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Pricing and Inference with Mixtures of Conditionally Normal Processes

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  • Henri Bertholon

    (Crest)

  • Alain Monfort

    (Crest)

  • Fulvio Pegoraro

    (Crest)

Abstract

We consider the problems of derivative pricing and inference when the stochastic discount factor has an exponentialaffineform and the geometric return of the underlying asset has a dynamics characterized by a mixture of conditionallyNormal processes. We consider both the static case in which the underlying process is a white noise distributed as amixture of Gaussian distributions (including extreme risks and jump diffusions) and the dynamic case in which theunderlying process is conditionally distributed as a mixture of Gaussian laws. Semi-parametric, non parametric andSwitching Regime situations are also considered. In all cases, the risk-neutral processes and explicit pricing formulasare obtained.

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  • Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2006-28
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    1. Kon, Stanley J, 1984. " Models of Stock Returns-A Comparison," Journal of Finance, American Finance Association, vol. 39(1), pages 147-165, March.
    2. Jondeau, Eric & Rockinger, Michael, 2001. "Gram-Charlier densities," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1457-1483, October.
    3. Vedat Akgiray & G. Geoffrey Booth, 1987. "Compound Distribution Models Of Stock Returns: An Empirical Comparison," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 10(3), pages 269-280, September.
    4. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, April.
    5. Garcia, Rene & Luger, Richard & Renault, Eric, 2003. "Empirical assessment of an intertemporal option pricing model with latent variables," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 49-83.
    6. Beine, Michel & Laurent, Sebastien, 2003. "Central bank interventions and jumps in double long memory models of daily exchange rates," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 641-660, December.
    7. Veronesi, Pietro, 2004. "The Peso problem hypothesis and stock market returns," Journal of Economic Dynamics and Control, Elsevier, vol. 28(4), pages 707-725, January.
    8. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    9. León, à ngel & Mencía, Javier & Sentana, Enrique, 2009. "Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(2), pages 176-192.
    10. Christoffersen, Peter & Heston, Steve & Jacobs, Kris, 2006. "Option valuation with conditional skewness," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 253-284.
    11. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    12. Bonomo, Marco & Garcia, Rene, 1996. "Consumption and equilibrium asset pricing: An empirical assessment," Journal of Empirical Finance, Elsevier, vol. 3(3), pages 239-265, September.
    13. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
    14. René Garcia & Richard Luger & Eric Renault, 2000. "Asymmetric Smiles, Leverage Effects and Structural Parameters," Working Papers 2000-57, Center for Research in Economics and Statistics.
    15. Dilip B. Madan & Frank Milne, 1994. "Contingent Claims Valued And Hedged By Pricing And Investing In A Basis," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 223-245.
    16. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393-393.
    17. Abel, Andrew B, 1994. "Exact Solutions for Expected Rates of Return under Markov Regime Switching: Implications for the Equity Premium Puzzle," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 26(3), pages 345-361, August.
    18. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. " Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    19. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous-Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    20. Pietro Veronesi, "undated". "How Does Information Quality Affect Stock Returns?," CRSP working papers 361, Center for Research in Security Prices, Graduate School of Business, University of Chicago.
    21. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    22. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    23. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    24. Christian Gourieroux & Alain Monfort & Vassilis Polimenis, 2002. "Affine Term Structure Models," Working Papers 2002-49, Center for Research in Economics and Statistics.
    25. Jondeau, Eric & Rockinger, Michael, 2000. "Reading the smile: the message conveyed by methods which infer risk neutral densities," Journal of International Money and Finance, Elsevier, vol. 19(6), pages 885-915, December.
    26. repec:crs:wpaper:9810 is not listed on IDEAS
    27. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    28. ROCKINGER, Michael & JONDEAU, Eric, 2000. "Entropy densities," HEC Research Papers Series 709, HEC Paris.
    29. Jin-Chuan Duan & Ivilina Popova & Peter Ritchken, 2002. "Option pricing under regime switching," Quantitative Finance, Taylor & Francis Journals, vol. 2(2), pages 116-132.
    30. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    31. Kyriakos Chourdakis & Elias Tzavalis, 2000. "Option Pricing under Discrete Shifts in Stock Returns," Working Papers 426, Queen Mary University of London, School of Economics and Finance.
    32. Ritchey, Robert J, 1990. "Call Option Valuation for Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 13(4), pages 285-296, Winter.
    33. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    34. Amin, Kaushik I & Ng, Victor K, 1993. " Option Valuation with Systematic Stochastic Volatility," Journal of Finance, American Finance Association, vol. 48(3), pages 881-910, July.
    35. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    36. Angelo Melino & Alan X. Yang, 2003. "State Dependent Preferences Can Explain the Equity Premium Puzzle," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 6(4), pages 806-830, October.
    37. Rockinger, Michael & Jondeau, Eric, 2002. "Entropy densities with an application to autoregressive conditional skewness and kurtosis," Journal of Econometrics, Elsevier, vol. 106(1), pages 119-142, January.
    38. Naik, Vasanttilak, 1993. " Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns," Journal of Finance, American Finance Association, vol. 48(5), pages 1969-1984, December.
    39. Harvey, Campbell R. & Siddique, Akhtar, 1999. "Autoregressive Conditional Skewness," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(04), pages 465-487, December.
    40. Baillie, Richard T & Bollerslev, Tim, 2002. "The Message in Daily Exchange Rates: A Conditional-Variance Tale," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 60-68, January.
    41. Benoit Mandelbrot, 1963. "New Methods in Statistical Economics," Journal of Political Economy, University of Chicago Press, vol. 71, pages 421-421.
    42. Stutzer, Michael, 1996. " A Simple Nonparametric Approach to Derivative Security Valuation," Journal of Finance, American Finance Association, vol. 51(5), pages 1633-1652, December.
    43. Jones, Christopher S., 2003. "The dynamics of stochastic volatility: evidence from underlying and options markets," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 181-224.
    44. Bollerslev, Tim & Ole Mikkelsen, Hans, 1999. "Long-term equity anticipation securities and stock market volatility dynamics," Journal of Econometrics, Elsevier, vol. 92(1), pages 75-99, September.
    45. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    46. C. Gourieroux & A. Monfort & V. Polimenis, 2006. "Affine Models for Credit Risk Analysis," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 4(3), pages 494-530.
    47. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
    48. Jing-zhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes," Journal of Finance, American Finance Association, vol. 59(3), pages 1405-1440, June.
    49. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    50. Pietro Veronesi, 2000. "How Does Information Quality Affect Stock Returns?," Journal of Finance, American Finance Association, vol. 55(2), pages 807-837, April.
    51. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-730, August.
    52. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    53. Bjørn Eraker & Michael Johannes & Nicholas Polson, 2003. "The Impact of Jumps in Volatility and Returns," Journal of Finance, American Finance Association, vol. 58(3), pages 1269-1300, June.
    54. Gourieroux, C. & Monfort, A., 2007. "Econometric specification of stochastic discount factor models," Journal of Econometrics, Elsevier, vol. 136(2), pages 509-530, February.
    55. Chernov, Mikhail & Ghysels, Eric, 2000. "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, Elsevier, vol. 56(3), pages 407-458, June.
    56. Pascal St-Amour & Stephen Gordon, 2000. "A Preference Regime Model of Bull and Bear Markets," American Economic Review, American Economic Association, vol. 90(4), pages 1019-1033, September.
    57. Markus Haas, 2004. "Mixed Normal Conditional Heteroskedasticity," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(2), pages 211-250.
    58. Epstein, Larry G & Zin, Stanley E, 1989. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework," Econometrica, Econometric Society, vol. 57(4), pages 937-969, July.
    59. Cecchetti, Stephen G & Lam, Pok-sang & Mark, Nelson C, 1990. "Mean Reversion in Equilibrium Asset Prices," American Economic Review, American Economic Association, vol. 80(3), pages 398-418, June.
    60. Luc Bauwens & Charles S. Bos & Herman K. van Dijk & Rutger D. van Oest, 2002. "Adaptive Polar Sampling," Computing in Economics and Finance 2002 307, Society for Computational Economics.
    61. Bjørn Eraker, 2004. "Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices," Journal of Finance, American Finance Association, vol. 59(3), pages 1367-1404, June.
    62. Engle, Robert F. & Mustafa, Chowdhury, 1992. "Implied ARCH models from options prices," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 289-311.
    63. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    64. Gurdip Bakshi & Nikunj Kapadia & Dilip Madan, 2003. "Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options," Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 101-143.
    65. Jondeau, Eric & Rockinger, Michael, 2003. "Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1699-1737, August.
    66. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    67. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1998. "Implied Volatility Functions: Empirical Tests," Journal of Finance, American Finance Association, vol. 53(6), pages 2059-2106, December.
    68. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    69. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
    70. René Garcia & Eric Ghysels & Éric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
    71. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-547, August.
    72. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    73. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    74. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(04), pages 419-438, December.
    75. Bates, David S., 1996. "Dollar jump fears, 1984-1992: distributional abnormalities implicit in currency futures options," Journal of International Money and Finance, Elsevier, vol. 15(1), pages 65-93, February.
    76. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    77. Chesney, Marc & Scott, Louis, 1989. "Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(03), pages 267-284, September.
    78. Mark Broadie & Mikhail Chernov & Michael Johannes, 2007. "Model Specification and Risk Premia: Evidence from Futures Options," Journal of Finance, American Finance Association, vol. 62(3), pages 1453-1490, June.
    79. Abel, Andrew B., 1999. "Risk premia and term premia in general equilibrium," Journal of Monetary Economics, Elsevier, vol. 43(1), pages 3-33, February.
    80. Pietro Veronesi, "undated". "How Does Information Quality Affect Stock Returns?," CRSP working papers 462, Center for Research in Security Prices, Graduate School of Business, University of Chicago.
    81. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
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    Cited by:

    1. Haas Markus, 2010. "Skew-Normal Mixture and Markov-Switching GARCH Processes," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 14(4), pages 1-56, September.
    2. Haas, Markus & Mittnik, Stefan & Paolella, Marc S., 2009. "Asymmetric multivariate normal mixture GARCH," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2129-2154, April.
    3. Mathieu Gatumel & Florian Ielpo, 2014. "The Number Of Regimes Across Asset Returns: Identification And Economic Value," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 1-25.
    4. Mencía, Javier, 2012. "Assessing the risk-return trade-off in loan portfolios," Journal of Banking & Finance, Elsevier, vol. 36(6), pages 1665-1677.
    5. Rombouts, Jeroen V.K. & Stentoft, Lars, 2014. "Bayesian option pricing using mixed normal heteroskedasticity models," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 588-605.
    6. León, à ngel & Mencía, Javier & Sentana, Enrique, 2009. "Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(2), pages 176-192.
    7. H. Bertholon & A. Monfort & F. Pegoraro, 2008. "Econometric Asset Pricing Modelling," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 6(4), pages 407-458, Fall.
    8. Cordis, Adriana S. & Kirby, Chris, 2014. "Discrete stochastic autoregressive volatility," Journal of Banking & Finance, Elsevier, vol. 43(C), pages 160-178.
    9. Rombouts, Jeroen V.K. & Stentoft, Lars, 2015. "Option pricing with asymmetric heteroskedastic normal mixture models," International Journal of Forecasting, Elsevier, vol. 31(3), pages 635-650.
    10. Rombouts, Jeroen V.K. & Stentoft, Lars, 2011. "Multivariate option pricing with time varying volatility and correlations," Journal of Banking & Finance, Elsevier, vol. 35(9), pages 2267-2281, September.
    11. Monfort, Alain & Pegoraro, Fulvio, 2012. "Asset pricing with Second-Order Esscher Transforms," Journal of Banking & Finance, Elsevier, vol. 36(6), pages 1678-1687.
    12. Mencía, Javier & Sentana, Enrique, 2015. "Volatility-related exchange traded assets: an econometric investigation," CEPR Discussion Papers 10444, C.E.P.R. Discussion Papers.
    13. Christophe Chorro & Florian Ielpo & Benoît Sévi, 2017. "The contribution of jumps to forecasting the density of returns," Documents de travail du Centre d'Economie de la Sorbonne 17006, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    14. Yin-Wong Cheung & Sang-Kuck Chung, 2011. "A Long Memory Model with Normal Mixture GARCH," Computational Economics, Springer;Society for Computational Economics, vol. 38(4), pages 517-539, November.
    15. León, à ngel & Mencía, Javier & Sentana, Enrique, 2009. "Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(2), pages 176-192.
    16. Christophe Chorro & Dominique Guegan & Florian Ielpo, 2008. "Option pricing under GARCH models with generalized hyperbolic innovations (I): methodology," Documents de travail du Centre d'Economie de la Sorbonne b08037, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    17. repec:dau:papers:123456789/6805 is not listed on IDEAS
    18. Steven Caudill & James Long, 2010. "Do former athletes make better managers? Evidence from a partially adaptive grouped-data regression model," Empirical Economics, Springer, vol. 39(1), pages 275-290, August.
    19. Julien Chevallier & Mathieu Gatumel & Florian Ielpo, 2013. "Understanding momentum in commodity markets," Applied Economics Letters, Taylor & Francis Journals, vol. 20(15), pages 1383-1402, October.
    20. Luiz Vitiello & Ser-Huang Poon, 2014. "Non-monotonic pricing kernel and an extended class of mixture of distributions for option pricing," Review of Derivatives Research, Springer, vol. 17(2), pages 241-259, July.

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    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • G1 - Financial Economics - - General Financial Markets

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