Estimation and hedging effectiveness of time‐varying hedge ratio: Flexible bivariate garch approaches

Bollerslev's ( 1990 , Review of Economics and Statistics, 52, 5–59) constant conditional correlation and Engle's (2002, Journal of Business & Economic Statistics, 20, 339–350) dynamic conditional correlation (DCC) bivariate generalized autoregressive conditional heteroskedasticity (BGARCH) models are usually used to estimate time‐varying hedge ratios. In this study, we extend the above model to more flexible ones to analyze the behavior of the optimal conditional hedge ratio based on two (BGARCH) models: (i) adopting more flexible bivariate density functions such as a bivariate skewed‐t density function; (ii) considering asymmetric individual conditional variance equations; and (iii) incorporating asymmetry in the conditional correlation equation for the DCC‐based model. Hedging performance in terms of variance reduction and also value at risk and expected shortfall of the hedged portfolio are also conducted. Using daily data of the spot and futures returns of corn and soybeans we find asymmetric and flexible density specifications help increase the goodness‐of‐fit of the estimated models, but do not guarantee higher hedging performance. We also find that there is an inverse relationship between the variance of hedge ratios and hedging effectiveness. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:71–99, 2010     

VIX futures pricing with conditional skewness

We develop a closed‐form VIX futures valuation formula based on the inverse Gaussian GARCH process by Christoffersen et al. that combines conditional skewness, conditional heteroskedasticity, and a leverage effect. The new model outperforms the benchmark in fitting the S&P 500 returns and the VIX futures prices. The fat‐tailed innovation underlying the model substantially reduced pricing errors during the 2008 financial crisis. The in‐ and out‐of‐sample pricing performance indicates that the new model should be a default modeling choice for pricing the medium‐ and long‐term VIX futures.     

What explains default risk premium during the financial crisis? Evidence from Japan

As is well documented, subprime mortgage markets carried significant default risk. This paper investigates the relationship between default risk premium, stock market conditions and macroeconomic variables during the financial crisis. Using iTraxx Japan Credit Default Swap (CDS) index spreads covering the period from March 2006 to November 2009, we employ a time-varying dynamic factor model with Markov regime switching to generate regime probabilities for default risk. We analyze the sensitivity of default risk premium changes to stock market conditions and macroeconomic variables by using two-state Markov switching models: a crisis regime sparked by rising loan defaults in the sub-prime mortgage market, and a non-crisis regime. We found strong evidence that the relationship between default risk premium changes, stock market and macroeconomic variables is regime-dependent. Our results suggest that during periods of crisis, CDS indices behave as a higher-risk indicator and become more sensitive to stock market conditions and macroeconomic variables. This paper examines the effects of the financial crisis in explaining the default risk premium. Understanding the determinants of default risk premium is important for financial analysts, economic policy makers and credit risk management.     

Sharing the value‐at‐risk under distributional ambiguity

This paper considers the problem of risk sharing, where a coalition of homogeneous agents, each bearing a random cost, aggregates their costs, and shares the value‐at‐risk of such a risky position. Due to limited distributional information in practice, the joint distribution of agents' random costs is difficult to acquire. The coalition, being aware of the distributional ambiguity, thus evaluates the worst‐case value‐at‐risk within a commonly agreed ambiguity set of the possible joint distributions. Through the lens of cooperative game theory, we show that this coalitional worst‐case value‐at‐risk is subadditive for the popular ambiguity sets in the distributionally robust optimization literature that are based on (i) convex moments or (ii) Wasserstein distance to some reference distributions. In addition, we propose easy‐to‐compute core allocation schemes to share the worst‐case value‐at‐risk. Our results can be readily extended to sharing the worst‐case conditional value‐at‐risk under distributional ambiguity.     

RATING BASED LÉVY LIBOR MODEL

In this paper, we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default‐free forward Libor rate and develop the rating based Libor market model to cover defaultable bonds with credit ratings. As driving processes for the dynamics of the default‐free and the predefault term structure of Libor rates, time‐inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented.     

CORRELATION UNDER STRESS IN NORMAL VARIANCE MIXTURE MODELS

We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modeling. For the special cases of jointly normally and t‐distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Fréchet or Gumbel distribution. It turns out that correlations in heavy‐tailed NVM models are less sensitive to stress than in medium‐ or light‐tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy. We also demonstrate how our results can be applied for more prudent stress testing.     

Portfolio Value-at-Risk with Heavy-Tailed Risk Factors

This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.     

Institutional quality and sovereign credit default swap spreads

We examine how the quality of political, legal, and regulatory institutions impacts sovereign risk premia. An improvement in institutional quality significantly lowers a country's sovereign credit default swap (CDS) spread, even after controlling for domestic and global macroeconomic factors. The incremental effect of institutional quality may also be economically important in explaining the variations in the level of sovereign CDS spreads. The basic results are robust to alternative model specifications, samples, control variables, measures of institutional quality, estimation methods, and controls for endogeneity. Overall, the evidence suggests that institutional quality may play a significant role in explaining sovereign CDS spreads.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

ARBITRAGE‐FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS

We develop an arbitrage‐free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on‐default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation‐induced contagion is illustrated with numerical examples.     

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     

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.     

The tail risk of emerging stock markets

We investigate tail risk in emerging stock markets at the country, regional and world levels, by comparing the investable and non-investable segments in terms of the expected shortfall of standardized returns and tail dependence on the world market. Employing the skewed Student-t GJR-GARCH model and the SJC copula, we show that most investable portfolios have lower tail risk but higher tail dependence than non-investable ones; emerging markets are likely more dependent on the world market during large joint losses than large joint gains; and tail dependence of the aggregate and investable markets on the world market varies across countries and regions.     

AN EXACT FORMULA FOR DEFAULT SWAPTIONS’ PRICING IN THE SSRJD STOCHASTIC INTENSITY MODEL

We develop and test a fast and accurate semi‐analytical formula for single‐name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.     

Credit risk pricing in a consumption-based equilibrium framework with incomplete accounting information

We present a consumption-based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where the default time is modeled as the first hitting time of a default boundary and bond investors have imperfect/partial information about the firm value. The imperfect information is generated by the underlying observed state variables and a noisy observation process of the firm value. In addition, the consumption, the volatility, and the firm value process are modeled to follow affine diffusion processes. Using the EZ equilibrium solution as the pricing kernel, we provide an equivalent pricing measure to compute the prices of financial derivatives as discounted values of the future payoffs given the incomplete information. The price of a zero-coupon bond is represented in terms of the solutions of a stochastic partial differential equation (SPDE) and a deterministic PDE; the self-contained proofs are provided for both this representation and the well-posedness of the involved SPDE. Furthermore, this SPDE is numerically solved, which yields some insights into the relationship between the structure of the yield spreads and the model parameters.     

The role of country-specific fundamentals in sovereign CDS spreads: Eastern European experiences

We investigate the determinants of sovereign CDS spreads on a sample of Eastern European data. A dynamic hierarchical factor model is used to aggregate information in indicators of economic fundamentals. CDS spreads are regressed on forecasts of factors. We find that domestic fundamentals explain more of CDS spread variance than global factors, largely due to their ability to explain differences in sovereign risk across countries. The effects on CDS spreads are found to be time-varying. In terms of economic significance, the factor of institutional-political strength stands out. We apply the model to study CDS spreads of Poland, Russia and Turkey.     

INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

通货膨胀对我国股市条件波动影响的统计检验

将通货膨胀因素引入到GARCH模型的方差方程中,以检验通货膨胀、通货膨胀变动以及通货膨胀不确定性对股市条件波动的影响。实证分析的结果表明,在统计意义上,通货膨胀及其变动对股市收益的条件波动几乎不会产生影响,而通货膨胀不确定性对股市波动的影响是显著的。这一结果意味着管理好通胀预期,降低通货膨胀不确定性,引导公众形成合理的通胀预期对促进股市平稳发展具有重要意义。     

Effect of corporate governance on default risk in financial versus nonfinancial firms: Canadian evidence

This paper investigates the influence of corporate governance variables on default risk of Canadian firms after the 2008 financial crisis. We provide evidence that important governance mechanisms have differential impacts between Canadian financial and nonfinancial firms. Ownership structure, (e.g., institutional ownership and insider ownership), has a significant impact on the default risk of financial firms but not on nonfinancial firms. Nonfinancial firms with more independent boards are associated with lower default risk, while financial firms with larger boards and more independent boards have higher default risk. In addition, although cross‐listing in the US reduces the default risk for Canadian nonfinancial firms, it actually increases the risk for Canadian financial firms during the postcrisis period. Copyright © 2016 ASAC. Published by John Wiley & Sons, Ltd.     

Credit Risk Assessment and Relationship Lending: An Empirical Analysis of German Small and Medium‐Sized Enterprises*

We estimate a logit scoring model for the prediction of the probability of default by German small and medium‐sized enterprises (SMEs) using a unique data set on SME loans in Germany. Our scoring model helps SMEs to gain knowledge about their default risk, which can be used to approximate their risk adequate cost of debt. This knowledge is likely to lead to a detection of hold‐up problems that German SMEs might be confronted with in their bank relationships. Furthermore, it allows them to monitor their bank’s pricing behavior and it reduces information asymmetries between lenders and borrowers. Finally, it can influence their future financing decisions toward capital market‐based financing.