign exchange volatility, slope of the term structure and money supply) with the exception of the unemployment rate, which is negatively related to IRS volatility. This finding is fairly consistent with the argument that the greater the macroeconomic risk the greater is the use of derivative instruments to hedge or speculate. The relationship between the macroeconomic risks and IRS volatility varies slightly across the different swap maturities but is robust to alternative volatility specifications. This linkage between swap market and macroeconomy has practical implications since market makers and hedgers use the swap rate as benchmark for pricing long‐term interest rates, corporate bonds and various other securities.     

Scheduled announcements and volatility patterns: The effects of monetary policy committee announcements on LIBOR and short sterling futures and options

Both the UK spot and futures markets in short‐term interest rates are found to react strongly to surprises in the scheduled announcements of the repo rate and RPI. Therefore, these announcements should also affect the market for options on short‐term interest rate futures. Because the repo rate and RPI announcements are scheduled, the options market can predict the days on which announcement shocks may hit, and build this information into its volatility expectations. It is argued that the volatility used in pricing options should alter over time in a predictable nonlinear manner that varies with contract maturity and the number of forthcoming announcements; but is independent of announcement content. The empirical results support this hypothesis. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:773–797, 2003     

An Analytic Solution for Interest Rate Swap Spreads

This paper argues that liquidity differences between government securities and short–term Eurodollar borrowings account for interest rate swap spreads. It then models the convenience of liquidity as a linear function of two mean–reverting state variables and values it. The interest rate swap spread for a swap of particular maturity is the annuitized equivalent of this value. It has a closed form solution: a simple integral. Special cases examined include the Vasicek and Cox–Ingersoll–Ross one–factor term structure models. Numerical values for the parameters in both special cases illustrate that many realistic ‘swap spread term structures’ can be replicated. Model parameters are estimated using weekly data on the term structure of swap spreads from several countries. The model fits the data well.     

On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite-dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite-dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

Treasury Bond risk and return, the implications for the hedging of consumption and lessons for asset pricing

All consumption-based models of asset pricing imply that the relation between the conditional mean and conditional volatility of any asset reflects the effectiveness of holding that asset as a hedge against intertemporal variation in the marginal utility of consumption. For Treasury Bonds of various maturities, we find significant positive relations. Our empirical findings support the conclusion that investors must sell bonds short to hedge shocks to marginal utility, because realized bond returns tend to be high (low) when investors least (most) desire an additional dollar of consumption. Implications for special cases of the general consumption-based model are also discussed.     

A two‐mean reverting‐factor model of the term structure of interest rates

This article presents a two‐factor model of the term structure of interest rates. It is assumed that default‐free discount bond prices are determined by the time to maturity and two factors, the long‐term interest rate, and the spread (i.e., the difference) between the short‐term (instantaneous) risk‐free rate of interest and the long‐term rate. Assuming that both factors follow a joint Ornstein‐Uhlenbeck process, a general bond pricing equation is derived. Closed‐form expressions for prices of bonds and interest rate derivatives are obtained. The analytical formula for derivatives is applied to price European options on discount bonds and more complex types of options. Finally, empirical evidence of the model's performance in comparison with an alternative two‐factor (Vasicek‐CIR) model is presented. The findings show that both models exhibit a similar behavior for the shortest maturities. However, importantly, the results demonstrate that modeling the volatility in the long‐term rate process can help to fit the observed data, and can improve the prediction of the future movements in medium‐ and long‐term interest rates. So it is not so clear which is the best model to be used. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23: 1075–1105, 2003     

Value at risk and conditional extreme value theory via markov regime switching models

This study develops a new conditional extreme value theory‐based (EVT) model that incorporates the Markov regime switching process to forecast extreme risks in the stock markets. The study combines the Markov switching ARCH (SWARCH) model (which uses different sets of parameters for various states to cope with the structural changes for measuring the time‐varying volatility of the return distribution) with the EVT to model the tail distribution of the SWARCH processed residuals. The model is compared with unconditional EVT and conditional EVT‐GARCH models to estimate the extreme losses in three leading stock indices: S&P 500 Index, Hang Seng Index and Hang Seng China Enterprise Index. The study found that the EVT‐SWARCH model outperformed both the GARCH and SWARCH models in capturing the non‐normality and in providing accurate value‐at‐risk forecasts in the in‐sample and out‐sample tests. The EVTSWARCH model, which exhibits the features of measuring the volatility of a heteroscedastic financial return series and coping with the non‐normality owing to structural changes, can be an alternative measure of the tail risk. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:155–181, 2008     

Information content of the fed funds rates

This study shows that the Fed Funds spot rate mostly affects the level of key interest rates while the Fed Funds futures rate tends to affect both the level and the volatility. Such effects are more concentrated on the shorter segment of the yield curve. In addition, only an unexpected change in the target rate affects both the level and the volatility of interest rates. Finally, the FOMC's increased disclosure of its policy stance has a calming effect on the volatility of key interest rates. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:753–774, 2005     

Samuelson hypothesis,arbitrage activity,and futures term premiums

The Samuelson hypothesis asserts that futures volatility increases as maturity decreases. On the basis of 10 US commodity futures and by capturing the dynamics of the futures volatility terms structure with three factors, we show that in most markets the slope factor is strongly negative in certain periods and at best only weakly negative in other periods. High inventory levels are found to correspond to flatter volatility term structures in seven futures. This finding is consistent with the linkage between carry arbitrage and the Samuelson hypothesis. We also find that a flatter volatility term structure corresponds to lower absolute futures term premiums.     

The Chinese interbank repo market: An analysis of term premiums

Because of the lack of short‐term government bonds, the interbank repo market in China has been providing the best information about market‐driven short‐term interest rates since its inception. This article examines the behavior of the repo rates of various terms and their term premiums. The work in this article supplements the study by F. Longstaff (2000), which reports supportive evidence for the pure expectations hypothesis over the short range of the term structure with the use of repo data from the United States. It is found that the pure expectations hypothesis is statistically rejected, although the term premiums are economically small. It is shown that the short‐term repo rate, repo rate volatility, repo market liquidity, and repo rate spreads are all important in determining the term premiums. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:153–167, 2006     

Can exchange seat prices predict financial market volatility?

There is considerable evidence that trading volume and volatility are positively related and that exchange seat prices are largely a function of trading volume. This article examines whether changes in seat prices at the Chicago Board of Trade (where stock index and interest rate futures account for the vast majority of trading volume) are useful in predicting changes in interest rate and stock market volatility. Exponential GARCH and transfer function models are used to demonstrate the power of changes in CBOT seat prices in predicting changes in interest rate and stock market volatility. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1206–1221, 2008     

Forecasting performance of extreme‐value volatility estimators

This study evaluates the forecasting performance of extreme‐value volatility estimators for the equity‐based Nifty Index using two‐scale realized volatility. This benchmark mitigates the effect of microstructure noise in the realized volatility. Extreme‐value estimates with relatively simple forecasting methods provide substantially better short‐term and long‐term forecasts, compared to historical volatility. The higher efficiency of extreme‐value estimators is primarily responsible for this improvement. The extent of possible improvement in forecasts is likely to be economically significant for applications like options pricing. By including extremevalue estimators, the forecasting performance of generalized autoregressive conditional heteroscedasticity (GARCH) can also be improved. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27: 1085–1105, 2007     

Applications of Eigenfunction Expansions in Continuous-Time Finance

We provide exact solutions for two closely related valuation problems in continuous-time finance. The first problem is to value generalized European-style options on stocks that pay dividends at a constant dollar rate. The second problem is to find the yield curve associated with the economy of R. C. Merton's "An Asymptotic Theory of Growth Under Uncertainty." In Merton's economic growth model, the interest rate process has a volatility linear in the rate level and a linear/quadratic drift. Both problems are solved by an eigenfunction expansion technique. The main technical difficulty is handling the problem of payoff functions that are not square-integrable with respect to the natural weight function of the models.