Variance Risk Premia

We propose a direct and robust method for quantifying the variance risk premium on financial assets. We theoretically and numerically show that the risk-neutral expected value of the return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. Ignoring the small approximation error, the difference between the realized variance and this synthetic variance swap rate quantifies the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks.

[1]  Marie Manthey,et al.  A GUIDE FOR , 1967 .

[2]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[3]  Richard J. Rendleman,et al.  STANDARD DEVIATIONS OF STOCK PRICE RATIOS IMPLIED IN OPTION PRICES , 1976 .

[4]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[5]  R. Litzenberger,et al.  SKEWNESS PREFERENCE AND THE VALUATION OF RISK ASSETS , 1976 .

[6]  Donald P. Chiras,et al.  The information content of option prices and a test of market efficiency , 1978 .

[7]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[8]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[9]  S. Kanemaki,et al.  A Theory for the , 1986 .

[10]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[11]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[12]  M. Brenner,et al.  A Simple Formula to Compute the Implied Standard Deviation , 1988 .

[13]  A. Kyle,et al.  Smart Money, Noise Trading and Stock Price Behavior , 1988 .

[14]  Sanford J. Grossman,et al.  Stock Market Volatility and the Crash , 1990 .

[15]  John Y. Campbell,et al.  No News is Good News: An Asymmetric Model of Changing Volatility in Stock Returns , 1991 .

[16]  K. Booth.  No news is good news--or is it? , 1991, Australian dental practice.

[17]  Walter N. Torous,et al.  The Effect of Volatility Changes on the Level of Stock Prices and Subsequent Expected Returns , 1991 .

[18]  T. Day,et al.  Stock market volatility and the information content of stock index options , 1992 .

[19]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[20]  Craig M. Lewis,et al.  Forecasting Futures Market Volatility , 1993 .

[21]  Stephen Figlewski,et al.  The Informational Content of Implied Volatility , 1993 .

[22]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[23]  Narasimhan Jegadeesh,et al.  Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency , 1993 .

[24]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[25]  C. Granger,et al.  A long memory property of stock market returns and a new model , 1993 .

[26]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[27]  Christopher G. Lamoureux,et al.  Forecasting Stock-Return Variance: Toward an Understanding of Stochastic Implied Volatilities , 1993 .

[28]  Bruno Dupire.  Pricing with a Smile , 1994 .

[29]  Philippe Jorion.  Predicting Volatility in the Foreign Exchange Market , 1995 .

[30]  J. Cox.  The Constant Elasticity of Variance Option Pricing Model , 1996 .

[31]  J. Jackwerth.  Recovering Risk Aversion from Option Prices and Realized Returns , 1998 .

[32]  C. Granger,et al.  Modeling volatility persistence of speculative returns: A new approach , 1996 .

[33]  D. Duffie,et al.  An Econometric Model of the Term Structure of Interest-Rate Swap Yields , 1997 .

[34]  Guojun Wu,et al.  Asymmetric Volatility and Risk in Equity Markets , 1997 .

[35]  Mark M. Carhart.  On Persistence in Mutual Fund Performance , 1997 .

[36]  K. Singleton,et al.  Specification Analysis of Affine Term Structure Models , 1997 .

[37]  K. Rouwenhorst,et al.  International Momentum Strategies , 1997 .

[38]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[39]  Liuren Wu,et al.  Predictable Changes in Yields and Forward Rates , 1998 .

[40]  Probability Theory III : Stochastic Calculus , 1998 .

[41]  N. Prabhala,et al.  The relation between implied and realized volatility , 1998 .

[42]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options , 1998 .

[43]  Jeff Fleming.  The quality of market volatility forecasts implied by S&P 100 index option prices , 1998 .

[44]  C. Whiteman,et al.  Endogenous term premia and anomalies in the term structure of interest rates: explaining the predictability smile , 1999 .

[45]  The Relation between Implied and Realized Volatility , 1999 .

[46]  Gurdip Bakshi,et al.  Do Call Prices and the Underlying Stock Always Move in the Same Direction , 1999 .

[47]  R. Engle,et al.  Empirical Pricing Kernels , 1999 .

[48]  K. Demeterfi,et al.  A Guide to Volatility and Variance Swaps , 1999 .

[49]  Stephen Taylor,et al.  Forecasting S&P 100 Volatility: The Incremental Information Content of Implied Volatilities and High Frequency Index Returns , 2000 .

[50]  O. Gwilym,et al.  Volatility forecasting in the framework of the option expiry cycle , 1999 .

[51]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

[52]  Tyler Shumway,et al.  Expected Option Returns , 2000 .

[53]  Nicholas G. Polson,et al.  The Impact of Jumps in Volatility and Returns , 2000 .

[54]  David S. Bates.  Post-'87 crash fears in the S&P 500 futures option market , 2000 .

[55]  Gurdip Bakshi,et al.  Pricing and hedging long-term options , 2000 .

[56]  S. Koopman,et al.  Forecasting the Varibility of Stock Index Returns with Stochastic Volatility Models and Implied Volatility , 2000 .

[57]  Mark Britten-Jones,et al.  Option Prices, Implied Price Processes, and Stochastic Volatility , 2000 .

[58]  Bjørn Eraker.  Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices , 2004 .

[59]  Francis X. Diebold,et al.  Modeling and Forecasting Realized Volatility , 2001 .

[60]  P. Carr,et al.  Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading , 2001 .

[61]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

[62]  Luca Benzoni,et al.  An Empirical Investigation of Continuous-Time Equity Return Models , 2001 .

[63]  Stephen L Taylor,et al.  Modelling S&P 100 Volatility: The Information Content of Stock Returns , 2001 .

[64]  Jun Pan.  The Jump-Risk Premia Implicit in Options : Evidence from an Integrated Time-Series Study , 2001 .

[65]  C. Hansen.  The relation between implied and realised volatility in the Danish option and equity markets , 2001 .

[66]  K. Singleton,et al.  Expectation puzzles, time-varying risk premia, and affine models of the term structure , 2002 .

[67]  Gurdip Bakshi,et al.  Delta-Hedged Gains and the Negative Market Volatility Risk Premium , 2001 .

[68]  John Krainer,et al.  Stock market volatility , 2002 .

[69]  Yacine Ait-Sahalia.  Closed-Form Likelihood Expansions for Multivariate Diffusions , 2002, 0804.0758.

[70]  Keith A. Lewis,et al.  What Type of Process Underlies Options ? , 2002 .

[71]  P. Carr,et al.  What Type of Process Underlies Options? A Simple Robust Test , 2003 .

[72]  Yacine Aït-Sahalia.  Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed‐form Approximation Approach , 2002 .

[73]  Jon Vilasuso.  Forecasting exchange rate volatility , 2002 .

[74]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study $ , 2002 .

[75]  B. Christensen,et al.  New evidence on the implied-realized volatility relation , 2002 .

[76]  Jinghong Shu,et al.  The relationship between implied and realized volatility of S&P 500 index , 2003 .

[77]  David S. Bates.  Empirical option pricing: a retrospection , 2003 .

[78]  Liuren Wu,et al.  Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes , 2003 .

[79]  C. Hafner.  On forecasting Exchange Rate Volatility , 2003 .

[80]  Nikunj Kapadia,et al.  Volatility Risk Premiums Embedded in Individual Equity Options , 2003 .

[81]  C. S. Jones.  The dynamics of stochastic volatility: evidence from underlying and options markets , 2003 .

[82]  Peter Carr,et al.  A Tale of Two Indices , 2004 .

[83]  Oleg Bondarenko.  Market Price of Variance Risk and Performance of Hedge Funds , 2004 .

[84]  Bjørn Eraker.  Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices , 2004 .

[85]  Yisong S. Tian,et al.  The Model-Free Implied Volatility and Its Information Content , 2005 .

[86]  Liuren Wu.  Variance Dynamics: Joint Evidence from Options and High-Frequency Returns , 2005 .

[87]  William H. Sackley.  The Cross-Section of Volatility and Expected Returns , 2006 .

[88]  R. Hodrick,et al.  The Cross-Section of Volatility and Expected Returns , 2006 .

[89]  Liuren Wu,et al.  Variance Risk Dynamics, Variance Risk Premia, and Optimal Variance Swap Investments , 2007 .

[90]  Michael S. Gibson,et al.  Dynamic Estimation of Volatility Risk Premia and Investor Risk Aversion from Option-Implied and Realized Volatilities , 2007 .

[91]  Mikhail Chernov.  On the Role of Risk Premia in Volatility Forecasting , 2007 .

[92]  P. Carr,et al.  Robust Replication of Volatility Derivatives , 2008 .

[93]  Michael Mecham,et al.  A long memory , 2008 .