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Now showing items 1 - 16 of 140

  • Remembering Peter Christoffersen (1967-2018)

    Chang, Bo Young   Jacobs, Kris   Ornthanalai, Chayawat   Figlewski, Stephen  

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  • Option valuation with observable volatility and jump dynamics

    Peter Christoffersen   Bruno Feunou   Yoontae Jeon  

    Abstract Under very general conditions, the total quadratic variation of a jump-diffusion process can be decomposed into diffusive volatility and squared jump variation. We use this result to develop a new option valuation model in which the underlying asset price exhibits volatility and jump intensity dynamics. The volatility and jump intensity dynamics in the model are directly driven by model-free empirical measures of diffusive volatility and jump variation. Because the empirical measures are observed in discrete intervals, our option valuation model is cast in discrete time, allowing for straightforward filtering and estimation of the model. Our model belongs to the affine class enabling us to derive the conditional characteristic function so that option values can be computed rapidly without simulation. When estimated on S&P500 index options and returns the new model performs well compared with standard benchmarks.
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  • Dynamic Dependence and Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    We characterize dependence and tail dependence in corporate credit using a new class of dynamic copula models which can capture dynamic dependence and asymmetry in large samples of firms. We also document important differences between the depen- dence dynamics for credit spreads and equity returns. Modeling a decade of weekly CDS spreads for 215 firms, we find that copula correlations are highly time-varying and persistent, and that they increase significantly in the financial crisis and have remained high since. Perhaps most importantly, tail dependence of CDS spreads increases even more than copula correlations during the crisis and remains high as well. The most important shocks to credit dependence occur in August of 2007 and in August of 2011, but interestingly these dates are not associated with significant changes to median credit spreads. The decrease in diversification potential caused by the increase in dependence and tail dependence is large. Finally, we find that the CDS volatility, correlation and tail dependence measures that we have constructed using the dynamic copula model are important determinants of credit spreads over time.
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  • Option-Based Estimation of Co-Skewness and Co-Kurtosis Risk Premia

    Peter Christoffersen   Kris Jacobs   Mehdi Karoui  

    We show that the price of risk for equity factors that are nonlinear in the market return are readily obtained using index option prices. We apply this insight to the price of co-skewness and co-kurtosis risk. The price of co-skewness risk corresponds to the spread between the physical and the risk-neutral second moments, and the price of co-kurtosis risk corresponds to the spread between the physical and the risk-neutral third moments. Our option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An out-of-sample analysis of factor models with co-skewness and co-kurtosis risk indicates that the new estimates of the price of risk improve the models' performance. The models using higher-order market moments also robustly outperform standard competitors such as the CAPM and the Fama-French model.
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  • Dynamic Dependence and Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    We characterize dependence and tail dependence in corporate credit using a new class of dynamic copula models which can capture dynamic dependence and asymmetry in large samples of firms. We also document important differences between the depen- dence dynamics for credit spreads and equity returns. Modeling a decade of weekly CDS spreads for 215 firms, we find that copula correlations are highly time-varying and persistent, and that they increase significantly in the financial crisis and have remained high since. Perhaps most importantly, tail dependence of CDS spreads increases even more than copula correlations during the crisis and remains high as well. The most important shocks to credit dependence occur in August of 2007 and in August of 2011, but interestingly these dates are not associated with significant changes to median credit spreads. The decrease in diversification potential caused by the increase in dependence and tail dependence is large. Finally, we find that the CDS volatility, correlation and tail dependence measures that we have constructed using the dynamic copula model are important determinants of credit spreads over time.
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  • Dynamic Dependence and Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    We characterize dependence and tail dependence in corporate credit using a new class of dynamic copula models which can capture dynamic dependence and asymmetry in large samples of... rms. We also document important di¤erences between the depen dence dynamics for credit spreads and equity returns. Modeling a decade of weekly CDS spreads for 215... rms, we... nd that copula correlations are highly time-varying and persistent, and that they increase signi... cantly in the... nancial crisis and have remained high since. Perhaps most importantly, tail dependence of CDS spreads increases even more than copula correlations during the crisis and remains high as well. The most important shocks to credit dependence occur in August of 2007 and in August of 2011, but interestingly these dates are not associated with signi... cant changes to median credit spreads. The decrease in diversi... cation potential caused by the increase in dependence and tail dependence is large. Finally, we... nd that the CDS volatility, correlation and tail dependence measures that we have constructed using the dynamic copula model are important determinants of credit spreads over time.
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  • Correlation dynamics and international diversification benefits

    Peter Christoffersen   Vihang Errunza   Kris Jacobs   Xisong Jin  

    Abstract Forecasting the evolution of security co-movements is critical for asset pricing and portfolio allocation. Hence, we investigate patterns and trends in correlations over time using weekly returns for developed markets (DMs) and emerging markets (EMs) over the period 1973–2012. We show that it is possible to model co-movements for many countries simultaneously using BEKK, DCC, and DECO models. Empirically, we find that correlations have trended upward significantly for both DMs and EMs. Based on a time-varying measure of diversification benefits, we find that it is not possible to circumvent the increasing correlations in a long-only portfolio by adjusting the portfolio weights over time. However, we do find some evidence that adding EMs to a DM-only portfolio increases diversification benefits.
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  • Nonlinear Kalman Filtering in Affine Term Structure Models

    Peter Christoffersen   Christian Dorion   Kris Jacobs  

    The extended Kalman filter, which linearizes the relationship between security prices and state variables, is widely used in fixed income applications. We investigate if the unscented Kalman filter should be used to capture nonlinearities, and compare the performance of the Kalman filter to that of the particle filter. We analyze the cross section of swap rates, which are mildly nonlinear in the states, and cap prices, which are highly nonlinear. When caps are used to filter the states, the unscented Kalman filter significantly outperforms its extended counterpart. The unscented Kalman filter also performs well when compared to the much more computationally intensive particle filter. These findings suggest that the unscented Kalman filter may prove to be a good approach for variety of problems in fixed income pricing.
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  • Dynamic Dependence and Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    We characterize dependence and tail dependence in corporate credit using a new class of dynamic copula models which can capture dynamic dependence and asymmetry in large samples of firms. We also document important differences between the dependence dynamics for credit spreads and equity returns. Modeling a decade of weekly CDS spreads for 215 firms, we find that copula correlations are highly time-varying and persistent, and that they increase significantly in the financial crisis and have remained high since. Perhaps most importantly, tail dependence of CDS spreads increases even more than copula correlations during the crisis and remains high as well. The most important shocks to credit dependence occur in August of 2007 and in August of 2011, but interestingly these dates are not associated with significant changes to median credit spreads. The decrease in diversification potential caused by the increase in dependence and tail dependence is large. Finally, we find that the CDS volatility, correlation and tail dependence measures that we have constructed using the dynamic copula model are important determinants of credit spreads over time.
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  • Factor Structure in Commodity Futures Return and Volatility

    Peter Christoffersen   Asger Lunde   Kasper V. Olesen  

    Using data on more than 750 million futures trades during 2004-2013, we analyze eight stylized facts of commodity price and volatility dynamics in the post financialization period. We pay particular attention to the factor structure in returns and volatility and to commodity market integration with the equity market. We find evidence of a factor structure in daily commodity futures returns. However, the factor structure in daily commodity futures volatility is even stronger than in returns. When computing model-free realized commodity betas with the stock market we find that they were high during 2008-2010 but have since returned to the pre-crisis level close to zero. The common factor in commodity volatility is nevertheless clearly related to stock market volatility. We conclude that, while commodity markets appear to again be segmented from the equity market when only returns are considered, commodity volatility indicates a nontrivial degree of market integration.
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  • Factor Structure in Commodity Futures Return and Volatility

    Peter Christoffersen   Asger Lunde   Kasper V. Olesen  

    Using data on more than 750 million futures trades during 2004-2013, we analyze eight stylized facts of commodity price and volatility dynamics in the post financialization period. We pay particular attention to the factor structure in returns and volatility and to commodity market integration with the equity market. We find evidence of a factor structure in daily commodity futures returns. However, the factor structure in daily commodity futures volatility is even stronger than in returns. When computing model-free realized commodity betas with the stock market we find that they were high during 2008-2010 but have since returned to the pre-crisis level close to zero. The common factor in commodity volatility is nevertheless clearly related to stock market volatility. We conclude that, while commodity markets appear to again be segmented from the equity market when only returns are considered, commodity volatility indicates a nontrivial degree of market integration.
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  • Option-Based Estimation of Co-Skewness and Co-Kurtosis Risk Premia

    Peter Christoffersen   Mathieu Fournier   Kris Jacobs  

    We show that the price of risk for equity factors that are nonlinear in the market return are readily obtained using index option prices. We apply this insight to the price of co-skewness and co-kurtosis risk. The price of co-skewness risk corresponds to the spread between the physical and the risk-neutral second moments, and the price of co-kurtosis risk corresponds to the spread between the physical and the risk-neutral third moments. Our option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An out-of-sample analysis of factor models with co-skewness and co-kurtosis risk indicates that the new estimates of the price of risk improve the models’ performance. Models with higher-order market moments also robustly outperform standard competitors such as the CAPM and the Fama-French model.
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  • Dynamic Dependence and Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    We characterize dependence and tail dependence in corporate credit using a new class of dynamic copula models which can capture dynamic dependence and asymmetry in large samples of firms. We also document important differences between the depen- dence dynamics for credit spreads and equity returns. Modeling a decade of weekly CDS spreads for 215 firms, we find that copula correlations are highly time-varying and persistent, and that they increase significantly in the financial crisis and have remained high since. Perhaps most importantly, tail dependence of CDS spreads increases even more than copula correlations during the crisis and remains high as well. The most important shocks to credit dependence occur in August of 2007 and in August of 2011, but interestingly these dates are not associated with significant changes to median credit spreads. The decrease in diversification potential caused by the increase in dependence and tail dependence is large. Finally, we find that the CDS volatility, correlation and tail dependence measures that we have constructed using the dynamic copula model are important determinants of credit spreads over time.
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  • The Factor Structure in Equity Options

    Peter Christoffersen   Mathieu Fournier   Kris Jacobs  

    Principal component analysis of equity options on Dow-Jones firms reveals a strong factor structure. The first principal component explains 77% of the variation in the equity volatility level, 77% of the variation in the equity option skew, and 60% of the implied volatility term structure across equities. Furthermore, the first principal component has a 92% correlation with S&P500 index option volatility, a 64% correlation with the index option skew, and a 80% correlation with the index option term structure. We develop an equity option valuation model that captures this factor structure. The model allows for stochastic volatility in the market return and also in the idiosyncratic part of firm returns. The model predicts that firms with higher betas have higher implied volatilities, and steeper moneyness and term structure slopes. We provide a tractable approach for estimating the model on a large set of index and equity option data on which the model provides a good fit. The equity option data support the cross-sectional implications of the estimated model.
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  • Dynamic Diversification in Corporate Credit

    Peter Christoffersen   Kris Jacobs   Xisong Jin  

    This paper documents cross-sectional dependence in CDS spreads, and compares it with de- pendence in equity returns. Our results are largely complementary to existing correlation and dependence estimates, which are typically based on historical default rates or factor models of equity returns, and to existing intensity-based studies, which characterize observable macro variables that induce realistic correlation patterns in default probabilities (see Duffee (1999) and Duffie, Saita and Wang (2007)). Importantly, we use econometric techniques that allow us to estimate a model with multivariate asymmetries and time-varying dependence using a long time series and a large cross-section of CDS spreads. We document six important stylized facts. First, copula correlations in CDS spreads vary substantially over our sample, with a significant increase following the financial crisis in 2007. Equity correlations also increase in the financial crisis, but somewhat later, and the increase is less significant and not as persistent. Second, our estimates indicate fat tails in the univariate distributions, but also multivariate non-normalities. Multivariate asymmetries seem to be less important for credit than they are for equities. Third, credit dependence is more persistent than equity persistence, and this greatly affects how major events such as the Quant Meltdown, the Lehman bankruptcy, and the U. S. sovereign debt downgrade affect subsequent dependence in credit and equity markets. Fourth, tail dependence increases more significantly over the sample than copula correlations. Fifth, economic variables explain a significant part of the time-series variation in dependence and tail dependence. Sixth, the dependence and tail dependence measures are related to the time-series variation in credit spreads, even after accounting for other well-known firm-level determinants of spreads. These stylized facts, and the increase in cross-sectional dependence in particular, have important implications for the management of portfolio credit risk. We illustrate these impli- cations by computing the diversification benefits from selling credit protection. The increase in cross-sectional dependence following the financial crisis has reduced diversification benefits, not unlike what happened in equity markets. When computing diversification benefits, taking non-normalities into account is more important for credit than for equity. Several other important implications of our results deserve further study. First, given the richness and complexity of the equity and credit dependence, it may prove interesting to explore the implications for the pricing of structured products. In particular it would be interesting to investigate if the CDO pricing model suggested by the estimated dynamics removes some of the observed correlation smile in CDO tranches. See Berd, Engle, and Voronov (2007) for an example of such an approach. Second, our estimates can be used to manage a portfolio of counterparty risks. Third, our approach can be used to integrate credit and equity dependence dynamics in a single portfolio exercise that allows for diversification across asset classes. Finally, a possible extension is to investigate alternative measures of credit portfolio risk (Vasicek 1987, 1991, 2002).
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  • Correlation Dynamics and International Diversification Benefits

    Peter Christoffersen   Vihang R. Errunza   Kris Jacobs  

    Forecasting the evolution of security co-movements is critical for asset pricing and portfolio allocation. Hence, we investigate patterns and trends in correlations over time using weekly re- turns for developed markets (DMs) and emerging markets (EMs) during the period 1973-2012. We show that it is possible to model co-movements for many countries simultaneously using BEKK, DCC, and DECO models. Empirically, we find that correlations have significantly trended upward for both DMs and EMs. Based on a time-varying measure of diversification benefit, we find that it is not possible in a long-only portfolio to circumvent the increasing correlations by adjusting the portfolio weights over time. However, we do find some evidence that adding EMs to a DM-only portfolio increases diversification benefits.
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