A Unified Valuation Framework for Variance Swaps under Non-Affine Stochastic Volatility Models

In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based variance swap prices. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as our pricing kernel candidate. Following standard assumptions on the time-varying GARCH parameters, we show that these quantities converge to discretely and continuously sampled variance swaps constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier-tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.

[1]  Jin E. Zhang,et al.  GARCH Option Pricing Models, the CBOE VIX, and Variance Risk Premium , 2013 .

[2]  Bruno Feunou,et al.  Série Scientifique Scientific Series 2009 s-32 Option Valuation with Conditional Heteroskedasticity and Non-Normality , 2009 .

[3]  Markus Leippold,et al.  National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No . 870 Inferring Volatility Dynamics and Risk Premia from the S & P 500 and VIX Markets , 2013 .

[4]  D. Filipović,et al.  Quadratic Variance Swap Models , 2014 .

[5]  Tak Kuen Siu,et al.  On Pricing Derivatives Under GARCH Models: A Dynamic Gerber-Shiu Approach , 2004, North American Actuarial Journal.

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

[7]  Hao Wang,et al.  Credit Default Swap Spreads and Variance Risk Premia , 2010 .

[8]  Chardin Wese Simen,et al.  Jump and variance risk premia in the S&P 500 , 2016 .

[9]  Carole Bernard,et al.  Prices and Asymptotics for Discrete Variance Swaps , 2013 .

[10]  Paul Schneider,et al.  Generalized Risk Premia , 2012 .

[11]  Philip Protter,et al.  Discretely sampled variance and volatility swaps versus their continuous approximations , 2011, Finance Stochastics.

[12]  Robert J. Elliott,et al.  Non-Gaussian GARCH Option Pricing Models and Their Diffusion Limits , 2013, Eur. J. Oper. Res..

[13]  Anatoliy Swishchuk.  Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities , 2013 .

[14]  Peter Carr,et al.  Volatility Derivatives , 2009 .

[15]  Daniel B. Nelson.  ARCH models as diffusion approximations , 1990 .

[16]  Emese Lazar,et al.  Option Valuation with Normal Mixture GARCH Models , 2008 .

[17]  R. Engle,et al.  A GARCH Option Pricing Model with Filtered Historical Simulation , 2008 .

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

[19]  Mark Broadie,et al.  The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps , 2008 .

[20]  Artur Sepp.  Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility , 2008 .

[21]  Robert J. Elliott,et al.  A Discrete Time Equivalent Martingale Measure , 1998 .

[22]  Jean-Guy Simonato,et al.  The Role of the Conditional Skewness and Kurtosis in VIX Index Valuation , 2017 .

[23]  Peter Forsyth,et al.  Pricing methods and hedging strategies for volatility derivatives , 2006 .

[24]  J. Duan.  THE GARCH OPTION PRICING MODEL , 1995 .

[25]  Valentina Corradi,et al.  Reconsidering the continuous time limit of the GARCH(1, 1) process , 2000 .

[26]  A. Yaron,et al.  What's Vol Got to Do With It , 2009 .

[27]  Dawn Hunter,et al.  An analytical approximation for the GARCH option pricing model , 2000 .

[28]  Song-Ping Zhu,et al.  A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY , 2010 .

[29]  Tail Risk Premia for Long-Term Equity Investors , 2016, 1602.00865.

[30]  Realized EGARCH, CBOE VIX and Variance Risk Premium , 2015 .

[31]  Carol Alexander,et al.  Model-Free Discretisation-Invariant Swap Contracts , 2016, 1602.00235.

[32]  Loriano Mancini,et al.  The Term Structure of Variance Swaps and Risk Premia , 2015 .

[33]  Yiwen Shen,et al.  Pricing the CBOE VIX Futures with the Heston–Nandi GARCH Model , 2017 .

[34]  Carole Bernard,et al.  Convergence of the Discrete Variance Swap in Time-Homogeneous Diffusion Models , 2013 .

[35]  Peter F. Christoffersen,et al.  Capturing Option Anomalies with a Variance-Dependent Pricing Kernel , 2011 .

[36]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

[37]  Saikat Nandi,et al.  Derivatives on Volatility: Some Simple Solutions Based on Observables , 2000 .

[38]  Jin-Chuan Duan,et al.  Augmented GARCH (p,q) process and its diffusion limit , 1997 .

[39]  Alexandru Badescu,et al.  GARCH option pricing: A semiparametric approach , 2008 .

[40]  J. Kanniainen,et al.  Estimating and Using GARCH Models with VIX Data for Option Valuation , 2014 .

[41]  George Dotsis,et al.  The Market Price of Risk of the Variance Term Structure , 2011 .

[42]  Peter Carr,et al.  Variance Risk Premiums , 2009 .

[43]  P. Hansen,et al.  Realized GARCH: A Joint Model of Returns and Realized Measures of Volatility , 2010 .

[44]  S. Heston,et al.  A Closed-Form GARCH Option Valuation Model , 2000 .

[45]  C. Alexander,et al.  On the Continuous Limit of GARCH , 2005 .

[46]  Carol Alexander,et al.  Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusions , 2012 .

[47]  Peter Christoffersen,et al.  GARCH Option Valuation: Theory and Evidence , 2013, The Journal of Derivatives.

[48]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[49]  S. Byun,et al.  The role of the variance premium in Jump-GARCH option pricing models , 2015 .

[50]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[51]  Eirini Konstantinidi,et al.  How Does the Market Variance Risk Premium Vary Over Time? Evidence from S&P 500 Variance Swap Investment Returns , 2014 .

[52]  B. Rémillard,et al.  Pricing Variance Options in a GARCH Setting , 2004 .