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     

An empirical comparison of continuous time models of the short term interest rate

This article tests the performance of a wide variety of well-known continuous time models—with particular emphasis on the Black, Derman, and Toy (1990; henceforth BDT) term structure model—in capturing the stochastic behavior of the short term interest rate volatility. Many popular interest rate models are nested within a more flexible time-varying BDT framework that allows us to compare the models and find the proper specification of the dynamics of short rates. The empirical results indicate that the equilibrium models that do not allow the drift and diffusion parameters to vary over time and parameterize the volatility only as a function of interest rate levels overemphasize the sensitivity of volatility to the level of interest rate and fail to model adequately the serial correlation in conditional variances. On the other hand, the GARCH-based arbitrage-free models with time-dependent parameters in the drift and diffusion functions define the volatility only as a function of unexpected information shocks and fail to capture adequately the relationship between interest rate levels and volatility. This study shows that the most successful models in capturing the dynamics of short term interest rates are those that introduce time-dependent parameters to the short rate process and define the conditional volatility as a function of both the interest rate levels and the last period's unexpected news. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 777–797, 1999     

Testing mean reversion in financial market volatility: Evidence from S&P 500 index futures

This article presents a comprehensive study of continuous time GARCH (generalized autoregressive conditional heteroskedastic) modeling with the thintailed normal and the fat‐tailed Student's‐t and generalized error distributions (GED). The study measures the degree of mean reversion in financial market volatility based on the relationship between discrete‐time GARCH and continuoustime diffusion models. The convergence results based on the aforementioned distribution functions are shown to have similar implications for testing mean reversion in stochastic volatility. Alternative models are compared in terms of their ability to capture mean‐reverting behavior of futures market volatility. The empirical evidence obtained from the S&P 500 index futures indicates that the conditional variance, log‐variance, and standard deviation of futures returns are pulled back to some long‐run average level over time. The study also compares the performance of alternative GARCH models with normal, Student's‐ t, and GED density in terms of their power to predict one‐day‐ahead realized volatility of index futures returns and provides some implications for pricing futures options. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1–33, 2008     

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.     

Asymmetric covariance in spot‐futures markets

This article studies how the spot‐futures conditional covariance matrix responds to positive and negative innovations. The main results of the article are achieved by obtaining the Volatility Impulse Response Function (VIRF) for asymmetric multivariate GARCH structures, extending Lin (1997) findings for symmetric GARCH models. This theoretical result is general and can be applied to analyze covariance dynamics in any financial system. After testing how multivariate GARCH models clean up volatility asymmetries, the Asymmetric VIRF is computed for the Spanish stock index IBEX‐35 and its futures contract. The empirical results indicate that the spot‐futures variance system is more sensitive to negative than positive shocks, and that spot volatility shocks have much more impact on futures volatility than vice versa. Additionally, evidence is obtained showing that optimal hedge ratios are insensitive to the well‐known asymmetric volatility behavior in stock markets. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1019–1046, 2003     

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     

Modeling nonlinear dynamics of daily futures price changes

The purpose of this article is to characterize linear and nonlinear serial dependence in daily futures price changes. The daily prices of four futures are included in this study: (i) S&P 500; (ii) Japanese yen; (iii) Deutsche mark; and (iv) Eurodollar. Our major empirical findings are: (i) Based on the results of nonlinearity tests (that is, the BDS, the Q², and the TAR-F tests), we found all futures price changes contain nonlinearity in the series; (ii) a GARCH model can explain the source of nonlinearity for three out of four series; (iii) a threshold autoregressive model and autoregressive volatility model can adequately represent nonlinear dynamics of S&P 500 series; and (iv) deterministic chaos is not evident in the scaled residuals from the nonlinear time series models. Hence we favor a statistical time series approach to represent the data-generating mechanism of futures price changes. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 325–351, 1999     

Stock return dynamics,option volume,and the information content of implied volatility

This article reports new empirical results on the information content of implied volatility, with respect to modeling and forecasting the volatility of individual firm returns. The 50 firms with the highest option volume on the Chicago Board Options Exchange between 1988 and 1995 are examined. First, the results indicate that the ability of implied volatility to subsume all relevant information about conditional variance depends on option trading volume. For the most active options in the sample, implied volatility reliably outperforms GARCH and subsumes all information in return shocks beyond the first lag. For these active options, implied volatility performs substantially better than indicated by the prior results of Lamoureux and Lastrapes ( 1993 ), despite significant methodological improvements in the time‐series volatility models in this study including the use of high‐frequency intraday return shocks. For the lower option‐volume firms in the sample, the performance of implied volatility deteriorates relative to time‐series volatility models. Finally, compared to a time‐series approach, the implied volatility of equity index options provides reliable incremental information about future firm‐level volatility. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:615–646, 2003     

Predicting Interest Rate Volatility Using Information on the Yield Curve

This study examines whether information on the yield curve is useful for predicting volatility of the yield curve. The information is used within dynamic models by specifying the covariance matrix of changes in yield factors as nonlinear functions of the factors. Using such models, it is found that the information (i) is useful for predicting volatility of the slope factor, achieving the accuracy comparable with the GARCH model; (ii) has incremental value for predicting volatility of the curvature factor when combined with a volatility‐specific factor; and (iii) does not much improve prediction of volatility of the level factor once the volatility‐specific factor is introduced.     

Volatility Spillovers and the Risk-Return Relation Between Stock and Foreign Exchange Markets in Brazil

We examine the evidence of mean and volatility spillovers between stock and foreign exchange markets in Brazil with multivariate GARCH models and nonlinear Granger causality tests. We also use a multivariate GARCH-in-mean model to assess the relationship between risk and return in these markets. The results indicate that the stock market leads the foreign exchange market in price formation and that nonlinear Granger causalities from the exchange market to the stock market do occur. Part of these nonlinear causalities are explained by volatility spillovers. We show that exchange rate volatility affects not only stock market volatility but also stock returns.     

Emerging Equity Market Volatility

Abstract

This paper considers two emerging markets that are under-researched, Kenya and Nigeria. It offers a comprehensive view of four time properties that emerged from the empirical time series literature on asset returns: (1) the predictability of returns from past observations; (2) the auto-regressive behavior of conditional volatility; (3) the asymmetric response of conditional volatility to innovations; and (4) the conditional variance risk premium. Results of the exponential GARCH (EGARCH) model indicate that asymmetric volatility found in the U.S. and other developed markets also characterized the Nigerian stock exchange. In Kenya, however, the asymmetric volatility coefficient is significant and positive, suggesting that positive shocks increase volatility more than negative shocks of an equal magnitude. The Nairobi Stock Exchange (KSE) returns series report negative but insignificant risk-premium parameters. In Nigeria (NSE), return series exhibit a significant and positive time-varying risk premium. The results also show that expected returns are predictable, that the auto-regressive return parameters (? ₁ ) are significant in both Kenya and Nigeria. Finally, the GARCH parameter (b) is statistically significant, indicating that volatility persistence is present in the two emerging markets studied.     

Time‐varying jump risk premia in stock index futures returns

This study tests the presence of time‐varying risk premia associated with extreme news events or jumps in stock index futures return. The model allows for a dynamic jump component with autoregressive jump intensity, long‐range dependence in volatility dynamics, and a volatility in mean structure separately for the normal and extreme news events. The results show significant jump risk premia in four stock market index futures returns including the DAX, FTSE, Nikkei, and S&P500 indices. Our results are robust to various specifications of conditional variance including the plain GARCH, component GARCH, and Fractionally Integrated GARCH models. We also find the time‐varying risk premium associated with normal news events is not significant across all indices. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:639–659, 2012     

Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations

On the basis of the theory of a wedge between the physical and risk‐neutral conditional volatilities in Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010), we develop a modification of the GARCH option pricing model with the filtered historical simulation proposed in Barone‐Adesi, G., Engle, R. F., & Mancini, L. (2008). The one‐day‐ahead conditional volatilities under physical and risk‐neutral measures are the same in the previous model, but should have been allowed to be different. Using extensive data on S&P 500 index options, our approach, which employs one‐day‐ahead risk‐neutral conditional volatility estimated from the cross‐section of the option prices (in contrast to the existing GARCH option pricing models), maintains theoretical consistency under conditional non‐normality, and improves the empirical performances. Remarkably, the risk‐neutral volatility dynamics are stable over time in this model. In addition, the comparison between the VIX index and the risk‐neutral integrated volatility economically validates our approach. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:1–28, 2013     

ON PROPERTIES OF ANALYTICALLY SOLVABLE FAMILIES OF LOCAL VOLATILITY DIFFUSION MODELS

We present some further developments in the construction and classification of new solvable one‐dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form. Our approach is based on so‐called diffusion canonical transformations that produce a large class of multiparameter nonlinear local volatility diffusion models that are mapped onto various simpler diffusions. Using an asymptotic analysis, we arrive at a rigorous boundary classification as well as a characterization with respect to probability conservation and the martingale property of the newly constructed diffusions. Specifically, we analyze and classify in detail four main families of driftless regular diffusion models that arise from the underlying squared Bessel process (the Bessel family), Cox–Ingersoll–Ross process (the confluent hypergeometric family), the Ornstein‐Uhlenbeck diffusion (the OU family), and the Jacobi diffusion (the hypergeometric family). We show that the Bessel family is a superset of the constant elasticity of variance model without drift. The Bessel family, in turn, is nested by the confluent hypergeometric family. For these two families we find further subfamilies of conservative strict supermartingales and nonconservative martingales with an exit boundary. For the new classes of nonconservative regular diffusions we also derive analytically exact first exit time densities that are given in terms of generalized inverse Gaussians and extensions. As for the two other new models, we show that the OU family of processes are conservative strict martingales, whereas the Jacobi family are nonconservative nonmartingales. Considered as asset price diffusion models, we also show that these models demonstrate a wide range of local volatility shapes and option implied volatility surfaces that include various pronounced skew and smile patterns.     

Forecasting volatility

The forecasting ability of the most popular volatility forecasting models is examined and an alternative model developed. Existing models are compared in terms of four attributes: (1) the relative weighting of recent versus older observations, (2) the estimation criterion, (3) the trade‐off in terms of out‐of‐sample forecasting error between simple and complex models, and (4) the emphasis placed on large shocks. As in previous studies, we find that financial markets have longer memories than reflected in GARCH(1,1) model estimates, but find this has little impact on outofsample forecasting ability. While more complex models which allow a more flexible weighting pattern than the exponential model forecast better on an in‐sample basis, due to the additional estimation error introduced by additional parameters, they forecast poorly out‐of‐sample. With the exception of GARCH models, we find that models based on absolute return deviations generally forecast volatility better than otherwise equivalent models based on squared return deviations. Among the most popular time series models, we find that GARCH(1,1) generally yields better forecasts than the historical standard deviation and exponentially weighted moving average models, though between GARCH and EGARCH there is no clear favorite. However, in terms of forecast accuracy, all are dominated by a new, simple, nonlinear least squares model, based on historical absolute return deviations, that we develop and test here. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:465–490, 2005     

我国推出股指期货对股票市场波动性影响的实证研究

本文以2006年11月1日至2010年12月27日的沪深300指数收盘价为原始数据,建立GARCH及EGARCH模型,进行实证研究,探究我国推出股指期货对股票市场波动性的影响。通过对该模型的分析,得出结论:股指期货在我国的推出,一定程度上降低了我国股票现货市场的波动性,对我国股票现货市场的健康发展起到了维稳作用。     

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     

Nonlinear dynamics and competing behavioral interpretations: Evidence from intra‐day FTSE‐100 index and futures data

Extant empirical research has reported nonlinear behavior within arbitrage relationships. In this article, the authors consider potential nonlinear dynamics within FTSE‐100 index and index‐futures. Such nonlinearity can be rationalized by the existence of transactions costs or through the interaction between informed and noise traders. They consider several empirical models designed to capture these alternative dynamics. Their empirical results provide evidence of a stationary basis term, and thus cointegration between index and index‐futures, and the presence of nonlinear dynamics within that relationship. The results further suggest that noise traders typically engage in momentum trading and are more prone to this behavior type when the underlying market is rising. Fundamental, or arbitrage, traders are characterized by heterogeneity, such that there is slow movement between regimes of behavior. In particular, fundamental traders act more quickly in response to small deviations from equilibrium, but are reluctant to act quickly in response to larger mispricings that are exposed to greater noise trader price risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:343–368, 2006     

Switching asymmetric GARCH and options on a volatility index

Few proposed types of derivative securities have attracted as much attention and interest as option contracts on volatility. Grunbichler and Longstaff (1996) is the only study that proposes a model to value options written on a volatility index. Their model, which is based on modeling volatility as a GARCH process, does not take into account the switching regime and asymmetry properties of volatility. We show that the Grunbichler and Longstaff (1996) model underprices a three‐month option by about 10%. A Switching Regime Asymmetric GARCH is used to model the generating process of security returns. The comparison between the switching regime model and the traditional uni‐regime model among GARCH, EGARCH, and GJR‐GARCH demonstrates that a switching regime EGARCH model fits the data best. Next, the values of European call options written on a volatility index are computed using Monte Carlo integration. When comparing the values of the option based on the Switching Regime Asymmetric GARCH model and the traditional GARCH specification, it is found that the option values obtained from the different processes are very different. This clearly shows that the Grunbichler‐Longstaff model is too stylized to be used in pricing derivatives on a volatility index. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:251–282, 2004     

APPROXIMATING GARCH-JUMP MODELS, JUMP-DIFFUSION PROCESSES, AND OPTION PRICING

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset prices and volatilities. We extend theory developed by Nelson (1990) and Duan (1997) by considering the limiting models for our approximating GARCH Jump process. Limiting cases of our processes consist of models where both asset price and local volatility follow jump diffusion processes with correlated jump sizes. Convergence of a few GARCH models to their continuous time limits is evaluated and the benefits of the models explored.