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The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments

Published online by Cambridge University Press:  12 August 2010

Daniel Egloff
Affiliation:
QuantCatalyst, Hardturmstrasse 101, 8005 Zurich, Switzerland. [email protected].
Markus Leippold
Affiliation:
University of Zurich, Swiss Banking Institute, Plattenstrasse 14, 8032 Zurich, Switzerland. [email protected].
Liuren Wu
Affiliation:
Baruch College, Zicklin School of Business, One Bernard Baruch Way, Box B10-225, New York, NY 10010. [email protected].

Abstract

This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index.

Type
Research Articles
Copyright
Copyright © Michael G. Foster School of Business, University of Washington 2010

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References

Aït-Sahalia, Y., and Kimmel, R.. “Maximum Likelihood Estimation of Stochastic Volatility Models.” Journal of Financial Economics, 83 (2007), 413452.CrossRefGoogle Scholar
Aït-Sahalia, Y., and Mancini, L.. “Out of Sample Forecasts of Quadratic Variation.” Journal of Econometrics, 147 (2008), 1733.Google Scholar
Aït-Sahalia, Y.; Mykland, P.; and Zhang, L.. “How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise.” Review of Financial Studies, 18 (2005), 351416.CrossRefGoogle Scholar
Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; and Ebens, H.. “The Distribution of Realized Stock Return Volatility.” Journal of Financial Economics, 61 (2001), 4376.Google Scholar
Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; and Labys, P.. “Modeling and Forecasting Realized Volatility.” Econometrica, 71 (2003), 579625.Google Scholar
Andersen, T.; Bollerslev, T.; and Meddahi, N.. “Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities.” Econometrica, 73 (2005), 279296.Google Scholar
Andreou, E., and Ghysels, E.. “Rolling-Sample Volatility Estimators: Some New Theoretical, Simulation, and Empirical Results.” Journal of Business and Economic Statistics, 20 (2002), 363376.CrossRefGoogle Scholar
Bakshi, G., and Kapadia, N.. “Delta-Hedged Gains and the Negative Market Volatility Risk Premium.” Review of Financial Studies, 16 (2003a), 527566.CrossRefGoogle Scholar
Bakshi, G., and Kapadia, N.. “Volatility Risk Premiums Embedded in Individual Equity Options: Some New Insights.” Journal of Derivatives, 11 (2003b), 4554.Google Scholar
Balduzzi, P.; Das, S.; and Foresi, S.. “The Central Tendency: A Second Factor in Bond Yields.” Review of Economics and Statistics, 80 (1998), 6272.CrossRefGoogle Scholar
Bandi, F. M., and Russell, J. R.. “Separating Microstructure Noise from Volatility.” Journal of Financial Economics, 79 (2006), 655692.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., and Shephard, N.. “Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics.” Econometrica, 72 (2004a), 885925.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., and Shephard, N.. “Power and Bipower Variation with Stochastic Volatility and Jumps.” Journal of Financial Econometrics, 2 (2004b), 137.Google Scholar
Bates, D. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies, 9 (1996), 69107.CrossRefGoogle Scholar
Bates, D. “Post-’87 Crash Fears in the S&P 500 Futures Option Market.” Journal of Econometrics, 94 (2000), 181238.CrossRefGoogle Scholar
Bates, D. “Empirical Option Pricing: A Retrospection.” Journal of Econometrics, 116 (2003), 387404.CrossRefGoogle Scholar
Bates, D. “The Market for Crash Risk.” Journal of Economic Dynamics and Control, 32 (2008), 22912321.CrossRefGoogle Scholar
Bergomi, L. “Smile Dynamics.” Risk, 17 (2004), 117123.Google Scholar
Bergomi, L. “Smile Dynamics II.” Risk, 18 (2005), 6773.Google Scholar
Bergomi, L. “Smile Dynamics III.” Risk, 21 (2008), 9096.Google Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (1973), 637654.CrossRefGoogle Scholar
Bollen, N. P., and Whaley, R. E.. “Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?Journal of Finance, 59 (2004), 711753.CrossRefGoogle Scholar
Bondarenko, O. “Why Are Put Options So Expensive?” Working Paper, University of Illinois at Chicago (2003).Google Scholar
Bondarenko, O. “Market Price of Variance Risk and Performance of Hedge Funds.” Working Paper, University of Illinois at Chicago (2004).CrossRefGoogle Scholar
Broadie, M.; Chernov, M.; and Johannes, M.. “Model Specification and Risk Premia: Evidence from Futures Options.” Journal of Finance, 62 (2007), 14531490.CrossRefGoogle Scholar
Buehler, H. “Consistent Variance Curve Models.” Finance and Stochastics, 10 (2006), 178203.CrossRefGoogle Scholar
Carr, P.; Jin, X.; and Madan, D.. “Optimal Investment in Derivative Securities.” Finance and Stochastics, 5 (2001), 3359.CrossRefGoogle Scholar
Carr, P.; Lee, R.; and Wu, L.. “Variance Swaps on Time-Changed Lévy Processes.” Finance and Stochastics, forthcoming (2010).Google Scholar
Carr, P., and Madan, D.. “Optimal Positioning in Derivative Securities.” Quantitative Finance, 1 (2001), 1937.CrossRefGoogle Scholar
Carr, P., and Sun, J.. “A New Approach for Option Pricing under Stochastic Volatility.” Review of Derivatives Research, 10 (2007), 87150.CrossRefGoogle Scholar
Carr, P., and Wu, L.. “Static Hedging of Standard Options.” Working Paper, New York University and Baruch College (2002).Google Scholar
Carr, P., and Wu, L.. “A Tale of Two Indices.” Journal of Derivatives, 13 (2006), 1329.Google Scholar
Carr, P., and Wu, L.. “Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions: Disentangling the Multi-Dimensional Variations in S&P 500 Index Options.” Working Paper, Bloomberg and Baruch College (2008).Google Scholar
Carr, P., and Wu, L.. “Variance Risk Premiums.” Review of Financial Studies, 22 (2009), 13111341.CrossRefGoogle Scholar
Chacko, G., and Viceira, L. M.. “Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets.” Review of Financial Studies, 18 (2005), 13691402.CrossRefGoogle Scholar
Chernov, M., and Ghysels, E.. “A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation.” Journal of Financial Economics, 56 (2000), 407458.CrossRefGoogle Scholar
Cox, D. R. “Some Statistical Methods Connected with Series of Events.” Journal of the Royal Statistical Society, Series B, 17 (1955), 129164.Google Scholar
Duanmu, Z. “Rational Pricing of Options on Realized Volatility: The Black Scholes Way.” Working Paper, Merrill Lynch (2004).Google Scholar
Duffie, D. Dynamic Asset Pricing Theory. Princeton, NJ: Princeton University Press (1992).Google Scholar
Duffie, D.; Pan, J.; and Singleton, K.. “Transform Analysis and Asset Pricing for Affine Jump Diffusions.” Econometrica, 68 (2000), 13431376.CrossRefGoogle Scholar
Dupire, B. “Model Art.” Risk, 6 (1993), 118124.Google Scholar
Engle, R. “Risk and Volatility: Econometric Models and Financial Practice.” American Economic Review, 94 (2004), 405420.CrossRefGoogle Scholar
Eraker, B. “Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices.” Journal of Finance, 59 (2004), 13671403.CrossRefGoogle Scholar
Gârleanu, N.; Pedersen, L. H.; and Poteshman, A. M.. “Demand-Based Option Pricing.” Review of Financial Studies, 22 (2009), 42594299.CrossRefGoogle Scholar
Hansen, P. R., and Lunde, A.. “Realized Variance and Market Microstructure Noise.” Journal of Business and Economic Statistics, 24 (2006), 127161.CrossRefGoogle Scholar
Heston, S. L. “Closed-Form Solution for Options with Stochastic Volatility, with Application to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327343.CrossRefGoogle Scholar
Isaenko, S. “Dynamic Equilibrium with Overpriced Put Options.” Economic Notes, 36 (2007), 126.CrossRefGoogle Scholar
Jackwerth, J. C. “Recovering Risk Aversion from Option Prices and Realized Returns.” Review of Financial Studies, 13 (2000), 433451.CrossRefGoogle Scholar
Jones, C. S. “The Dynamics of Stochastic Volatility: Evidence from Underlying and Options Markets.” Journal of Econometrics, 116 (2003), 181224.CrossRefGoogle Scholar
Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems.” Transactions of the ASME–Journal of Basic Engineering, 82 (1960), 3545.CrossRefGoogle Scholar
Liu, J., and Pan, J.. “Dynamic Derivative Strategies.” Journal of Financial Economics, 69 (2003), 401430.CrossRefGoogle Scholar
Liu, J.; Pan, J.; and Wang, T.. “An Equilibrium Model of Rare-Event Premia and Its Implication for Option Smirks.” Review of Financial Studies, 18 (2005), 131164.Google Scholar
Merton, R. C. “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory, 3 (1971), 373413.CrossRefGoogle Scholar
Newey, W. K., and West, K. D.. “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55 (1987), 703708.Google Scholar
Oomen, R. C. A. “Properties of Bias-Corrected Realized Variance under Alternative Sampling Schemes.” Journal of Financial Econometrics, 3 (2005), 555577.CrossRefGoogle Scholar
Pan, J. “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study.” Journal of Financial Economics, 63 (2002), 350.Google Scholar
Santa-Clara, P., and Saretto, A.. “Option Strategies: Good Deals and Margin Calls.” Journal of Financial Markets, 12 (2009), 391417.CrossRefGoogle Scholar
Sharpe, W. F. “Mutual Fund Performance.” Journal of Business, 39 (1966), 119138.CrossRefGoogle Scholar
Wu, L. “Variance Dynamics: Joint Evidence from Options and High-Frequency Returns.” Journal of Econometrics, forthcoming (2010).Google Scholar
Zhang, L.; Mykland, P. A.; and Aït-Sahalia, Y.. “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data.” Journal of the American Statistical Association, 100 (2005), 13941411.Google Scholar