Option Pricing with Asymmetric Heteroskedastic Normal Mixture Models

We propose an asymmetric GARCH in mean mixture model and provide a feasible method for option pricing within this general framework by deriving the appropriate risk neutral dynamics. We forecast the out-of-sample prices of a large sample of options on the S&P 500 index from January 2006 to December 2011, and compute dollar losses and implied standard deviation losses. We compare our results to those of existing mixture models and other benchmarks like component models and jump models. Using the model confidence set test, the overall dollar root mean squared error of the best performing benchmark model is significantly larger than that of the best mixture model.

[1]  Peter Christoffersen,et al.  Série Scientifique Scientific Series Option Valuation with Conditional Skewness Option Valuation with Conditional Skewness , 2022 .

[2]  A. Monfort,et al.  Econometric Asset Pricing Modelling , 2008 .

[3]  Louis O. Scott.  Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

[4]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

[5]  Dongcheol Kim,et al.  Alternative Models for the Conditional Heteroscedasticity of Stock Returns , 1994 .

[6]  Jeroen V. K. Rombouts,et al.  Mixed Exponential Power Asymmetric Conditional Heteroskedasticity , 2007 .

[7]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study , 2001 .

[8]  Peter F. Christoffersen,et al.  Option Valuation with Long-Run and Short-Run Volatility Components , 2008 .

[9]  Alan L. Tucker,et al.  The Probability Distribution of Foreign Exchange Price Changes: Tests of Candidate Processes , 1988 .

[10]  Peter Christoffersen,et al.  Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options , 2009 .

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

[12]  Alain Monfort,et al.  Econometric specification of stochastic discount factor models , 2007 .

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

[14]  James B. Wiggins.  Option values under stochastic volatility: Theory and empirical estimates , 1987 .

[15]  David S. Bates.  The Crash of ʼ87: Was It Expected? The Evidence from Options Markets , 1991 .

[16]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[17]  Gurdip Bakshi,et al.  Stochastic Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount Factors in International Economies , 2006 .

[18]  Stephen Gray.  Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process , 1996 .

[19]  Stanley J. Kon.  Models of Stock Returns—A Comparison , 1984 .

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

[21]  Sylvia Frühwirth-Schnatter,et al.  Finite Mixture and Markov Switching Models , 2006 .

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

[23]  Saikat Nandi.  Pricing and hedging index options under stochastic volatility: an empirical examination , 1996 .

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

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

[26]  Kaushik I. Amin,et al.  Option Valuation with Systematic Stochastic Volatility , 1993 .

[27]  E. Stein,et al.  Stock Price Distributions with Stochastic Volatility: An Analytic Approach , 1991 .

[28]  W. Li,et al.  On a mixture autoregressive model , 2000 .

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

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

[31]  Valentina Corradi,et al.  Predicting the volatility of the S&P-500 stock index via GARCH models: the role of asymmetries , 2005 .

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

[33]  Alain Monfort,et al.  Pricing and Inference with Mixtures of Conditionally Normal Processes , 2007 .

[34]  Peter H. Ritchken,et al.  An empirical comparison of GARCH option pricing models , 2006 .

[35]  M. Yor,et al.  Stochastic Volatility for Levy Processes , 2001 .

[36]  R. Engle,et al.  A Permanent and Transitory Component Model of Stock Return Volatility , 1993 .

[37]  Peter Reinhard Hansen,et al.  The Model Confidence Set , 2010 .

[38]  P. Boothe,et al.  The statistical distribution of exchange rates: Empirical evidence and economic implications , 1987 .

[39]  L. Glosten,et al.  On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks , 1993 .

[40]  Luc Bauwens,et al.  Bayesian Clustering of Many Garch Models , 2003 .

[41]  Lars Stentoft,et al.  American Option Pricing Using GARCH Models and the Normal Inverse Gaussian Distribution , 2007 .

[42]  Peter Christoffersen,et al.  Dynamic jump intensities and risk premiums: Evidence from S&P500 returns and options , 2012 .

[43]  Peter F. Christoffersen,et al.  Which GARCH Model for Option Valuation? , 2002, Manag. Sci..

[44]  Lars Stentoft,et al.  Multivariate Option Pricing with Time Varying Volatility and Correlations , 2010 .

[45]  D. Shanno,et al.  Option Pricing when the Variance Is Changing , 1987, Journal of Financial and Quantitative Analysis.

[46]  J. Davidson.  Stochastic Limit Theory , 1994 .

[47]  Luc Bauwens,et al.  Theory and Inference for a Markov Switching GARCH Model , 2007 .

[48]  B. Hansen.  Autoregressive Conditional Density Estimation , 1994 .

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

[50]  Ming-Shiun Pan,et al.  The distribution of currency futures price changes: A two-piece mixture of normals approach , 1995 .

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

[52]  Lars Stentoft,et al.  Pricing American options when the underlying asset follows GARCH processes , 2005 .

[53]  P. Carr,et al.  Stochastic Skew in Currency Options , 2004 .

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

[55]  James D. Hamilton,et al.  Normalization in Econometrics , 2004 .

[56]  Peter Christoffersen,et al.  Volatility Components, Affine Restrictions, and Nonnormal Innovations , 2008 .

[57]  Jin-Chuan Duan,et al.  Jump Starting GARCH: Pricing and Hedging Options With Jumps in Returns and Volatilities , 2006 .

[58]  Campbell R. Harvey,et al.  Current Version : April 6 , 1999 Autoregressive Conditional Skewness , 1999 .

[59]  Eric Jondeau,et al.  Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements , 2003 .

[60]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[61]  G. Durham.  SV mixture models with application to S&P 500 index returns ☆ , 2007 .

[62]  Lars Stentoft,et al.  The Value of Multivariate Model Sophistication: An Application to Pricing Dow Jones Industrial Average Options , 2012 .

[63]  Jin-Chuan Duan,et al.  APPROXIMATING GARCH‐JUMP MODELS, JUMP‐DIFFUSION PROCESSES, AND OPTION PRICING , 2006 .

[64]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[65]  Ralf Becker,et al.  Are combination forecasts of S&P 500 volatility statistically superior? , 2008 .

[66]  Jeroen V. K. Rombouts,et al.  Multivariate Mixed Normal Conditional Heteroskedasticity , 2006, Comput. Stat. Data Anal..

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

[68]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

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

[70]  Marc S. Paolella,et al.  Mixed Normal Conditional Heteroskedasticity , 2004 .

[71]  Wai Keung Li,et al.  On a Mixture Autoregressive Conditional Heteroscedastic Model , 2001 .

[72]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.